text:
The easy-solving of math problems will help you to understand all the difficult problems. So, begin your practice now and be on the top list to score good marks in the exam. Use Go Math 6 Standard Answer Key Chapter 4 Model Ratios handy solutions to learn deep maths online or offline. Come and fall in love with maths by practicing the problems from HMH Go Math 6th Grade Chapter 4 Model Ratios Answer Key. Hit the links and start your preparation now.

Lesson 1: Investigate • Model Ratios

Lesson 2: Ratios and Rates

Lesson 3: Equivalent Ratios and Multiplication Tables

Lesson 4: Problem Solving • Use Tables to Compare Ratios

Lesson 5: Algebra • Use Equivalent Ratios

Mid-Chapter Checkpoint

Lesson 6: Find Unit Rates

Lesson 7: Algebra • Use Unit Rates

Lesson 8: Algebra • Equivalent Ratios and Graphs

Chapter 4 Review/Test

### Share and Show – Page No. 213

Write the ratio of yellow counters to red counters.

Question 1.

Type below:
___________

1: 2

Explanation:
There are one yellow counter and two red counters.
So, the ratio is 1:2

Question 2.

Type below:
___________

5: 3

Explanation:
There are 5 yellow counter and 3 red counters.
So, the ratio is 5:3

Draw a model of the ratio.

Question 3.
3 : 2
Type below:
___________

Explanation:
As the ratio is 3:2, we can draw three yellow counters and 2 red counters.

Question 4.
1 : 5
Type below:
___________

Explanation:
As the ratio is 1:5, we can draw 1 yellow counter and 5 red counters.

Use the ratio to complete the table.

Question 5.
Wen is arranging flowers in vases. For every 1 rose she uses, she uses 6 tulips. Complete the table to show the ratio of roses to tulips.

Type below:
___________

Explanation:
There is 1 box for every 6 Tulips.
The ratio is 1 : 6.
Each time the number of boxes increases by 1, the number of Tulips increases by 6
So, for 2 boxes, 6 + 6 = 12 Tulips
For 3 boxes, 12 + 6 = 18 Tulips
For 4 boxes, 18 + 6 = 24 Tulips

Question 6.
On the sixth-grade field trip, there are 8 students for every 1 adult. Complete the table to show the ratio of students to adults.

Type below:
___________

Explanation:
There is 1 adult out of 8 Students.
The ratio is 8:1.
Each time the number of students increases double, the number of adults becomes double.
So, for 8 + 8 = 16 students, 2 Adults available
For 16 + 8 = 24 students, 2 + 1 = 3 Adults available
For 24 + 8 = 32 students, 3 + 1 = 4 Adults available

Question 7.
Zena adds 4 cups of flour for every 3 cups of sugar in her recipe. Draw a model that compares cups of flour to cups of sugar.
Type below:
___________

Explanation:
Zena adds 4 cups of flour for every 3 cups of sugar in her recipe.
For every 3 cups of sugar, she adds 4 cups of flour.
For 6 cups of sugar, she adds 8 cups of flour
For 9 cups of sugar, she adds 12 cups of flour
For 12 cups of sugar, she adds 16 cups of flour

### Draw Conclusions – Page No. 214

Hikers take trail mix as a snack on long hikes because it is tasty, nutritious, and easy to carry. There are many different recipes for trail mix, but it is usually made from different combinations of dried fruit, raisins, seeds, and nuts. Tanner and his dad make trail mix that has 1 cup of raisins for every 3 cups of sunflower seeds.

Question 8.
Model Mathematics Explain how you could model the ratio that compares cups of raisins to cups of sunflower seeds when Tanner uses 2 cups of raisins.
Type below:
___________

Explanation:
Hikers take trail mix as a snack on long hikes because it is tasty, nutritious, and easy to carry. There are many different recipes for trail mix, but it is usually made from different combinations of dried fruit, raisins, seeds, and nuts. Tanner and his dad make trail mix that has 1 cup of raisins for every 3 cups of sunflower seeds.
For 2 cups of raisins, he needs 3 + 3 = 6 cups of sunflower seeds

The table shows the ratio of cups of raisins to cups of sunflower seeds for different amounts of trail mix. Model each ratio as you complete the table.

Question 9.
Describe the pattern you see in the table.
Type below:
___________

Multiply Raisins by 3 to get number of Sunflower Seeds.

Question 10.
Draw Conclusions What conclusion can Tanner draw from this pattern?
Type below:
___________

He needs 3 times as many seeds as raisins

Question 11.
What is the ratio of cups of sunflower seeds to cups of trail mix when Tanner uses 4 cups of raisins?
Type below:
___________

4:12

Explanation:
If tanner uses 4 cups of raisins, he needs 12 cups of sunflower seeds.

### Model Ratios – Page No. 215

Write the ratio of gray counters to white counters.

Question 1.

Type below:
___________

3:4

Explanation:
There are 3 gray counter and 4 white counters.
So, the ratio is 3:4

Question 2.

Type below:
___________

4:1

Explanation:
There are 4 gray counter and 1 white counter.
So, the ratio is 4:1

Question 3.

Type below:
___________

2:3

Explanation:
There are 2 gray counter and 3 white counters.
So, the ratio is 2:3

Draw a model of the ratio.

Question 4.
5 : 1
Type below:
___________

Explanation:
As the ratio is 5:1, we can draw 5 yellow counters and 1 red counter.

Question 5.
6 : 3
Type below:
___________

Explanation:
As the ratio is 6:3, we can draw 6 yellow counters and 3 red counters.

Use the ratio to complete the table.

Question 6.
Marc is assembling gift bags. For every 2 pencils he places in the bag, he uses 3 stickers. Complete the table to show the ratio of pencils to stickers.

Type below:
___________

Explanation:
Marc is assembling gift bags. For every 2 pencils he places in the bag, he uses 3 stickers.
For 4 pencils, he uses 3 + 3 = 6 stickers
For 6 pencils, he uses 6 + 3 = 9 stickers
For 8 pencils, he uses 9 + 3 = 12 stickers

Question 7.
Singh is making a bracelet. She uses 5 blue beads for every 1 silver bead. Complete the table to show the ratio of blue beads to silver beads

Type below:
___________

Explanation:
Singh is making a bracelet. She uses 5 blue beads for every 1 silver bead.
For 2 silver bead, she uses 5 + 5 = 10 blue beads.
For 3 silver bead, she uses 10 + 5 = 15 blue beads.
For 4 silver bead, she uses 15 + 5 = 20 blue beads.

Problem Solving

Question 8.
There are 4 quarts in 1 gallon. How many quarts are in 3 gallons?
______ quarts

12 quarts

Explanation:
There are 4 quarts in 1 gallon. If there are 3 gallons, he uses 3 × 4 = 12 quarts

Question 9.
Martin mixes 1 cup lemonade with 4 cups cranberry juice to make his favorite drink. How much cranberry juice does he need if he uses 5 cups of lemonade?
______ cups

20 cups

Explanation:
Martin mixes 1 cup lemonade with 4 cups cranberry juice to make his favorite drink. If he uses 5 cups of lemonade, 5 × 4 = 20 cups

Question 10.
Suppose there was 1 centerpiece for every 5 tables. Use counters to show the ratio of centerpieces to tables. Then make a table to find the number of tables if there are 3 centerpieces.
Type below:
___________

Explanation:
Suppose there was 1 centerpiece for every 5 tables.
If there are 3 centerpieces, 5 × 3 = 15 tables

### Lesson Check – Page No. 216

Question 1.
Francine is making a necklace that has 1 blue bead for every 6 white beads. How many white beads will she use if she uses 11 blue beads?

Explanation:
Francine is making a necklace that has 1 blue bead for every 6 white beads.
11 × 6 = 66 white beads

Question 2.
A basketball league assigns 8 players to each team. How many players can sign up for the league if there are 24 teams?
______ players

192 players

Explanation:
A basketball league assigns 8 players to each team.
If there are 24 teams, 24 × 8 = 192 players to each team

Spiral Review

Question 3.
Louis has 45 pencils and 75 pens to divide into gift bags at the fair. He does not want to mix the pens and pencils. He wants to place an equal amount in each bag. What is the greatest number of pens or pencils he can place in each bag?
______

Louis can form at most 15 bags, each of them will contain 3 pencils and 5 pens.

Explanation:
Louis has 45 pencils and 75 pens to divide into gift bags at the fair. He does not want to mix the pens and pencils. He wants to place an equal amount in each bag.
Factor both these numbers:
45 = 3·3·5;
75 = 3·5·5.
The greatest common factor (write all common factors and multiply them) is 3·5=15. Then:
45=15·3;
75=15·5.
Louis can form at most 15 bags, each of them will contain 3 pencils and 5 pens.

Question 4.
Of the 24 students in Greg’s class, $$\frac{3}{8}$$ ride the bus to school. How many students ride the bus?
______ students

9 students

Explanation:
Of the 24 students in Greg’s class, $$\frac{3}{8}$$ ride the bus to school.
3/8 x 24= 9

Question 5.
Elisa made 0.44 of the free throws she attempted. What is that amount written as a fraction in simplest form?
$$\frac{□}{□}$$

$$\frac{11}{25}$$

Explanation:
Elisa made 0.44 of the free throws she attempted.
0.44 = 44/100
44/100 = 22/50 = 11/25
11/25

Question 6.
On a coordinate plane, the vertices of a rectangle are (–1, 1), (3, 1), (–1, –4), and (3, –4). What is the perimeter of the rectangle?
______ units

18 units

Explanation:
On a coordinate plane, the vertices of a rectangle are (–1, 1), (3, 1), (–1, –4), and (3, –4).
|-1| = 1
The distance from (–1, 1), (3, 1) is 1 + 0 + 0 + 3 = 4
|-4| = 4
The distance from (3, 1), (3, –4) is 1 + 0 + 0 + 4 = 5
perimeter of the rectangle = 4 + 5 + 5 + 4 = 18

### Share and Show – Page No. 219

Question 1.
Write the ratio of the number of red bars to blue stars.

$$\frac{□}{□}$$

$$\frac{8}{3}$$

Explanation:
There are 8 stars and 3 red boxes.
So, the ratio is 8:3

Write the ratio in two different ways.

Question 2.
8 to 16
Type below:
___________

$$\frac{8}{16}$$
8:16

Explanation:
8 to 16 as a fraction 8/16
8 to 16 with a colon 8:16

Question 3.
$$\frac{4}{24}$$
Type below:
___________

4 to 24
4:24

Explanation:
$$\frac{4}{24}$$ using words 4 to 24
$$\frac{4}{24}$$ with a colon 4:24

Question 4.
1 : 3
Type below:
___________

1 to 3
$$\frac{1}{3}$$

Explanation:
1 : 3 using words 1 to 3
1 : 3 as a fraction 1/3

Question 5.
7 to 9
Type below:
___________

$$\frac{7}{9}$$
7:9

Explanation:
7 to 9 as a fraction 7/9
7 to 9 with a colon 7:9

Question 6.
Marilyn saves $15 per week. Complete the table to find the rate that gives the amount saved in 4 weeks. Write the rate in three different ways. Type below: ___________ Answer: Explanation: Marilyn saves$15 per week.
for 4 weeks, $15 × 4 =$60

Write the ratio in two different ways.

Question 7.
$$\frac{16}{40}$$
Type below:
___________

16 to 40
16:40

Explanation:
$$\frac{16}{40}$$ using words 16 to 40
$$\frac{16}{40}$$ with a colon 16:40

Question 8.
8 : 12
Type below:
___________

8 to 12
$$\frac{8}{12}$$

Explanation:
8 : 12 using words 8 to 12
8 : 12 as a fraction $$\frac{8}{12}$$

Question 9.
4 to 11
Type below:
___________

$$\frac{4}{11}$$
4:11

Explanation:
4 to 11 as a fraction $$\frac{4}{11}$$
4 to 11 with a colon 4:11

Question 10.
2 : 13
Type below:
___________

2 to 13
$$\frac{2}{13}$$

Explanation:
2 : 13 using words 2 to 13
2 : 13 as a fraction $$\frac{2}{13}$$

Question 11.
There are 24 baseball cards in 4 packs. Complete the table to find the rate that gives the number of cards in 2 packs. Write this rate in three different ways.

Type below:
___________

Explanation:
There are 24 baseball cards in 4 packs.
For 2 packs, (2 × 24)/4 = 12
For 1 pack, (1× 24)/4 = 6

Question 12.
Make Connections Explain how the statement “There is $$\frac{3}{4}$$ cup per serving” represents a rate.
Type below:
___________

There is a 3/4 cup of whatever in one serving. If that serving amount changed to 2, then the 3/4 would be multiplied by 2. If there is half a serving, then it would be divided by 2. There is a constant change and not one that is always changing.

### Problem Solving + Applications – Page No. 220

Use the diagram of a birdhouse for 13–15.

