# Into Math Grade 6 Module 5 Lesson 5 Answer Key Solve Ratio and Rate Problems Using Proportional Reasoning

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## HMH Into Math Grade 6 Module 5 Lesson 5 Answer Key Solve Ratio and Rate Problems Using Proportional Reasoning

I Can find and use equivalent ratios using a table, model, or double number line to solve a real-world problem.

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1. Lemons are sold in bags of 6 lemons for $4. If you bought 24 lemons, how much would you spend? A. Write a ratio of 6 lemons to the cost of 6 lemons in dollars. Answer: Explanation: Lemons are sold in bags of 6 lemons for$4
24 lemons = x
x × 6 = 4 × 24
x × 6 = 96
x = 96/6
x = 16
Thus you spend $16 for 24 lemons. B. What is the unknown value you are looking for? __________________ Answer: We are looking to find the cost of 24 lemons. C. One way to find the unknown number is to use a table. Complete the table using equivalent ratios. Answer: D. Look at the results in the table. What do you notice about the number of bags of lemons, the number of lemons, and the cost? __________________ E. Another way to find the unknown number is to find a factor that generates the equivalent ratio. When both quantities are multiplied by the same number, the result is an equivalent ratio. Find the unknown number using this method. Answer: F. What is the cost of 24 lemons? __________________ Answer: The cost of 24 lemons is$16.

Turn and Talk How could you use the relationship between 6 lemons and $4 to find the relationship between 24 lemons and the cost for 24 lemons? Answer: 6 lemons =$4
24 lemons = x
x × 6 = 4 × 24
x × 6 = 96
x = 96/6
x = 16
Thus you spend $16 for 24 lemons. 2. Jarrah wants to make a batch of slime. The diagram shows the ratio of hot water to cold water needed to make the slime. He wants to use 18 parts of water in total. How many parts of cold water does he need to make the slime? A. Look at the tape diagram. What is the total parts of water shown? _________________________________ Answer: Hot water has 6 parts = 6 × 2 = 12 parts Cold water has 3 parts = 3 × 2 = 6 parts Total parts of water shown are 12 + 6 = 18 B. Write the ratio of total parts of water to parts of cold water. Answer: Total parts of water = 18 Parts of cold water = 6 18/6 = 3 The ratio of total parts of water to parts of cold water is 3 : 1. C. What is the relationship between the quantities in the ratio in Part B? _________________________________ Answer: The relationship between the quantities in the ratio in Part B is 3 parts is equal to 1. D. How can you use the relationship between the quantities in Part B to find the parts of cold water needed for a total of 18 parts of water? _________________________________ _________________________________ E. How many parts of cold water will Jarrah need if he has a total of 18 parts of water? _________________________________ Answer: 6 parts of cold water Jarrah need if he has a total of 18 parts of water. F. If Jarrah wanted to make a larger batch of slime using 36 total parts of water, how many parts of cold water would he need? _________________________________ Answer: 6 × 2 = 12 parts of cold water Jarrah needs 12 parts of cold water to make a larger batch of slime using 36 total parts of water. G. How many parts of hot water will Jarrah need if he uses a total of 18 parts of water? _________________________________ Answer: 12 parts of hot water Jarrah need if he uses a total of 18 parts of water. H. How many parts of hot water will Jarrah need if he uses 36 total parts of water? _________________________________ Answer: 12 × 2 = 24 parts of hot water 24 parts of hot water Jarrah needs if he uses a total of 36 parts of water. Turn ana