Question 13.
Write the ratio of AB to BC in three different ways.
Type below:
___________

28 : 12, 28 to 12, $$\frac{2}{13}$$

Explanation:
AB = 28 in
BC = 12 in
AB : BC = 28 : 12, 28 to 12, $$\frac{2}{13}$$

Question 14.
Write the ratio of the shortest side length of triangle ABC to the perimeter of the triangle in three different ways.
Type below:
___________

12 : 64, 12 to 64, $$\frac{12}{64}$$

Explanation:
the shortest side length of triangle ABC = 12 in
the perimeter of the triangle 12 + 28 + 24 = 64
12 : 64, 12 to 64, $$\frac{12}{64}$$

Question 15.
Represent a Problem Write the ratio of the perimeter of triangle ABC to the longest side length of the triangle in three different ways.
Type below:
___________

64 : 28, 64 to 28, $$\frac{64}{28}$$

Explanation:
the ratio of the perimeter of triangle ABC = 12 + 28 + 24 = 64
the longest side length of the triangle = 28 in
64 : 28, 64 to 28, $$\frac{64}{28}$$

Question 16.
Leandra places 6 photos on each page in a photo album. Find the rate that gives the number of photos on 2 pages. Write the rate in three different ways.
Type below:
___________

6 : 12, 6 to 12, $$\frac{6}{12}$$

Explanation:
Leandra places 6 photos on each page in a photo album.
For 2 pages, 6 × 2 = 12 in
6 : 12, 6 to 12, $$\frac{6}{12}$$

Question 17.
What’s the Question? The ratio of total students in Ms. Murray’s class to students in the class who have an older brother is 3 to 1. The answer is 1:2. What is the question?
Type below:
___________

What is the ratio of students in the class who don’t have an older brother to students in the class with an older brother.

Question 18.
What do all unit rates have in common?
Type below:
___________

A rate is a ratio that is used to compare different kinds of quantities. A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.

Question 19.
Julia has 2 green reusable shopping bags and 5 purple reusable shopping bags. Select the ratios that compare the number of purple reusable shopping bags to the total number of reusable shopping bags. Mark all that apply.

• 5 to 7
• 5 : 7
• 5 : 2
• $$\frac{2}{5}$$
• 2 to 7
• $$\frac{5}{7}$$

Type below:
___________

5 to 7, 5 : 7, $$\frac{5}{7}$$

Explanation:
the number of purple reusable shopping bags = 5
the total number of reusable shopping bags = 5 + 2 = 7
5 to 7, 5 : 7, $$\frac{5}{7}$$

### Ratios and Rates – Page No. 221

Write the ratio in two different ways.

Question 1.
$$\frac{4}{5}$$
Type below:
___________

4 to 5
4 : 5

Explanation:
$$\frac{4}{5}$$ using words 4 to 5
$$\frac{4}{5}$$ with a colon 4 : 5

Question 2.
16 to 3
Type below:
___________

$$\frac{16}{3}$$
16 : 3

Explanation:
16 to 3 as a fraction $$\frac{16}{3}$$
16 to 3 with a colon 16 : 3

Question 3.
9 : 13
Type below:
___________

9 to 13
$$\frac{9}{13}$$

Explanation:
9 : 13 using words 9 to 13
9 : 13 as a fraction $$\frac{9}{13}$$

Question 4.
$$\frac{15}{8}$$
Type below:
___________

15 to 8
15 : 8

Explanation:
$$\frac{15}{8}$$ using words 15 to 8
$$\frac{15}{8}$$ with a colon 15 : 8

Question 5.
There are 20 light bulbs in 5 packages. Complete the table to find the rate that gives the number of light bulbs in 3 packages. Write this rate in three different ways.

Type below:
___________

Explanation:
There are 20 light bulbs in 5 packages.
For 1 package, 4 light bulbs available
For 2 package, 8 light bulbs available
For 3 package, 12 light bulbs available
For 4 package, 16 light bulbs available

Problem Solving

Question 6.
Gemma spends 4 hours each week playing soccer and 3 hours each week practicing her clarinet. Write the ratio of hours spent practicing clarinet to hours spent playing soccer three different ways.
Type below:
___________

$$\frac{3}{4}$$, 3 : 4, 3 to 4

Explanation:
Gemma spends 4 hours each week playing soccer and 3 hours each week practicing her clarinet.
3/4, 3 : 4, 3 to 4

Question 7.
Randall bought 2 game controllers at Electronics Plus for $36. What is the unit rate for a game controller at Electronics Plus? Type below: ___________ Answer: $$\frac{18}{1}$$ Explanation: Randall bought 2 game controllers at Electronics Plus for$36. $36/2 =$18/1 is the unit rate for a game controller at Electronics Plus

Question 8.
Explain how to determine if a given rate is also a unit rate.
Type below:
___________

when rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. If you have a multiple-unit rate such as 120 student for every 3 buses, an want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term

### Lesson Check – Page No. 222

Question 1.
At the grocery store, Luis bought 10 bananas and 4 apples. What are three different ways to write the ratio of apples to bananas?
Type below:
___________

4 : 10, 4 to 10, $$\frac{4}{10}$$

Explanation:
At the grocery store, Luis bought 10 bananas and 4 apples. 4/10, 4 : 10, 4 to 10

Question 2.
Rita checked out 7 books from the library. She had 2 non-fiction books. The rest were fiction. What are three different ways to write the ratio of non-fiction to fiction?
Type below:
___________

2 to 5, 2 : 5, $$\frac{2}{5}$$

Explanation:
Rita checked out 7 books from the library. She had 2 non-fiction books. The rest were fiction.
fiction = 5
2 to 5, 2 : 5, $$\frac{2}{5}$$

Spiral Review

Question 3.
McKenzie bought 1.2 pounds of coffee for $11.82. What was the cost per pound?$ ______

$9.85 Explanation: McKenzie bought 1.2 pounds of coffee for$11.82. $11.82/1.2 =$9.85

Question 4.
Pedro has a bag of flour that weighs $$\frac{9}{10}$$ pound. He uses $$\frac{2}{3}$$ of the bag to make gravy. How many pounds of flour does Pedro use to make gravy?
$$\frac{□}{□}$$ pound

$$\frac{3}{5}$$ pound

Explanation:
Pedro has a bag of flour that weighs $$\frac{9}{10}$$ pound. He uses $$\frac{2}{3}$$ of the bag to make gravy.
$$\frac{9}{10}$$ × $$\frac{2}{3}$$ = 3/5

Question 5.
Gina draws a map of her town on a coordinate plane. The point that represents the town’s civic center is 1 unit to the right of the origin and 4 units above it. What are the coordinates of the point representing the civic center?
Type below:
___________

(-1, 4)

Explanation:
Gina draws a map of her town on a coordinate plane. The point that represents the town’s civic center is 1 unit to the right of the origin and 4 units above it.
(-1, 4)

Question 6.
Stefan draws these shapes. What is the ratio of triangles to stars?

Type below:
___________

2 to 5

Explanation:
There are 2 triangles and 5 stars. So, the ratio is 2 : 5

### Share and Show – Page No. 225

Write two equivalent ratios.

Question 1.
Use a multiplication table to write two ratios that are equivalent to $$\frac{4}{7}$$.
Type below:
___________

$$\frac{4}{7}$$ = $$\frac{8}{14}$$, $$\frac{12}{21}$$

Explanation:
The original ratio is 4/7. Shade the row for 4 and the row for 7 on the multiplication table.
The column for 2 shows there are 2 ∙ 4, when there are 2 ∙ 7. So, 4/7 equal to 8/14
The column for 3 shows there are 3 ∙ 4, when there are 3 ∙ 7. So, 4/7 equal to 12/21

Question 2.

Type below:
___________

Explanation:
The original ratio is 3/7. Shade the row for 3 and the row for 7 on the multiplication table.
The column for 2 shows there are 2 ∙ 3, when there are 2 ∙ 7. So, 3/7 equal to 6/14
The column for 3 shows there are 3 ∙ 3, when there are 3 ∙ 7. So, 3/7 equal to 9/21

Question 3.

Type below:
___________

Explanation:
The original ratio is 5/2. Shade the row for 5 and the row for 2 on the multiplication table.
The column for 2 shows there are 2 ∙ 5 when there are 2 ∙ 2. So, 5/2 equal to 10/4
The column for 3 shows there are 3 ∙ 5 when there are 3 ∙ 2. So, 5/2 equal to 15/6

Question 4.

Type below:
___________

Explanation:
The original ratio is 2/10. Shade the row for 2 and the row for 10 on the multiplication table.
The column for 1 shows there are 1 ∙ 2 when there are 5 ∙ 2. So, 2/10 equal to 1/5
The column for 3 shows there are 1 ∙ 3 when there are 5 ∙ 3. So, 2/10 equal to 3/15

Question 5.
$$\frac{4}{5}$$
Type below:
___________

$$\frac{4}{5}$$ = $$\frac{8}{10}$$, $$\frac{12}{15}$$

Explanation:
The original ratio is 4/5. Shade the row for 4 and the row for 5 on the multiplication table.
The column for 2 shows there are 2 ∙ 4, when there are 2 ∙ 5. So, 4/5 equal to 8/10
The column for 3 shows there are 3 ∙ 4, when there are 3 ∙ 5. So, 4/5 equal to 12/15

Question 6.
$$\frac{12}{30}$$
Type below:
___________

$$\frac{12}{30}$$ = $$\frac{24}{60}$$, $$\frac{36}{90}$$

Explanation:
The original ratio is 12/30. Shade the row for 12 and the row for 30 on the multiplication table.
The column for 2 shows there are 2 ∙ 12 when there are 2 ∙ 30. So, 12/30 equal to 24/60
The column for 3 shows there are 3 ∙ 12 when there are 3 ∙ 30. So, 12/30 equal to 36/90

Question 7.
$$\frac{2}{9}$$
Type below:
___________

$$\frac{2}{9}$$ = $$\frac{4}{18}$$, $$\frac{6}{27}$$

Explanation:
The original ratio is 2/9. Shade the row for 2 and the row for 9 on the multiplication table.
The column for 2 shows there are 2 ∙ 2, when there are 2 ∙ 9. So, 2/9 equal to 4/18
The column for 3 shows there are 3 ∙ 2, when there are 3 ∙ 9. So, 2/9 equal to 6/27

Write two equivalent ratios.

Question 8.

Type below:
___________

Explanation:
The original ratio is 9/8. Shade the row for 9 and the row for 8 on the multiplication table.
The column for 2 shows there are 2 ∙ 9 when there are 2 ∙ 8. So, 9/8 equal to 18/16
The column for 3 shows there are 3 ∙ 9 when there are 3 ∙ 8. So, 9/8 equal to 27/24

Question 9.

Type below:
___________

Explanation:
The original ratio is 5/4. Shade the row for 5 and the row for 4 on the multiplication table.
The column for 2 shows there are 2 ∙ 5 when there are 2 ∙ 4. So, 5/4 equal to 10/8
The column for 3 shows there are 3 ∙ 5 when there are 3 ∙ 4. So, 5/4 equal to 15/20

Question 10.

Type below:
___________

Explanation:
The original ratio is 6/9. Shade the row for 6 and the row for 9 on the multiplication table.
The column for 1 shows there are 1 ∙ 3 when there are 1. 4.5. So, 5/4 equal to 3/4.5
The column for 3 shows there are 3 ∙ 3 when there are 3 ∙ 4.5. So, 5/4 equal to 9/13.5

Question 11.
$$\frac{8}{7}$$
Type below:
___________

$$\frac{8}{7}$$ = $$\frac{16}{14}$$, $$\frac{24}{21}$$

Explanation:
The original ratio is 8/7. Shade the row for 8 and the row for 7 on the multiplication table.
The column for 2 shows there are 2 ∙ 8, when there are 2 ∙ 7. So, 8/7 equal to 16/14
The column for 3 shows there are 3 ∙ 8, when there are 3 ∙ 7. So, 8/7 equal to 24/21

Question 12.
$$\frac{2}{6}$$
Type below:
___________

$$\frac{2}{6}$$ = $$\frac{4}{12}$$, $$\frac{6}{18}$$

Explanation:
The original ratio is 2/6. Shade the row for 2 and the row for 6 on the multiplication table.
The column for 2 shows there are 2 ∙ 2, when there are 2 ∙ 6. So, 2/6 equal to 4/12
The column for 3 shows there are 3 ∙ 2, when there are 3 ∙ 6. So, 2/6 equal to 6/18

Question 13.
$$\frac{4}{11}$$
Type below:
___________

$$\frac{4}{11}$$ = $$\frac{8}{22}$$, $$\frac{12}{33}$$

Explanation:
The original ratio is 4/11. Shade the row for 4 and the row for 11 on the multiplication table.
The column for 2 shows there are 2 ∙ 4, when there are 2 ∙ 11. So, 4/11 equal to 8/22
The column for 3 shows there are 3 ∙ 4, when there are 3 ∙ 11. So, 4/11 equal to 12/33

Determine whether the ratios are equivalent.

Question 14.
$$\frac{2}{3} \text { and } \frac{8}{12}$$
___________

Yes

Explanation:
2/3 × 4/4 = 8/12
So, 2/3 is equal to 8/12

Question 15.
$$\frac{8}{10} \text { and } \frac{6}{10}$$
___________

No

Explanation:
8/10 ÷ 2/2 = 4/5
8/10 is not equal to 6/10

Question 16.
$$\frac{16}{60} \text { and } \frac{4}{15}$$
___________

yes

Explanation:
16/60 ÷ 4/4 = 4/15
16/60 is equal to 4/15

Question 17.
$$\frac{3}{14} \text { and } \frac{8}{28}$$
___________

No

Explanation:
3/14 is not equal to 8/28

### Problem Solving + Applications – Page No. 226

Use the multiplication table for 18 and 19.

Question 18.
In Keith’s baseball games this year, the ratio of times he has gotten on base to the times he has been at bat is $$\frac{4}{14}$$. Write two ratios that are equivalent to $$\frac{4}{14}$$.
Type below:
___________

$$\frac{4}{14}$$ = $$\frac{8}{28}$$, $$\frac{2}{7}$$

Explanation:
4/14
multiply both numbers by 2
8/28
divide both numbers by 2
2/7

Question 19.
Pose a Problem Use the multiplication table to write a new problem involving equivalent ratios. Then solve the problem.
Type below:
___________

The ratio of times he has gotten on base to the times he has been at bat is $$\frac{6}{9}$$. Write two ratios that are equivalent to $$\frac{6}{9}$$
.multiply both numbers by 2 = 12/18
multiply both numbers by 3 = 18/ 27

Question 20.
Describe how to write an equivalent ratio for $$\frac{9}{27}$$ without using a multiplication table.
Type below:
___________

$$\frac{9}{27}$$ = $$\frac{18}{54}$$, $$\frac{3}{9}$$

Explanation:
$$\frac{9}{27}$$
multiply both numbers by 2, 18/54
divide both numbers by 3
3/9

Question 21.
Write a ratio that is equivalent to $$\frac{6}{9} \text { and } \frac{16}{24}$$.
$$\frac{□}{□}$$

$$\frac{2}{3}$$

Explanation:
$$\frac{6}{9} \text { and } \frac{16}{24}$$
$$\frac{2}{3}$$ is the equivalent ratio to $$\frac{6}{9} \text { and } \frac{16}{24}$$

Question 22.
Determine whether each ratio is equivalent to $$\frac{1}{3}, \frac{5}{10}, \text { or } \frac{3}{5}$$. Write the ratio in the correct box.

Type below:
___________

3/9, 7/21, 18/30, 10/30

Explanation:
2/4 = 1/2
3/9 = 1/3
7/21 = 1/3
18/30 = 3/5
10/30 = 1/3
6/10 = 2/5
8/16 = 4/8 = 1/2

### Equivalent Ratios and Multiplication Tables – Page No. 227

Write two equivalent ratios.

Question 1.
Use a multiplication table to write two ratios that are equivalent to $$\frac{5}{3}$$.
Type below:
___________

$$\frac{5}{3}$$ = $$\frac{10}{6}$$, $$\frac{15}{9}$$

Explanation:
The original ratio is 5/3. Shade the row for 5 and the row for 3 on the multiplication table.
The column for 2 shows there are 2 ∙ 5, when there are 2 ∙ 3. So, 5/3 equal to 10/6
The column for 3 shows there are 3 ∙ 5, when there are 3 ∙ 3. So, 5/3 equal to 15/9

Question 2.

Type below:
___________

Explanation:
The original ratio is 6/7. Shade the row for 6 and the row for 7 on the multiplication table.
The column for 2 shows there are 2 ∙ 6 when there are 2 ∙ 7. So, 6/7 equal to 12/14
The column for 3 shows there are 3 ∙ 6 when there are 3 ∙ 7. So, 6/7 equal to 18/21

Question 2.

Type below:
___________

Explanation:
The original ratio is 3/2. Shade the row for 3 and the row for 2 on the multiplication table.
The column for 2 shows there are 2 ∙ 3 when there are 2 ∙ 2. So, 3/2 equal to 6/4
Multiply 3/2 with 4/4 = 12/8

Question 4.
$$\frac{6}{8}$$
Type below:
___________

$$\frac{6}{8}$$ =$$\frac{12}{16}$$, $$\frac{18}{24}$$

Explanation:
The original ratio is 6/8. Shade the row for 6 and the row for 8 on the multiplication table.
The column for 2 shows there are 2 ∙ 6, when there are 2 ∙ 8. So, 6/8 equal to 12/16
The column for 3 shows there are 3 ∙ 6, when there are 3 ∙ 8. So, 6/8 equal to 18/24

Question 5.
$$\frac{11}{1}$$
Type below:
___________

$$\frac{11}{1}$$ = $$\frac{22}{2}$$, $$\frac{33}{3}$$

Explanation:
The original ratio is 11/1. Shade the row for 11 and the row for 1 on the multiplication table.
The column for 2 shows there are 2 ∙ 11, when there are 2 ∙ 1. So, 11/1 equal to 22/2
The column for 3 shows there are 3 ∙ 11, when there are 3 ∙ 1. So, 11/1 equal to 33/3

Determine whether the ratios are equivalent.

Question 6.
$$\frac{2}{3} \text { and } \frac{5}{6}$$.
___________

No

Explanation:
2/3 is not equal to 5/6

Question 7.
$$\frac{5}{10} \text { and } \frac{1}{6}$$.
___________

No

Explanation:
5/10 is not equal to 1/6

Question 8.
$$\frac{8}{3} \text { and } \frac{32}{12}$$.
___________

Yes

Explanation:
8/3 × 4/4 = 32/12
8/3 is equal to 32/12

Question 9.
$$\frac{9}{12} \text { and } \frac{3}{4}$$.
___________

Yes

Explanation:
9/12 ÷ 3/3 = 3/4
9/12 is equal to 3/4

Problem Solving

Question 10.
Tristan uses 7 stars and 9 diamonds to make a design. Write two ratios that are equivalent to $$\frac{7}{9}$$.
Type below:
___________

$$\frac{7}{9}$$  = $$\frac{14}{18}$$ , $$\frac{21}{27}$$

Explanation:
Tristan uses 7 stars and 9 diamonds to make a design.
$$\frac{7}{9}$$
The original ratio is 7/9. Shade the row for 7 and the row for 9 on the multiplication table.
The column for 2 shows there are 2 ∙ 7, when there are 2 ∙ 9. So, 7/9 equal to 14/18
The column for 3 shows there are 3 ∙ 7, when there are 3 ∙ 9. So, 7/9 equal to 21/27

Question 11.
There are 12 girls and 16 boys in Javier’s math class. There are 26 girls and 14 boys in Javier’s choir class. Are the ratios of girls to boys in the two classes equivalent? Explain.
Type below:
___________

No, the ratio 26/14 is not equal to the ratio 12/16

Question 12.
Explain how to determine whether two ratios are equivalent.
Type below:
___________

If any ratio is multiplied or divided by the same number, then the ratios are equivalent.

### Lesson Check – Page No. 228

Question 1.
A pancake recipe calls for 4 cups of flour and 3 cups milk. Does a recipe calling for 2 cups flour and 1.5 cups milk use the same ratio of flour to milk?
___________

A muffin recipe that calls for 2 cups flour and 1.5 cups milk

Explanation:
A pancake recipe calls for 4 cups of flour and 3 cups milk. A muffin recipe that calls for 2 cups flour and 1.5 cups milk.

Question 2.
___________

The bracelet has 14 red and 19 gold, so the ratio between red and gold is 14/19. We cannot simplify this ratio as there are not common factors between 14 and 19, because 19 is a prime number.
As there are 84 red and 133 gold the ratio will be 84/133. For this ratio to be equal to 14/19 it should be that 84 is multiple of 14 and 133 multiple of 19, and both multiples must the same,
84/133 is not equal to 14/19

Spiral Review

Question 3.
Scissors come in packages of 3. Glue sticks come in packages of 10. Martha wants to buy the same number of each. What is the fewest glue sticks Martha can buy?
_____ glue sticks

30 glue sticks

Explanation:
Scissors come in packages of 3. Glue sticks come in packages of 10. Martha wants to buy the same number of each.
3 × 10 = 30 glue sticks

Question 4.
Cole had $$\frac{3}{4}$$ hour of free time before dinner. He spent $$\frac{2}{3}$$ of the time playing the guitar. How long did he play the guitar?
$$\frac{□}{□}$$ hour

$$\frac{1}{2}$$ hour

Explanation:
Cole had $$\frac{3}{4}$$ hour of free time before dinner. He spent $$\frac{2}{3}$$ of the time playing the guitar.
$$\frac{2}{3}$$ × $$\frac{3}{4}$$ = 1/2 hour

Question 5.
Delia has 3 $$\frac{5}{8}$$ yards of ribbon. About how many $$\frac{1}{4}$$-yard-long pieces can she cut?

Explanation:
Length of yards of ribbon is 3 5/8 = 29/8
Length of yards of ribbon pieces need to be cut is 1/4
Number of yards = 29/8 ÷ 1/4 = 14.5 = 14

Question 6.
Which point is located at –1.1?

Type below:
___________

B

Explanation:
-1.1 is in between -1 and -2
-1.1 is close to -1
So, the answer is point B

### Share and Show – Page No. 231

Question 1.
In Jawan’s school, 4 out of 10 students chose basketball as a sport they like to watch, and 3 out of 5 students chose football. Is the ratio of students who chose basketball (4 to 10) equivalent to the ratio of students who chose football (3 to 5)?
Type below:
___________

the ratio of students who chose basketball (4 to 10) is not equivalent to the ratio of students who chose football (3 to 5)

Explanation:
In Jawan’s school, 4 out of 10 students chose basketball as a sport they like to watch, and 3 out of 5 students chose football.
4/10 = 0.4
3/5 = 0.6
0.4 is not equal to 0.6
The ratio of students who chose basketball (4 to 10) is not equivalent to the ratio of students who chose football (3 to 5)

Question 2.
What if 20 out of 50 students chose baseball as a sport they like to watch? Is this ratio equivalent to the ratio for either basketball or football? Explain.
Type below:
___________

The baseball ratio is equal to the basketball ratio

Explanation:
If 20 out of 50 students chose baseball, 20/50 = 2/5
2/5 × 2/2 = 4/10
The baseball ratio is equal to the basketball ratio.

Question 3.
Look for Structure The table shows the results of the quizzes Hannah took in one week. Did Hannah get the same score on her math and science quizzes? Explain.

Type below:
___________

Hannah didn’t get the same score on her math and science quizzes

Explanation:
Social Studies = 4/5
Math = 8/10 = 0.8
Science = 3/4 = 0.75
English = 10/12
Math = 8/10
Divide the 8/10 with 2/2 = 8/10 ÷ 2/2 = 4/5
Hannah didn’t get the same score on her math and science quizzes

Question 4.
Did Hannah get the same score on the quizzes in any of her classes? Explain.
Type below:
___________

The ratio of Social Studies is equal to the ratio of Math

Explanation:
Social Studies = 4/5 = 0.8
Math = 8/10 = 0.8
Science = 3/4 = 0.75
English = 10/12 = 0.8333
The ratio of Social Studies is equal to the ratio of Math

### On Your Own – Page No. 232

Question 5.
For every $10 that Julie makes, she saves$3. For every $15 Liam makes, he saves$6. Is Julie’s ratio of money saved to money earned equivalent to Liam’s ratio of money saved to money earned?
Type below:
___________

Julie’s ratio of money saved to money earned is not equivalent to Liam’s ratio of money saved to money earned.

Explanation:
No. Julie’s ratio is 3:10 or 30 percent towards her savings while Lion’s is 6:15 which is 40 percent towards savings.

Question 6.
A florist offers three different bouquets of tulips and irises. The list shows the ratios of tulips to irises in each bouquet. Determine the bouquets that have equivalent ratios.

Type below:
___________

The ratio of Spring Mix is equal to the ratio of Splash of Sun

Explanation:
Spring Mix = 4/6 = 0.66
Morning Melody = 9/12 = 0.75
Splash of Sun = 10/15 =0.66
The ratio of Spring Mix is equal to the ratio of Splash of Sun

Question 7.
The ratio of boys to girls in a school’s soccer club is 3 to 5. The ratio of boys to girls in the school’s chess club is 13 to 15. Is the ratio of boys to girls in the soccer club equivalent to the ratio of boys to girls in the chess club? Explain
Type below:
___________

No

Explanation:
They are not equivalent because you can not reduce 13 any further because it is a prime number and if you multiply 3 by 3 and 5 by 3 you would get 9:15 as the equivalent ratio.

Question 8.
Analyze Thad, Joey, and Mia ran in a race. The finishing times were 4.56 minutes, 3.33 minutes, and 4.75 minutes. Thad did not finish last. Mia had the fastest time. What was each runner’s time?
Type below:
___________

Mia = 3.33 minutes
Joey = 4.75 minutes

Explanation:
Mia had the fastest time. 3.33 minutes
Thad did not finish last. So, Joey = 4.75 minutes

Question 9.
Fernando donates $2 to a local charity organization for every$15 he earns. Cleo donates $4 for every$17 she earns. Is Fernando’s ratio of money donated to money earned equivalent to Cleo’s ratio of money donated to money earned? Explain.
Type below:
___________

Fernando’s ratio of money donated to money earned is not equivalent to Cleo’s ratio of money donated to money earned

Explanation:
Fernando donates $2 to a local charity organization for every$15 he earns.
$2/$15 = 0.1333
Cleo donates $4 for every$17 she earns. $4/$17 = 0.2359
Fernando’s ratio of money donated to money earned is not equivalent to Cleo’s ratio of money donated to money earned

### Problem Solving Use Tables to Compare Ratios – Page No. 233

Question 1.
Sarah asked some friends about their favorite colors. She found that 4 out of 6 people prefer blue, and 8 out of 12 people prefer green. Is the ratio of friends who chose blue to the total asked equivalent to the ratio of friends who chose green to the total asked?
Type below:
___________

Yes, 4/6 is equivalent to 8/12

Explanation:

4/6 = 0.666
8/12 = 0.666

Question 2.
Lisa and Tim make necklaces. Lisa uses 5 red beads for every 3 yellow beads. Tim uses 9 red beads for every 6 yellow beads. Is the ratio of red beads to yellow beads in Lisa’s necklace equivalent to the ratio in Tim’s necklace?
Type below:
___________

The ratio of red beads to yellow beads in Lisa’s necklace is not equivalent to the ratio in Tim’s necklace

Explanation:
Lisa and Tim make necklaces. Lisa uses 5 red beads for every 3 yellow beads.
5/3 = 1.666
Tim uses 9 red beads for every 6 yellow beads. 9/6 = 1.5
The ratio of red beads to yellow beads in Lisa’s necklace is not equivalent to the ratio in Tim’s necklace

Question 3.
Mitch scored 4 out of 5 on a quiz. Demetri scored 8 out of 10 on a quiz. Did Mitch and Demetri get equivalent scores?
Type below:
___________

Mitch and Demetri get equivalent scores

Explanation:
Mitch scored 4 out of 5 on a quiz. 4/5 = 0.8
Demetri scored 8 out of 10 on a quiz. = 8/10 = 0.8
Mitch and Demetri get equivalent scores

Question 4.
Use tables to show which of these ratios are equivalent : $$\frac{4}{6}$$, $$\frac{10}{25}, \text { and } \frac{6}{15}$$.
Type below:
___________

$$\frac{10}{25}, \text { and } \frac{6}{15}$$ are equal

Explanation:
$$\frac{4}{6}$$ = 0.6666
$$\frac{10}{25}$$ = 0.4
$$\frac{6}{15}$$ = 0.4
$$\frac{10}{25}, \text { and } \frac{6}{15}$$ are equal

### Page No. 234

Question 1.
Mrs. Sahd distributes pencils and paper to students in the ratio of 2 pencils to 10 sheets of paper. Three of these ratios are equivalent to $$\frac{2}{10}$$. Which one is NOT equivalent?
$$\frac{1}{5} \frac{7}{15} \frac{4}{20} \frac{8}{40}$$
Type below:
___________

$$\frac{7}{15}$$ is not equal $$\frac{2}{10}$$

Explanation:
Mrs. Sahd distributes pencils and paper to students in the ratio of 2 pencils to 10 sheets of paper. Three of these ratios are equivalent to $$\frac{2}{10}$$ = 0.2
$$\frac{1}{5}$$ = 0.2
$$\frac{7}{15}$$ = 0.4666
$$\frac{4}{20}$$ = 0.2
$$\frac{8}{40}$$ = 0.2
$$\frac{7}{15}$$ is not equal $$\frac{2}{10}$$

Question 2.
Keith uses 18 cherries and 3 peaches to make a pie filling. Lena uses an equivalent ratio of cherries to peaches when she makes pie filling. Can Lena use a ratio of 21 cherries to 6 peaches? Explain.
Type below:
___________

No, she cannot use a ratio of 21 cherries to 6 peaches

Explanation:
Keith uses 18 cherries and 3 peaches to make a pie filling. 18/3 = 6
Lena uses a ratio of 21 cherries to 6 peaches, 21/6 = 3.5
No, she cannot use a ratio of 21 cherries to 6 peaches

Spiral Review

Question 3.
What is the quotient $$\frac{3}{20} \div \frac{7}{10}$$?
Type below:
___________

$$\frac{3}{14}$$

Explanation:
$$\frac{3}{20} \div \frac{7}{10}$$
3/20 × 10/7 = 3/14

Question 4.
Which of these numbers is greater than – 2.25 but less than –1?
1 -1.5 0 -2.5
Type below:
___________

Explanation:
1 lies between 0 to 1
-1.5 lies between -1 and -2. It is greater than -2.25 and also less than -1
0 lies between -1 to 1
-2.5 lies between -2 and -3. -2.5 is less than -2.25

Question 5.
Alicia plots a point at (0, 5) and (0, –2). What is the distance between the points?
Type below:
___________

7 units

Explanation:
Alicia plots a point at (0, 5) and (0, –2).
The given points have the same x-coordinates.
|-2| = 2
5 + 0 = 5
0 + 2 = 2
5 + 2 = 7
The distance is 7 units

Question 6.
Morton sees these stickers at a craft store. What is the ratio of clouds to suns?

Type below:
___________

3 : 2

Explanation:
there are 3 clouds and 2 suns. So, the ratio is 3 to 2.

### Share and Show – Page No. 237

Use equivalent ratios to find the unknown value.

Question 1.
$$\frac{?}{10}=\frac{4}{5}$$
_____

$$\frac{8}{10}$$ = $$\frac{4}{5}$$

Explanation:
Use common denominators to write equivalent ratios.
10 is a multiple of 5, so 10 is a common denominator.
Multiply the 4 and denominator by 2 to write the ratios using a common denominator.
4/5 × 2/2 = 8/10
The denominators are the same, so the numerators are equal to each other.
So, the unknown value is 8/10 = 4/5
$$\frac{8}{10}$$

Question 2.
$$\frac{18}{24}=\frac{6}{?}$$
_____

$$\frac{6}{8}$$ = $$\frac{18}{24}$$

Explanation:
Write an equivalent ratio with 18 in the numerator.
Divide 18 by 6 to get 3
So, divide the denominator by 24 as well.
24/3 = 8
The numerators are the same, so the denominators are equal to each other.
So, the unknown value is 6/8 = 18/24
$$\frac{6}{8}$$

Question 3.
$$\frac{3}{6}=\frac{15}{?}$$
_____

$$\frac{15}{30}$$

Explanation:
Write an equivalent ratio with 15 in the numerator.
Multiply 3 with 5 to get 15
So, Multiply 6 with 5 to get the denominator of unknown number.
6 × 5 = 30
The numerators are the same, so the denominators are equal to each other.
So, the unknown value is 3/6 = 15/30
$$\frac{15}{30}$$

Question 4.
$$\frac{?}{5}=\frac{8}{10}$$
_____

$$\frac{4}{5}$$

Explanation:
Write an equivalent ratio with 10 in the denominator.
Divide 10 by 2 to get 5
So, divide the numerator 8 as well.
8/2 = 4
The denominators are the same, so the numerators are equal to each other.
So, the unknown value is 8/10 = 4/5
$$\frac{4}{5}$$

Question 5.
$$\frac{7}{4}=\frac{?}{12}$$
_____

$$\frac{21}{12}$$

Explanation:
Write an equivalent ratio with 12 in the denominator.
Multiply 4 with 3 to get 12
So, Multiply 7 with 3 to get the numerator of unknown number.
7 × 3 = 21
The denominators are the same, so the numerators are equal to each other.
So, the unknown value is 21/12 = 7/4
$$\frac{21}{12}$$

Question 6.
$$\frac{10}{?}=\frac{40}{12}$$
_____

$$\frac{10}{3}$$

Explanation:
Write an equivalent ratio with 40 in the numerator.
Divide 40 by 4 to get 10
So, divide the denominator 12 as well.
12/4 = 3
The numerators are the same, so the denominators are equal to each other.
So, the unknown value is 10/3 = 40/12
$$\frac{10}{3}$$

Use equivalent ratios to find the unknown value.

Question 7.
$$\frac{2}{6}=\frac{?}{30}$$
_____

$$\frac{10}{30}$$

Explanation:
Use common denominators to write equivalent ratios.
30 is a multiple of 6, so 30 is a common denominator.
Multiply the 6 and denominator by 5 to write the ratios using a common denominator.
2/6 × 5/5 =10/30
The denominators are the same, so the numerators are equal to each other.
So, the unknown value is 10/30 = 2/6
$$\frac{10}{30}$$

Question 8.
$$\frac{5}{?}=\frac{55}{110}$$
_____

$$\frac{5}{10}$$

Explanation:
Write an equivalent ratio with 55 in the numerator.
Divide 55 with 11 to get 5
So, Divide 110 with 11 to get the denominator of unknown number.
110/11 = 10
The numerators are the same, so the denominators are equal to each other.
So, the unknown value is 5/10 = 55/110
$$\frac{5}{10}$$

Question 9.
$$\frac{3}{9}=\frac{9}{?}$$
_____

$$\frac{9}{27}$$

Explanation:
Write an equivalent ratio with 9 in the numerator.
Multiply 3 with 3 to get 9
So, Multiply 9 with 3 to get the denominator of unknown number.
9 × 3 = 27
The numerators are the same, so the denominators are equal to each other.
So, the unknown value is 9/27 = 3/9
$$\frac{9}{27}$$

Question 10.
$$\frac{?}{6}=\frac{16}{24}$$
_____

$$\frac{4}{6}$$

Explanation:
Use common denominators to write equivalent ratios.
Divide 24 with 4 to get 6.
So, divide 16 with 4 to know the unknown number of numerator
16/4 = 4
The denominators are the same, so the numerators are equal to each other.
So, the unknown value is 4/6 = 16/24
$$\frac{4}{6}$$

Question 11.
Mavis walks 3 miles in 45 minutes. How many minutes will it take Mavis to walk 9 miles?
_____ minutes

135 minutes

Explanation:
Mavis walks 3 miles in 45 minutes.
For 9 miles, (9 × 45)/3 = 135 minutes

Question 12.
The ratio of boys to girls in a choir is 3 to 8. There are 32 girls in the choir. How many members are in the choir?
_____ members

12 members

Explanation:
The ratio of boys to girls in a choir is 3 to 8.
3/8 × 4/ 4 = 12/32
So, if there are 32 girls in the choir, there will be 12 boys present.

Question 13.
Use Reasoning Is the unknown value in $$\frac{2}{3}=\frac{?}{18}$$ the same as the unknown value in $$\frac{3}{2}=\frac{18}{?}$$? Explain.
Type below:
___________

12

Explanation:
$$\frac{2}{3}=\frac{?}{18}$$
2/3 × 6/6 = 12/18
the unknown value is 12
$$\frac{3}{2}=\frac{18}{?}$$
3/2 × 6/6 = 18/12
the unknown value is 12

### Problem Solving + Applications – Page No. 238

Solve by finding an equivalent ratio.

Question 14.
It takes 8 minutes for Sue to make 2 laps around the go-kart track. How many laps can Sue complete in 24 minutes?
_____ laps

6 laps

Explanation:
It takes 8 minutes for Sue to make 2 laps around the go-kart track.
For 24 minutes, (24 × 2)/8 = 48/8 =6

Question 15.
The width of Jay’s original photo is 8 inches. The length of the original photo is 10 inches. He prints a smaller version that has an equivalent ratio of width to length. The width of the smaller version is 4 inches less than the width of the original. What is the length of the smaller version?
_____ inches

5 inches

Explanation:
The width of Jay’s original photo is 8 inches. The length of the original photo is 10 inches.
8/10
He prints a smaller version that has an equivalent ratio of width to length. The width of the smaller version is 4 inches less than the width of the original.
4/s
8/10 ÷ 2/2 = 4/5
5 inches

Question 16.
Ariel bought 3 raffle tickets for $5. How many tickets could Ariel buy for$15?
_____ tickets

9 tickets

Explanation:
Ariel bought 3 raffle tickets for $5. For$15, ($15 × 3)/$5 = 45/5 = 9

Question 17.
What’s the Error? Greg used the steps shown to find the unknown value. Describe his error and give the correct solution.
$$\frac{2}{6}=\frac{?}{12}$$
$$\frac{2+6}{6+6}=\frac{?}{12}$$
$$\frac{8}{12}=\frac{?}{12}$$
The unknown value is 8.
Type below:
___________

Greg added 6 to the numerator and denominator which is not correct to find the unknown value.
$$\frac{2}{6}=\frac{?}{12}$$
2/6 × 2/2 = 4/12
4 is the unknown value.

Question 18.
Courtney bought 3 maps for $10. Use the table of equivalent ratios to find how many maps she can buy for$30.

Type below:
___________

Explanation:
3/10 × 3/3 = 9/30

### Use Equivalent Ratios – Page No. 239

Use equivalent ratios to find the unknown value.

Question 1.
$$\frac{4}{10}=\frac{?}{40}$$
_____

$$\frac{16}{40}$$

Explanation:
4/10 × 4/4 = 16/40

Question 2.
$$\frac{3}{24}=\frac{33}{?}$$
_____

$$\frac{33}{264}$$

Explanation:
3/24 × 11/11 = 33/264

Question 3.
$$\frac{7}{?}=\frac{21}{27}$$
_____

$$\frac{7}{9}$$

Explanation:
21/27 ÷ 3/3 = 7/9

Question 4.
$$\frac{?}{9}=\frac{12}{54}$$
_____

$$\frac{2}{9}$$

Explanation:
12/54 ÷ 6/6 = 2/9

Question 5.
$$\frac{3}{2}=\frac{12}{?}$$
_____

$$\frac{12}{8}$$

Explanation:
3/2 × 4/4 = 12/8

Question 6.
$$\frac{4}{5}=\frac{?}{40}$$
_____

$$\frac{32}{40}$$

Explanation:
4/5 × 8/8 = 32/40

Question 7.
$$\frac{?}{2}=\frac{45}{30}$$
_____

$$\frac{3}{2}$$

Explanation:
45/30 ÷ 15/15 = 3/2

Question 8.
$$\frac{45}{?}=\frac{5}{6}$$
_____

$$\frac{45}{54}$$

Explanation:
5/6 × 9/9 = 45/54

Problem Solving

Question 9.
Honeybees produce 7 pounds of honey for every 1 pound of beeswax they produce. Use equivalent ratios to find how many pounds of honey are produced when 25 pounds of beeswax are produced.
_____ pounds

175 pounds

Explanation:
Honeybees produce 7 pounds of honey for every 1 pound of beeswax they produce.
7/1
25 pounds of beeswax, 25 × 7 = 175 pounds

Question 10.
A 3-ounce serving of tuna provides 21 grams of protein. Use equivalent ratios to find how many grams of protein are in 9 ounces of tuna.
_____ grams of protein

63 grams of protein

Explanation:
A 3-ounce serving of tuna provides 21 grams of protein.
For 9 ounces of tuna, (21 × 9)/3 = 63

Question 11.
Explain how using equivalent ratios is like adding fractions with unlike denominators.
Type below:
___________

Equivalent ratios have different numbers but represent the same relationship. In this tutorial, you’ll see how to find equivalent ratios by first writing the given ratio as a fraction. And it cannot be the same by adding tow fraction with different ratio

### Lesson Check – Page No. 240

Question 1.
Jaron paid $2.70 for 6 juice boxes. How much should Jaron expect to pay for 18 juice boxes?$ _____

$8.1 Explanation: Jaron paid$2.70 for 6 juice boxes. For 6 boxes he paid $2.70. For 18 juice boxes, (18 ×$2.70)/6 = $8.1 Question 2. A certain shade of orange paint is made by mixing 3 quarts of red paint with 2 quarts of yellow paint. To make more paint of the same shade, how many quarts of yellow paint should be mixed with 6 quarts of red paint? _____ quarts Answer: 4 quarts Explanation: A certain shade of orange paint is made by mixing 3 quarts of red paint with 2 quarts of yellow paint. 3 quarts of red paint is mixed with 2 quarts of yellow paint So, 6 quarts of red paint is mixed with 6/3 × 2 = 4 quarts of yellow paint Spiral Review Question 3. What is the quotient $$2 \frac{4}{5} \div 1 \frac{1}{3}$$? ______ $$\frac{□}{□}$$ Answer: 2$$\frac{1}{10}$$ Explanation: 2 4/5 = 14/5 = 2.8 1 1/3 = 4/3 = 1.333 2.8/1.333 = 2 1/10 Question 4. What is the quotient $$-2 \frac{2}{3}$$? ______ $$\frac{□}{□}$$ Answer: 1$$\frac{11}{16}$$ Explanation: −4 1/2 ÷ -2 2/3 1 11/16 Question 5. On a map, a clothing store is located at (–2, –3). A seafood restaurant is located 6 units to the right of the clothing store. What are the coordinates of the restaurant? Type below: ___________ .Answer: (4, -3) Explanation: On a map, a clothing store is located at (–2, –3). A seafood restaurant is located 6 units to the right of the clothing store. |-2| = 2 2 + 0 = 2 0+4 = 4 2 + 4 = 6 units Question 6. Marisol plans to make 9 mini-sandwiches for every 2 people attending her party. Write a ratio that is equivalent to Marisol’s ratio. Type below: ___________ Answer: 27/6 and 45/10 Explanation: Marisol plans to make 9 mini-sandwiches for every 2 people attending her party. 9/2 × 3/3 = 27/6 9/2 × 5/5 = 45/10 ### Mid-Chapter Checkpoint – Vocabulary – Page No. 241 Choose the best term from the box to complete the sentence. Question 1. A _____ is a rate that makes a comparison to 1 unit. Type below: ___________ Answer: rate Question 2. Two ratios that name the same comparison are _____ . Type below: ___________ Answer: Equivalent Ratios Concepts and Skills Question 3. Write the ratio of red circles to blue squares. Type below: ___________ Answer: 3 : 5 Explanation: There are 3 red counter and 5 square boxes. So, the ratio is 3 : 5 Write the ratio in two different ways. Question 4. 8 to 12 Type below: ___________ Answer: $$\frac{8}{12}$$ 8 : 12 Explanation: 8 to 12 as a fraction $$\frac{8}{12}$$ 8 to 12 with a colon 8 : 12 Question 5. 7 : 2 Type below: ___________ Answer: $$\frac{7}{2}$$ 7 to 2 Explanation: 7 : 2 as a fraction $$\frac{7}{2}$$ 7 : 2 using words 7 to 2 Question 6. $$\frac{5}{9}$$ Type below: ___________ Answer: 5 to 9 5 : 9 Explanation: $$\frac{5}{9}$$ using words 5 to 9 $$\frac{5}{9}$$ with a colon 5 : 9 Question 7. 11 to 3 Type below: ___________ Answer: $$\frac{11}{3}$$ 11 : 3 Explanation: 11 to 3 as a fraction $$\frac{11}{3}$$ 11 to 3 with a colon 11 : 3 Write two equivalent ratios. Question 8. $$\frac{2}{7}$$ Type below: ___________ Answer: $$\frac{2}{7}$$ = $$\frac{4}{14}$$, $$\frac{6}{21}$$ Explanation: The original ratio is 2/7. Shade the row for 2 and the row for 7 on the multiplication table. The column for 2 shows there are 2 ∙ 2, when there are 2 ∙ 7. So, 2/7 equal to 4/14 The column for 3 shows there are 3 ∙ 2, when there are 3 ∙ 7. So, 2/7 equal to 6/21 Question 9. $$\frac{6}{5}$$ Type below: ___________ Answer: $$\frac{6}{5}$$ = $$\frac{12}{10}$$, $$\frac{18}{15}$$ Explanation: The original ratio is 6/5. Shade the row for 6 and the row for 5 on the multiplication table. The column for 2 shows there are 2 ∙ 6, when there are 2 ∙ 5. So, 6/5 equal to 12/10 The column for 3 shows there are 3 ∙ 6, when there are 3 ∙ 5. So, 6/5 equal to 18/15 Question 10. $$\frac{9}{12}$$ Type below: ___________ Answer: $$\frac{9}{12}$$ = $$\frac{18}{24}$$, $$\frac{27}{36}$$ Explanation: The original ratio is 9/12. Shade the row for 9 and the row for 12 on the multiplication table. The column for 2 shows there are 2 ∙ 9, when there are 2 ∙ 12. So, 9/12 equal to 18/24 The column for 3 shows there are 3 ∙ 9, when there are 3 ∙ 12. So, 9/12 equal to 27/36 Question 11. $$\frac{18}{6}$$ Type below: ___________ Answer: $$\frac{18}{6}$$ = $$\frac{36}{12}$$, $$\frac{54}{18}$$ Explanation: The original ratio is 18/6. Shade the row for 18 and the row for 6 on the multiplication table. The column for 2 shows there are 2 ∙ 18, when there are 2 ∙ 6. So, 18/6 equal to 36/12 The column for 3 shows there are 3 ∙ 18, when there are 3 ∙ 6. So, 18/6 equal to 54/18 Find the unknown value. Question 12. $$\frac{15}{?}=\frac{5}{10}$$ Type below: ___________ Answer: 30 Explanation: 5/10 × 3/3 = 15/30 So, the unknown number is 30 Question 13. $$\frac{?}{9}=\frac{12}{3}$$ Type below: ___________ Answer: 36 Explanation: 12/3 × 3/3 = 36/9 So, the unknown number is 36 Question 14. $$\frac{48}{16}=\frac{?}{8}$$ Type below: ___________ Answer: 24 Explanation: 48/16 ÷ 2/2 = 24/8 So, the unknown number is 24 Question 15. $$\frac{9}{36}=\frac{3}{?}$$ Type below: ___________ Answer: 12 Explanation: 9/36 ÷ 3/3 = 3/12 So, the unknown number is 12 ### Page No. 242 Question 16. There are 36 students in the chess club, 40 students in the drama club, and 24 students in the film club. What is the ratio of students in the drama club to students in the film club? Type below: ___________ Answer: 40 to 24 Explanation: There are 36 students in the chess club, 40 students in the drama club, and 24 students in the film club. The ratio of students in the drama club to students in the film club 40 to 24 Question 17. A trail mix has 4 cups of raisins, 3 cups of dates, 6 cups of peanuts, and 2 cups of cashews. Which ingredients are in the same ratio as cashews to raisins? Type below: ___________ Answer: dates to peanuts Explanation: A trail mix has 4 cups of raisins, 3 cups of dates, 6 cups of peanuts, and 2 cups of cashews. cashews to raisins = 2/4 = 1/2 dates to peanuts = 3/6 = 1/2 Question 18. There are 32 adults and 20 children at a school play. What is the ratio of children to people at the school play? Type below: ___________ Answer: 5 to 13 Explanation: There are 32 adults and 20 children at a school play. people = 32 + 20 = 52 the ratio of children to people at the school play = 20/52 = 5/13 Question 19. Sonya got 8 out of 10 questions right on a quiz. She got the same score on a quiz that had 20 questions. How many questions did Sonya get right on the second quiz? How many questions did she get wrong on the second quiz? Type below: ___________ Answer: 4 wrong Explanation: 8/10 = x/20 So, 10 × 2 = 20, so 8 × 2=16 so she got 16 out of 20 right and 20 – 16 = 4 She got 4 wrong. ### Share and Show – Page No. 245 Write the rate as a fraction. Then find the unit rate. Question 1. Sara drove 72 miles on 4 gallons of gas. _____ miles/gallon Answer: 18 miles/gallon Explanation: Sara drove 72 miles on 4 gallons of gas. 72/4 Divide 72/4 with 4/4 72/4 ÷ 4/4 = 18 Question 2. Dean paid$27.00 for 4 movie tickets.
$_____ per ticket Answer:$6.75 per ticket

Explanation:
Dean paid $27.00 for 4 movie tickets.$27.00/4
Divide $27.00/4 with 4/4$27.00/4 ÷ 4/4 = $6.75 Question 3. Amy and Mai have to read Bud, Not Buddy for a class. Amy reads 20 pages in 2 days. Mai reads 35 pages in 3 days. Who reads at a faster rate? ___________ Answer: Mai reads at a faster rate Explanation: Amy and Mai have to read Bud, Not Buddy for a class. Amy reads 20 pages in 2 days. 20/2 = 10 pages for each day Mai reads 35 pages in 3 days. 35/3 = 11.66 pages for each day Mai reads at a faster rate Question 4. An online music store offers 5 downloads for$6.25. Another online music store offers 12 downloads for $17.40. Which store offers the better deal? ___________ Answer: An online music store offers 5 downloads for$6.25 offers the better deal

Explanation:
An online music store offers 5 downloads for $6.25.$6.25/5 = $1.25 Another online music store offers 12 downloads for$17.40.
$17.40/12 =$1.45
An online music store offers 5 downloads for $6.25 offers the better deal On Your Own Write the rate as a fraction. Then find the unit rate. Question 5. A company packed 108 items in 12 boxes. Type below: ___________ Answer: 9 Explanation: A company packed 108 items in 12 boxes. 108/12 Divide 108/12 with 12/12 108/12 ÷ 12/12 = 9 Question 6. There are 112 students for 14 teachers. Type below: ___________ Answer: 8 Explanation: There are 112 students for 14 teachers. 112/14 Divide 112/14 with 14/14 112/14 ÷ 14/14 = 8 Question 7. Geoff charges$27 for 3 hours of swimming lessons. Anne charges $31 for 4 hours. How much more does Geoff charge per hour than Anne?$ _____

$1.25 Explanation: Geoff charges$27 for 3 hours of swimming lessons.
$27/3 =$9 for an hour
Anne charges $31 for 4 hours.$31/4 = $7.75$9 – $7.75 =$1.25
Geoff charge $1.25 per hour more than Anne Question 8. Compare One florist made 16 bouquets in 5 hours. A second florist made 40 bouquets in 12 hours. Which florist makes bouquets at a faster rate? Type below: ___________ Answer: A second florist made 40 bouquets in 12 hours at a faster rate Explanation: Compare One florist made 16 bouquets in 5 hours. 16/5 = 3.2 A second florist made 40 bouquets in 12 hours. 40/12 = 3.333 A second florist made 40 bouquets in 12 hours at a faster rate Tell which rate is faster by comparing unit rates. Question 9. $$\frac{160 \mathrm{mi}}{2 \mathrm{hr}} \text { and } \frac{210 \mathrm{mi}}{3 \mathrm{hr}}$$ Type below: ___________ Answer: 160mi/2hr Explanation: 160mi/2hr ÷ 2/2 = 80mi/hr 210mi/3hr = 70mi/hr 80mi/hr > 70mi/hr Question 10. $$\frac{270 \mathrm{ft}}{9 \mathrm{min}} \text { and } \frac{180 \mathrm{ft}}{9 \mathrm{min}}$$ Type below: ___________ Answer: 270ft/9min Explanation: 270ft/9min = 30ft/min 180ft/9min = 20ft/min 30ft/min > 20ft/min Question 11. $$\frac{250 \mathrm{m}}{10 \mathrm{s}} \text { and } \frac{120 \mathrm{m}}{4 \mathrm{s}}$$ Type below: ___________ Answer: 250m/10s Explanation: 250m/10s = 25m/s 120m/4s = 20m/s 25m/s > 20m/s ### Unlock the Problem – Page No. 246 Question 12. Ryan wants to buy treats for his puppy. If Ryan wants to buy the treats that cost the least per pack, which treat should he buy? Explain. a. What do you need to find? Type below: ___________ Answer: We need to find that cost the least per pack Question 12. b. Find the price per pack for each treat. Type below: ___________ Answer: Pup bites =$5.76/4 ÷ 4/4 = $1.44 Doggie Treats =$7.38/6 ÷ 6/6 = $1.23 Pupster snacks =$7.86/6 ÷ 6/6 = $1.31 Nutri-Biscuits =$9.44/8 ÷ 8/8 = $1.18 Question 12. c. Complete the sentences The treat with the highest price per pack is _____. The treat with the lowest price per pack is _____. Ryan should buy _____ because _____. Type below: ___________ Answer: The treat with the highest price per pack is Pup bites. The treat with the lowest price per pack is Nutri-Biscuits. Ryan should buy Nutri-Biscuits because it has the least cost. Question 13. Reason Abstractly What information do you need to consider in order to decide whether one product is a better deal than another? When might the lower unit rate not be the best choice? Explain. Type below: ___________ Answer: We will consider the low cost in order to decide whether one product is a better deal than another. The lower unit rate is not the best choice. Because it will show the highest cost. Question 14. Select the cars that get a higher mileage per gallon of gas than a car that gets 25 miles per gallon. Mark all that apply. Options: a. Car A 22 miles per 1 gallon b. Car B 56 miles per 2 gallons c. Car C 81 miles per 3 gallons d. Car D 51 miles per 3 gallons Answer: b. Car B 56 miles per 2 gallons c. Car C 81 miles per 3 gallons Explanation: 22/1 = 22 56/2 = 28 81/3 = 27 51/3 = 17 ### Find Unit Rates – Page No. 247 Write the rate as a fraction. Then find the unit rate. Question 1. A wheel rotates through 1,800º in 5 revolutions. Type below: ___________ Answer: Explanation: A wheel rotates through 1,800º in 5 revolutions. 1,800º/5 revolutions 1,800º/5 revolutions ÷ 5/5 = 360º/1revolution Question 2. There are 312 cards in 6 decks of playing cards. Type below: ___________ Answer: 52 cards/1 deck of playing cards Explanation: There are 312 cards in 6 decks of playing cards. 312/6 ÷ 6/6 = 52 cards/1 deck of playing cards Question 3. Bana ran 18.6 miles of a marathon in 3 hours. Type below: ___________ Answer: 6.2 miles/hour Explanation: Bana ran 18.6 miles of a marathon in 3 hours. 18.6 miles/ 3 hours ÷ 3/3 = 6.2 miles/hour Question 4. Cameron paid$30.16 for 8 pounds of almonds.
Type below:
___________

$3.77/1 pound Explanation: Cameron paid$30.16 for 8 pounds of almonds.
$30.16/8 pounds ÷ 8/8 =$3.77/1 pound

Compare unit rates.

Question 5.
An online game company offers a package that includes 2 games for $11.98. They also offer a package that includes 5 games for$24.95. Which package is a better deal?
_____ package

5 game package

Explanation:
An online game company offers a package that includes 2 games for $11.98.$11.98/2 = $5.99 They also offer a package that includes 5 games for$24.95.
$24.95/5 =$4.99

Question 6.
At a track meet, Samma finished the 200-meter race in 25.98 seconds. Tom finished the 100-meter race in 12.54 seconds. Which runner ran at a faster average rate?
___________

Tom

Explanation:
At a track meet, Samma finished the 200-meter race in 25.98 seconds.
200/25.98 seconds = 7.698 – meter/1 sec
Tom finished the 100-meter race in 12.54 seconds.
100 – meter/12.54 seconds = 7.974 – meter/1 sec

Problem Solving

Question 7.
Sylvio’s flight is scheduled to travel 1,792 miles in 3.5 hours. At what average rate will the plane have to travel to complete the trip on time?
Type below:
___________

512 miles per hour

Explanation:
Sylvio’s flight is scheduled to travel 1,792 miles in 3.5 hours.
1,792 miles/3.5 hours ÷ 3.5/3.5 = 512 miles per hour

Question 8.
Rachel bought 2 pounds of apples and 3 pounds of peaches for a total of $10.45. The apples and peaches cost the same amount per pound. What was the unit rate? Type below: ___________ Answer:$2.09 per pound

Explanation:
Rachel bought 2 pounds of apples and 3 pounds of peaches for a total of $10.45. The apples and peaches cost the same amount per pound. 2 + 3 = 5$10.45/5 = $2.09 per pound Question 9. Write a word problem that involves comparing unit rates. Type below: ___________ Answer: At a track meet, Samma finished the 200-meter race in 25.98 seconds. Tom finished the 100-meter race in 12.54 seconds. Which runner ran at a faster average rate? At a track meet, Samma finished the 200-meter race in 25.98 seconds. 200/25.98 seconds = 7.698 – meter/1 sec Tom finished the 100-meter race in 12.54 seconds. 100 – meter/12.54 seconds = 7.974 – meter/1 sec Tom ### Lesson Check – Page No. 248 Question 1. Cran–Soy trail mix costs$2.99 for 5 ounces, Raisin–Nuts mix costs $3.41 for 7 ounces, Lots of Cashews mix costs$7.04 for 8 ounces, and Nuts for You mix costs $2.40 for 6 ounces. List the trail mix brands in order from the least expensive to the most expensive. Type below: ___________ Answer: Nuts for You, Raisin–Nuts, Cran–Soy trail mix, Lots of Cashews mix Explanation: Cran–Soy trail mix costs$2.99 for 5 ounces,
$2.99/5 =$0.598
Raisin–Nuts mix costs $3.41 for 7 ounces,$3.41/7 = $0.487 Lots of Cashews mix costs$7.04 for 8 ounces,
$7.04/8 =$0.88
and Nuts for You mix costs $2.40 for 6 ounces.$2.40/6 = $0.4 Question 2. Aaron’s heart beats 166 times in 120 seconds. Callie’s heart beats 88 times in 60 seconds. Emma’s heart beats 48 times in 30 seconds. Galen’s heart beats 22 times in 15 seconds. Which two students’ heart rates are equivalent? Type below: ___________ Answer: Callie and Galen Explanation: Aaron’s heart beats 166 times in 120 seconds. 166/120 = 1.3833 Callie’s heart beats 88 times in 60 seconds. 88/60 = 1.4666 Emma’s heart beats 48 times in 30 seconds. 48/30 = 1.6 Galen’s heart beats 22 times in 15 seconds. 22/15 = 1.4666 Spiral Review Question 3. Courtlynn combines $$\frac{7}{8}$$ cup sour cream with $$\frac{1}{2}$$ cup cream cheese. She then divides the mixture between 2 bowls. How much mixture does Courtlynn put in each bowl? $$\frac{□}{□}$$ cup Answer: $$\frac{11}{16}$$ cup Explanation: Courtlynn combines $$\frac{7}{8}$$ cup sour cream with $$\frac{1}{2}$$ cup cream cheese. 7/8 + 1/2 = 11/8 11/8 ÷ 2 = 11/8 × 1/2 = 11/16 cup Question 4. Write a comparison using < or > to show the relationship between |-$$\frac{2}{3}$$| and – $$\frac{5}{6}$$. Type below: ___________ Answer: > Explanation: |-$$\frac{2}{3}$$| = 2/3 = 0.666 – $$\frac{5}{6}$$ = -0.8333 |-$$\frac{2}{3}$$| > – $$\frac{5}{6}$$ Question 5. There are 18 tires on one truck. How many tires are on 3 trucks of the same type? _____ tires Answer: 54 tires Explanation: There are 18 tires on one truck. For 3 trucks, (3 × 18)/1 = 54 tires Question 6. Write two ratios that are equivalent to $$\frac{5}{6}$$. Type below: ___________ Answer: $$\frac{5}{6}$$ = $$\frac{10}{12}$$, $$\frac{15}{18}$$ Explanation: 5/6 × 2/2 = 10/12 5/6 × 3/3 = 15/18 ### Share and Show – Page No. 251 Use a unit rate to find the unknown value. Question 1. $$\frac{10}{?}=\frac{6}{3}$$ _____ Answer: 5 Explanation: 6/3 ÷ 3/3 = 2/1 2/1 × 5/5 = 10/1 The unknown value is 5 Question 2. $$\frac{6}{8}=\frac{?}{20}$$ _____ Answer: 15 Explanation: 6/8 ÷ 8/8 = 0.75/1 0.75/1 × 20/20 = 15/20 The unknown value is 15 On Your Own Use a unit rate to find the unknown value. Question 3. $$\frac{40}{8}=\frac{45}{?}$$ _____ Answer: 9 Explanation: 40/8 ÷ 8/8 = 5/1 5/1 × 9/9 = 45/9 The unknown value is 9 Question 4. $$\frac{42}{14}=\frac{?}{5}$$ _____ Answer: 15 Explanation: 42/14 ÷ 14/14 = 3/1 3/1 × 5/5 = 15/5 The unknown value is 15 Question 5. $$\frac{?}{2}=\frac{56}{8}$$ _____ Answer: 14 Explanation: 56/8 ÷ 8/8 = 7/1 7/1 × 2/2 = 14/2 The unknown value is 14 Question 6. $$\frac{?}{4}=\frac{26}{13}$$ _____ Answer: 8 Explanation: 26/13 ÷ 13/13 = 2/1 2/1 × 4/4 = 8/4 The unknown value is 8 Practice: Copy and Solve Draw a bar model to find the unknown value. Question 7. $$\frac{4}{32}=\frac{9}{?}$$ _____ Answer: Explanation: 4/32 ÷ 32/32 = 0.125/1 0.125/1 × 72/72 = 9/72 The unknown value is 72 Question 8. $$\frac{9}{3}=\frac{?}{4}$$ _____ Answer: 12 Explanation: 9/3 ÷ 3/3 = 3/1 3/1 × 4/4 = 12/4 The unknown value is 12 Question 9. $$\frac{?}{14}=\frac{9}{7}$$ _____ Answer: Explanation: 9/7 ÷ 7/7 = 1.2857/1 1.2857/1 × 14/14 = 18/14 The unknown value is 18 Question 10. $$\frac{3}{?}=\frac{2}{1.25}$$ _____ Answer: 1.875 Explanation: 2/1.25 ÷ 1.25/1.25 = 1.6/1 1.6/1 × 1.875/1.875 = 3/1.875 The unknown value is 1.875 Question 11. Communicate Explain how to find an unknown value in a ratio by using a unit rate. Type below: ___________ Answer: Firstly, Identify the known ratio, where both values are known. Then, Identify the ratio with one known value and one unknown value. Next, Use the two ratios to create a proportion. Finally, Cross-multiply to solve the problem. Question 12. Savannah is tiling her kitchen floor. She bought 8 cases of tile for$192. She realizes she bought too much tile and returns 2 unopened cases to the store. What was her final cost for tile?
$_____ Answer:$144

Explanation:
If Peri earned $36, ($36 × 3)/$27 = 4 ### Use Unit Rates – Page No. 253 Use a unit rate to find the unknown value. Question 1. $$\frac{34}{7}=\frac{?}{7}$$ _____ Answer: 34 Explanation: 34/7 ÷ 7/7 = 4.8571/1 4.8571/1 × 7/7 = 34 The unknown value is 34 Question 2. $$\frac{16}{32}=\frac{?}{14}$$ _____ Answer: 7 Explanation: 16/32 ÷ 32/32 = 0.5/1 0.5/1 × 14/14 = 7/1 The unknown value is 7 Question 3. $$\frac{18}{?}=\frac{21}{7}$$ _____ Answer: 6 Explanation: 21/7 ÷ 7/7 = 3/1 3/1 × 6/6 = 18/6 The unknown value is 6 Question 4. $$\frac{?}{16}=\frac{3}{12}$$ _____ Answer: 4 Explanation: 3/12 ÷ 12/12 = 0.25/1 0.25/1 × 16/16 = 4 The unknown value is 4 Draw a bar model to find the unknown value. Question 5. $$\frac{15}{45}=\frac{6}{?}$$ _____ Answer: 18 Explanation: 15/45 ÷ 45/45 = 1/3 1/3 × 6/6 = 6/18 The unknown value is 18 Question 6. $$\frac{3}{6}=\frac{?}{7}$$ _____ Answer: 3.5 Explanation: 3/6 ÷ 6/6 = 1/2 1/2 × 3.5/3.5 = 3.5/7 The unknown value is 3.5 Problem Solving Question 7. To stay properly hydrated, a person should drink 32 fluid ounces of water for every 60 minutes of exercise. How much water should Damon drink if he rides his bike for 135 minutes? _____ fluid ounces Answer: 72 fluid ounces Explanation: To stay properly hydrated, a person should drink 32 fluid ounces of water for every 60 minutes of exercise. If he rides his bike for 135 minutes, (135 × 32)/60 = 72 Question 8. Lillianne made 6 out of every 10 baskets she attempted during basketball practice. If she attempted to make 25 baskets, how many did she make? _____ baskets Answer: 15 baskets Explanation: Lillianne made 6 out of every 10 baskets she attempted during basketball practice. If she attempted to make 25 baskets, (25 × 6)/10 = 15 baskets Question 9. Give some examples of real-life situations in which you could use unit rates to solve an equivalent ratio problem. Type below: ___________ Answer: 1) If a 10-ounce box of cereal costs$3 and a 20-ounce box of cereal costs $5, the 20 ounce box is the better value because each ounce of cereal is cheaper. 2) Yoda Soda is the intergalactic party drink that will have all your friends saying, “Mmmmmm, good this is!” You are throwing a party, and you need 555 liters of Yoda Soda for every 121212 guests. If you have 363636 guests, how many liters of Yoda Soda do you need? ### Lesson Check – Page No. 254 Question 1. Randi’s school requires that there are 2 adult chaperones for every 18 students when the students go on a field trip to the museum. If there are 99 students going to the museum, how many adult chaperones are needed? _____ chaperones Answer: 11 chaperones Explanation: Randi’s school requires that there are 2 adult chaperones for every 18 students when the students go on a field trip to the museum. If there are 99 students going to the museum, (99 × 2)/18 = 11 chaperones Question 2. Landry’s neighbor pledged$5.00 for every 2 miles he swims in a charity swim-a-thon. If Landry swims 3 miles, how much money will his neighbor donate?
$_____ Answer:$7.5

Explanation:
Landry’s neighbor pledged $5.00 for every 2 miles he swims in a charity swim-a-thon. If Landry swims 3 miles, 15/2 =$7.5

Spiral Review

Question 3.
Describe a situation that could be represented by –8.
Type below:
___________

In Alaska the normal temperature in December was 3 degrees. Scientist predicted that by February the temperature would drop 11 degrees. What is the predicted temperature for February? The answer is -8.

Question 4.
What are the coordinates of point G?

Type below:
___________

(-2, 0.5)

Explanation:
The x-coordinate is -2
The y-coordinate is 0.5

Question 5.
Gina bought 6 containers of yogurt for $4. How many containers of yogurt could Gina buy for$12?
_____ containers

18 containers

Explanation:
Gina bought 6 containers of yogurt for $4. For$12, ($12 × 6)/$4 = 18

Question 6.
A bottle containing 64 fluid ounces of juice costs $3.84. What is the unit rate?$ _____

$0.06 Explanation: A bottle containing 64 fluid ounces of juice costs$3.84.
$3.84/64 =$0.06

### Share and Show – Page No. 257

A redwood tree grew at a rate of 4 feet per year. Use this information for 1–3.

Question 1.
Complete the table of equivalent ratios for the first 5 years.

Type below:
___________

Explanation:
A redwood tree grew at a rate of 4 feet per year.
For 2 years, 2 × 4 = 8ft
For 3 years, 3 × 4 = 12ft
For 4 years, 4 × 4 = 16ft
For 5 years, 5 × 4 = 20ft

Question 2.
Write ordered pairs, letting the x-coordinate represent time in years and the y-coordinate represent height in feet.
Type below:
___________

(1, 4), (2, 8), (3, 12), (4, 16), (5, 20)

Question 3.
Use the ordered pairs to graph the tree’s growth over time.
Type below:
___________

The graph shows the rate at which Luis’s car uses gas, in miles per gallon. Use the graph for 4–8.

Question 4.
Complete the table of equivalent ratios.
Type below:
___________

30/1, 60/2, 90/3, 120/4, 150/5

Question 5.
Find the car’s unit rate of gas usage.
Type below:
___________

30mi/gal

Question 6.
How far can the car go on 5 gallons of gas?
_____ miles

150 miles

Explanation:
the car go on 5 gallons of gas, 150/5

Question 7.
Estimate the amount of gas needed to travel 50 miles.
Type below:
___________

5/3

Explanation:
30/1,
50/30 = 5/3

Question 8.
Ellen’s car averages 35 miles per gallon of gas. If you used equivalent ratios to graph her car’s gas usage, how would the graph differ from the graph of Luis’s car’s gas usage?
Type below:
___________

The distance is high for Ellen’s car’s gas usage compared to Luis’s car’s gas usage per one gal

Explanation:
35/1 × 2/2 = 70/2
35/1 × 3/3 = 105/3
35/1 × 4/4 = 140/4
35/1 × 5/5 = 175/5

### Problem Solving + Applications – Page No. 258

Question 9.
Look for Structure The graph shows the depth of a submarine over time. Use equivalent ratios to find the number of minutes it will take the submarine to descend 1,600 feet.

_____ minutes

8 minutes

Explanation:
200/1 × 8/8 = 1600/8

Question 10.
The graph shows the distance that a plane flying at a steady rate travels over time. Use equivalent ratios to find how far the plane travels in 13 minutes.

_____ miles

91 miles

Explanation:
7/1 × 13/13 = 91/13

Question 11.
Sense or Nonsense? Emilio types at a rate of 84 words per minute. He claims that he can type a 500-word essay in 5 minutes. Is Emilio’s claim sense or nonsense? Use a graph to help explain your answer.
Type below:
___________

He said that he can write 84 in 60sec ,500 words will be written in 500×60/84=357 it’s a nonsense

Question 12.
The Tuckers drive at a rate of 20 miles per hour through the mountains. Use the ordered pairs to graph the distance traveled over time.

Type below:
___________

### Equivalent Ratios and Graphs – Page No. 259

Christie makes bracelets. She uses 8 charms for each bracelet. Use this information for 1–3.

Question 1.
Complete the table of equivalent ratios for the first 5 bracelets.
Type below:
___________

Explanation:

Question 2.
Write ordered pairs, letting the x-coordinate represent the number of bracelets and the y-coordinate represent the number of charms.
Type below:
___________

(1, 8), (2, 16), (3, 24), (4, 32), (5, 40)

Question 3.
Use the ordered pairs to graph the charms and bracelets.
Type below:
___________

The graph shows the number of granola bars that are in various numbers of boxes of Crunch N Go. Use the graph for 4–5.

Question 4.
Complete the table of equivalent ratios.

Type below:
___________

Question 5.
Find the unit rate of granola bars per box.
Type below:
___________

10 bars/1 box

Problem Solving

Question 6.
Look at the graph for Christie’s Bracelets. How many charms are needed for 7 bracelets?
_____ charms

56 charms

Question 7.
Look at the graph for Crunch N Go Granola Bars. Stefan needs to buy 90 granola bars. How many boxes must he buy?
_____ boxes

9 boxes

Question 8.
Choose a real-life example of a unit rate. Draw a graph of the unit rate. Then explain how another person could use the graph to find the unit rate.
Type below:
___________

Sam prepares 4 bracelets per month. How many bracelets does she prepare in a span of 6 months?
For 1 month, 1 × 4 = 4 bracelets
For 2 months, 2 × 4 = 8 bracelets
For 3 months, 3 × 4 = 12 bracelets
For 4 months, 4 × 4 = 16 bracelets
For 5 months, 5 × 4 = 20 bracelets

### Lesson Check – Page No. 260

Question 1.
A graph shows the distance a car traveled over time. The x-axis represents time in hours, and the y-axis represents distance in miles. The graph contains the point (3, 165). What does this point represent?
Type below:
___________

Explanation:
In 3 hours the car traveled 165 miles.
(3,165) is (x,y) so 3 = x and 165 = y, and
3=x=time in hours
165=y= miles…. soooo
In 3 hours the car traveled 165 miles

Question 2.
Maura charges $11 per hour to babysit. She makes a graph comparing the amount she charges (the y-coordinate) to the time she babysits (the x-coordinate). Which ordered pair shown is NOT on the graph? (4, 44) (11, 1) (1, 11) (11, 12) Type below: ___________ Answer: (11, 1) Explanation: It is not 11,1 because she charges 11 hours per hour (y coordinate) and x would be time to babysit. so it can’t be 11,1 Spiral Review Question 3. List 0, –4, and 3 from least to greatest. Type below: ___________ Answer: -4, 0, 3 Question 4. What two numbers can be used in place of the ? to make the statement true? |?| = $$\frac{8}{9}$$ Type below: ___________ Answer: –$$\frac{8}{9}$$, $$\frac{8}{9}$$ Explanation: |-$$\frac{8}{9}$$| = $$\frac{8}{9}$$ |$$\frac{8}{9}$$| = $$\frac{8}{9}$$ Question 5. Morgan plots the point (4, –7) on a coordinate plane. If she reflects the point across the y-axis, what are the coordinates of the reflected point? Type below: ___________ Answer: (-4, -7) Explanation: Morgan plots the point (4, –7) on a coordinate plane. If she reflects the point across the y-axis, it will be (-4, -7) Question 6. Jonathan drove 220 miles in 4 hours. Assuming he drives at the same rate, how far will he travel in 7 hours? _____ miles Answer: 385 miles Explanation: Jonathan drove 220 miles in 4 hours. If he travel in 7 hours, (7 × 220)/4 = 385 miles ### Chapter 4 Review/Test – Page No. 261 Question 1. Kendra has 4 necklaces, 7 bracelets, and 5 rings. Draw a model to show the ratio that compares rings to bracelets Type below:___________ Answer: Question 2. There are 3 girls and 2 boys taking swimming lessons. Write the ratio that compares the girls taking swimming lessons to the total number of students taking swimming lessons. Type below: ___________ Answer: 3 : 5 Explanation: There are 3 girls and 2 boys taking swimming lessons. the total number of students taking swimming lessons = 5 3 : 5 Question 3. Luis adds 3 strawberries for every 2 blueberries in his fruit smoothie. Draw a model to show the ratio that compares strawberries to blueberries. Type below: ___________ Answer: Question 4. Write the ratio 3 to 10 in two different ways. Type below: ___________ Answer: 3/10, 3 : 10 Question 5. Alex takes 3 steps every 5 feet he walks. As Alex continues walking, he takes more steps and walks a longer distance. Complete the table by writing two equivalent ratios. Type below: ___________ Answer: Explanation: Alex takes 3 steps every 5 feet he walks. As Alex continues walking, he takes more steps and walks a longer distance. 3/5 × 2/2 = 6/10 3/5 × 3/3 = 9/15 ### Page No. 262 Question 6. Sam has 3 green apples and 4 red apples. Select the ratios that compare the number of red apples to the total number of apples. Mark all that apply. Options: a. 4 to 7 b. 3 to 7 c. 4 : 7 d. 4 : 3 e. $$\frac{3}{7}$$ f. $$\frac{4}{7}$$ Answer: a. 4 to 7 c. 4 : 7 f. $$\frac{4}{7}$$ Explanation: Sam has 3 green apples and 4 red apples. the total number of apples = 3 + 4 = 7 4 : 7 Question 7. Jeff ran 2 miles in 12 minutes. Ju Chan ran 3 miles in 18 minutes. Did Jeff and Ju Chan run the same number of miles per minute? Complete the tables of equivalent ratios to support your answer. Type below: ___________ Answer: Explanation: 2/12 × 2/2 = 4/24 2/12 × 3/3 = 6/39 2/12 × 4/4 = 8/48 3/18 × 2/2 = 6/36 3/18 × 3/3 = 9/24 3/18 × 4/4 = 12/72 Question 8. Jen bought 2 notebooks for$10. Write the rate as a fraction. Then find the unit rate.
Type below:
___________

$10/2 unit rate =$5

Explanation:
Jen bought 2 notebooks for $10.$10/2 ÷ 2/2 = $5 ### Page No. 263 Question 9. Determine whether each ratio is equivalent to $$\frac{1}{2}, \frac{2}{3}, \text { or } \frac{4}{7}$$. Write the ratio in the correct box. Type below: ___________ Answer: Explanation: 1/2 × 2/2 = 4/8 7/14 ÷ 2/2 = 1/2 20/35 ÷ 5/5 = 4/7 40/80 ÷ 40/40 = 1/2 8/14 ÷ 2/2 = 4/7 4/6 ÷ 2/2 = 2/3 8/12 ÷ 4/4 = 2/3 Question 10. Amos bought 5 cantaloupes for$8. How many cantaloupes can he buy for $24? Show your work. _____ cantaloupes Answer: 15 cantaloupes Explanation: Amos bought 5 cantaloupes for$8.
For $24, ($24 × 5)/$8 = 15 Question 11. Camille said $$\frac{4}{5}$$ is equivalent to $$\frac{24}{30}$$. Check her work by making a table of equivalent ratios. Type below: ___________ Answer: Question 12. A box of oat cereal costs$3.90 for 15 ounces. A box of rice cereal costs $3.30 for 11 ounces. Which box of cereal costs less per ounce? Use numbers and words to explain your answer. Type below: ___________ Answer: A box of oat cereal costs$3.90 for 15 ounces.
$3.90/15 =$0.26
For $42, (42 × 5)/35 = 6 Question 14. Use a unit rate to find the unknown value. Type below: ___________ Answer: Explanation: (9 × 42)/14 = 3 Question 15. Jenna saves$3 for every $13 she earns. Vanessa saves$6 for every $16 she earns. Is Jenna’s ratio of money saved to money earned equivalent to Vanessa’s ratio of money saved to money earned? Type below: ___________ Answer: No, 3/13 = 6/26. Vanessa ratio is 6/16 Question 16. The Hendersons are on their way to a national park. They are traveling at a rate of 40 miles per hour. Use the ordered pairs to graph the distance traveled over time Type below: ___________ Answer: ### Page No. 265 Question 17. Abby goes to the pool to swim laps. The graph shows how far Abby swam over time. Use equivalent ratios to find how far Abby swam in 7 minutes _____ meters Answer: 350 meters Explanation: 50/1 × 7/7 = 350/7 Question 18. Caleb bought 6 packs of pencils for$12.
Part A
How much will he pay for 9 packs of pencils? Use numbers and words to explain your answer
$_____ Answer:$18

Explanation:
Caleb bought 6 packs of pencils for $12. 6/12 = 1/2 × 9/9 = 9/18 So,$18 is the answer

Question 18.
Part B
Describe how to use a bar model to solve the problem.
Type below:
___________

Take the known ratio and identify the unknown value using known ratio.

### Page No. 266

Question 19.
A rabbit runs 35 miles per hour. Select the animals who run at a faster unit rate per hour than the rabbit. Mark all that apply.
Options:
a. Reindeer: 100 miles in 2 hours
b. Ostrich: 80 miles in 2 hours
c. Zebra: 90 miles in 3 hours
d. Squirrel: 36 miles in 3 hours

a. Reindeer: 100 miles in 2 hours
b. Ostrich: 80 miles in 2 hours

Explanation:
A rabbit runs 35 miles per hour.
35/1
100/2 = 50/1
80/2 = 40/1
90/3 = 30/1
36/3 = 12/1

Question 20.
Water is filling a bathtub at a rate of 3 gallons per minute.
Part A
Complete the table of equivalent ratios for the first five minutes of the bathtub filling up.

Type below:
___________

Question 20.
Part B
Emily said there will be 36 gallons of water in the bathtub after 12 minutes. Explain how Emily could have found her answer
Type below:
___________


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# ALSoundDetection ¶

NAOqi Audio - Overview | API

## What it does ¶

The ALSoundDetection module detects significative sounds in incoming audio buffers. This detection is based on the audio signal level and as such behaves similarly on any type of sound (provided it is loud enough).

## How it works ¶

The processing is made on the front microphone signal. The raw signal is first smoothed by calculating the signal mean on a moving window.

A peak based detection is then applied on this averaged signal to provide the user (through ALMemory in  SoundDetected  ) with the index of the detected sound start.

The end of that sound is eventually detected when the level of the averaged signal comes back to its original value (before the peak detection).

The  SoundDetected  key shows from 1 to n sounds detected in the last audio buffer:

[[index_1, type_1, confidence_1, time_1],
...,
[index_n, type_n, confidence_n, time_n]]

Where :

• index is the indexes of starting and ending points of a detected sound.
• type contains 1 for starting point and 0 for ending point.
• confidence gives an estimate of the probability [0;1] that the sound detected by the module corresponds to a real sound.
• time is the detection time in micro seconds.

## Getting started ¶

Using Choregraphe, the  SoundDetected  key can be retrieved and used (like any others) as a stimulating input to any other box by right clicking on diagrams left border: Add input from ALMemory .

format: plain


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# Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve

20 Aug 2020 Kezuka Yukako

Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $\mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-3})$... (read more)

PDF Abstract

# Code Add Remove Mark official

No code implementations yet. Submit your code now

# Categories

• NUMBER THEORY


text:
# Exponential distribution question

1. May 25, 2014

### oneamp

Hi. I notice that some values of X on the exponential distribution PDF have a value of around 1. I understand the integral ends up being one, since those values of X are less than 1. But P(X) at those points still gets to 1, or thereabouts. How does that make sense, that the probability of a value, say 0.00001, is about 1, and the others complete the integral to 1?

I hope this question makes sense.

Thank you.

2. May 25, 2014

### FactChecker

Do you mean the probability of 0<X< 0.00001 is almost 1 or that the PDF at x=0.0001 is about 1?

In the first case, the PDF values must be very large for the integral on the interval 0<X<0.00001 to be nearly 1. Then, yes, there is very little probability of higher values of X.

In the second case the integral over 0<X<0.00001 is about 0.00001. That leaves a lot of probability (0.9999) for higher values of X.

3. May 26, 2014

### oneamp

4. May 26, 2014

### homeomorphic

Imagine a rectangle that is very tall. Say it's 10, 000 meters tall. Does that mean it has a big area?

Well, what if its width is only 0.0000000000000000000000000000000000000000000000001 meters?

Obviously, its area is extremely small, even though it's very tall. If you multiply that tiny number by 10, 000 to get the area, it's still ridiculously small. For essentially the same reason, it's quite possible for a graph to be much taller than 1 for a while, while still having a total area of 1.

5. May 26, 2014

### oneamp

Thank you

6. May 26, 2014

### FactChecker

Remember that those are probability density graphs, not probability graphs. t
The densities are per unit of X. so they can get very large for a short interval of X values. The probabilities themselves never get over 1.

7. May 26, 2014

Thank you


text:
# Homework Help: DSP - Differential Equation for linear system H

1. Dec 30, 2011

### SMOF

Hello.

I need to find a differential equation for a linear system, H, which is characterised by the impulse response h(t) = e-2t, where t denotes time.

But I am not sure how to go about this. Would anyone be able to point me in the correct direction?

Seán

2. Jan 3, 2012

### rude man

1. Transform your impulse equation into the frequency (Laplace) domain: h(t) --> H(s)

2. Inverse-transform the resultant H(s) transfer function back to the time domain for your diff. eq.

Hint: H(s) = Y(s)/X(s) where X is transform of time input x(t) and Y is transform of output y(t).

3. Jan 3, 2012

### SMOF

Hello,

I will have a go at this in the morning, and I will let you know how I get on.

Thanks a lot. I just wasn't sure how to tackle it.

Yours
Seán

4. Jan 4, 2012

### SMOF

Hello again.

Can I just make sure I have this right?

I got the Laplace transfer of h(t), which is H(s) = 1/(s+a).

So, now I need to get the transform of the input, which is

0.5 cos(ω2 - ω1)t) + 0.5 cos(ω2 + ω1)t)

But I am really not sure about the Laplace transform of this. I was thinking it was ...0.5 s/(s2 - ω2) ...but I don't think that is right because in the time domain it is ω2 - ω1, and I am not too sure how to get this in the frequency domain.

Seán

5. Jan 4, 2012

### MisterX

You're supposed to describe system behavior for any x(t) input. There should be no specific expression for "transform of the input" for this problem besides "X(s)".

The goal is to have and equation for the system relating the output y and its derivatives to the input x and its derivatives.

6. Jan 4, 2012

### SMOF

Hello,

I am afriad I am getting lost on this one.

So, if I can ask, once I have the transform of the impulse, where do I go from there?

I thought I needed to get the transform of the input to the system, and from that, get an equation for the output by relating the input to the system H.

Seán

7. Jan 4, 2012

### MisterX

Do you know what is the transform of the derivative of a function in terms of that function's transform?

8. Jan 4, 2012

### SMOF

I'm sorry, I don't.

Seán

9. Jan 4, 2012

### MisterX

equation 11 is the general rule for the nth derivative. For the first derivative, this would be as follows.

$x(t) \stackrel{\mathcal{L}}{\leftrightarrow} X(s)$

$\frac{dx(t)}{dt} \stackrel{\mathcal{L}}{\leftrightarrow} sX(s) - x(0)$

10. Jan 4, 2012

### SMOF

Hey,

Okay, yes ...I get that ....but (and I am very sorry if this sounds rude, I don't mean it to be) I don't understand how this helps me.

I am more used to the hands on stuff ...building different things ....so, the maths behind it all is very new to me.

But from what you said earlier, that I need to relate the input to the output ...I don't actually know what the output of the system is ...or, at least I don't know if I know it.

If I knew specific steps to take, I might be able to figure it out. Kind of what like Rude Man suggested, except it seem to me like the inverse transform from step one would just take me back to where I was at the start.

Again, sorry for not getting this ...maybe I have just been trying to figure it out for a bit, I just have a bit of a mental block.

Seán

11. Jan 4, 2012

### SMOF

Oh, I think I may have got a bit further on ...I will try it and get back either way :)

Seán

12. Jan 4, 2012

### rude man

The input in your case is an impulse = δ(t). That's what's meant by 'impulse response'.
y(t) is your output = exp(-2t), and Y(s) = 1/(s+2) as you have correctly ascertained.

So the question now is, I know what my response is if the input is δ(t), so what is the response to a general input x(t)? If the input is x(t), what is the output y(t)?

Well, you got the transform of the output right, but you blew it big-time on the input.
What does your Laplace table say for X(s) if x(t) = δ(t)?

Once you have the transform of δ(t) right = X(s), you can then say Y(s)/X(s) = transfer function of your system = H(s).

Do that next.

13. Jan 4, 2012

### SMOF

Hello,

Well, for the table δ(t) = 1. So X(s) = 1.

If 1/(s+2) = Y(s), then H(s) = (1/(s+2))/1 ...?

14. Jan 4, 2012

### rude man

δ(t) is not 1, δ(t) = δ(t). But L{δ(t)} = 1, that's right. And H(s) is exactly what you said. So write an equation with X(s), Y(s) and H(s)?

15. Jan 4, 2012

### SMOF

Do you mean something like Y(s) = H(s)X(s)?

16. Jan 4, 2012

### rude man

Yes I do. You know what H(s) is now, so substitute in your equation.

17. Jan 4, 2012

### SMOF

Okay, well, if

X(s) = 1
Y(s) = 1/(s+2)
H(s) = (1/(s+2))/1

I get

$\frac{1}{s+2}$ = $\frac{1/s+2}{1}$$\times$1

Which is the same as

$e^{-2t}$ = $\frac{e^{-2t}}{1}$$\times$1

I think.

18. Jan 4, 2012

### MisterX

No, X(s) and Y(s) represent arbitrary functions, constrained only by the transfer function Y(s)/X(s). You don't get to decide this X(s) is equal to 1!

Think of Y(s)/X(s), as describing a relationship between the output in the input in general. The impulse response h(t) and corresponding Y(s)/X(s) characterize the response of a linear system to any input.

When X(s) = 1 (a transformed impulse), Y(s) becomes the transformed impulse response, of course. But the solution you seek is a differential equation for the system with any input, and so it is not proper to have X(s) equal to 1.

Last edited: Jan 4, 2012
19. Jan 5, 2012

### rude man

You have determined what H(s) is. You did that by knowing that if x(t) = δ(t), so that
X(s) = 1, so since H(s) = Y(s)/X(s), and Y(s) you have determined to be 1/(s+2) if X(s) = 1, now you know Y(s)/X(s) for any x(t) which, when transformed is X(s). So, stay with your equation: Y(s) = H(s)X(s). That is the key to coming up with your sought-after differential equation.

The next step is transforming that algebraic equation in the frequency domain (never mind for now what I mean by that) into a differential equation in the time domain.

Clue: use cross-multiplication in your equation!

20. Jan 5, 2012

### SMOF

Hello.

I get what you mean about needing to find a solution for any input. It is finally sinking in.

So, since I need to find a silution for any x(t), which is X(s), so would a better form of the equation to work with be

X(s) = Y(s)/H(s)?

I thought to go from the frequency domain (Laplace) to the time domain, I just needed to use the Laplace tables in reverse basically. I take it that is not the case?

And if I stick with Y(s) = H(s)X(s) ....how do I cross multiply this? Anytime I have cross multiplied something it has been in the form of (a/b) x (c/d).

Seán


text:
##### Actions

An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [1]. Let $A$ be an associative ring with unit element and let $\mathop{\rm GL} ( n , A )$ be the group of invertible $( n \times n )$- matrices over $A$. There are natural imbeddings

$$\mathop{\rm GL} ( 1, A) \subset \dots \subset \mathop{\rm GL} ( n , A) \subset \dots ;$$

$g \in \mathop{\rm GL} ( n, A)$ goes to

$$\left ( let  \mathop{\rm GL} ( A) = \cup _ {i=} 1 ^ \infty \mathop{\rm GL} ( i, A) . A matrix differing from the identity matrix in a single non-diagonal entry is called an [[elementary matrix]]. The subgroup  E( A) \subset \mathop{\rm GL} ( A)  generated by all elementary matrices coincides with the commutator group of  \mathop{\rm GL} ( A) . The commutator quotient group  K _ {1} A = \mathop{\rm GL} ( A) / E( A)  is called the Whitehead group of the ring  A . Let  [- 1] \in K _ {1} A  be the element corresponding to the matrix$$ \left \|

It has order 2. The quotient group $\overline{K}\; _ {1} ( A) = K _ {1} A/ \{ 0, [- 1] \}$ is called the reduced Whitehead group of the ring $A$.

Let $\Pi$ be a multiplicative group and let $\mathbf Z [ \Pi ]$ be its group ring over $\mathbf Z$. There is a natural homomorphism $j: \Pi \rightarrow \overline{K}\; _ {1} \mathbf Z [ \Pi ]$ coming from the inclusion of $\Pi \subset \mathop{\rm GL} ( 1, \mathbf Z [ \Pi ])$. The quotient group $\mathop{\rm Wh} ( \Pi ) = \overline{K}\; _ {1} \mathbf Z [ \Pi ] / j ( \Pi )$ is called the Whitehead group of the group $\Pi$.

Given a homomorphism of groups $f : \Pi _ {1} \rightarrow \Pi _ {2}$, there is a natural induced homomorphism $\mathop{\rm Wh} ( f ) : \mathop{\rm Wh} ( \Pi _ {1} ) \rightarrow \mathop{\rm Wh} ( \Pi _ {2} )$, such that $\mathop{\rm Wh} ( g \circ f ) = \mathop{\rm Wh} ( g) \circ \mathop{\rm Wh} ( f )$ for $g : \Pi _ {2} \rightarrow \Pi _ {3}$. Thus $\mathop{\rm Wh}$ is a covariant functor from the category of groups into the category of Abelian groups. If $f : \Pi \rightarrow \Pi$ is an inner automorphism, then $\mathop{\rm Wh} ( f ) = \mathop{\rm id} _ { \mathop{\rm Wh} ( \Pi ) }$.

The Whitehead group of the fundamental group of a space is independent of the choice of a base point and is essential for the definition of an important invariant of mappings, the Whitehead torsion.

#### References

 [1] J.H.C. Whitehead, "Simple homotopy types" Amer. J. Math. , 72 (1950) pp. 1–57 [2] J.W. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426 [3] J.W. Milnor, "Introduction to algebraic -theory" , Princeton Univ. Press (1971)

If $A$ is commutative, the determinant and, hence, the special linear groups $\mathop{\rm SL} ( n, A)$ are defined. Using these instead of the $\mathop{\rm GL} ( n, A)$ one obtains the special Whitehead group $SK _ {1} ( A)$. One has $K _ {1} ( A) = U( A) \oplus SK _ {1} ( A)$ where $U( A)$ is the group of units of $A$.


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# Tag Info

0

tl dr; Putting it all together Little background first Anki uses latex compiler to create .dvi intermediate documents from the input LaTeX code, which are then converted to .png bitmap image using dvipng converter. Anki is only able to work with .png picture format! Now, TikZ is (in vast majority of cases) only able to be compiled with pdflatex compiler, ...

1

It's a problem due to how the conversion works. The file is actually compiled twice, but one time essentially without any package and so when the aux file is read in this phase, the command is not defined. Workaround: \providecommand{\pgfsyspdfmark}[3]{} \documentclass[convert={convertexe={convert}}]{standalone} \usepackage[utf8]{inputenc} \usepackage{...

1

As I'm sure you know when you have used chemfig already the bonds have optional arguments. The manual says: 6 Customization of bonds There is a fifth and last optional argument for bonds which is found after the fourth comma: [,,,,<tikz code>] This means all you have to do is adding the TikZ code for not-drawing: \chemfig{-C-[,,,,draw=...

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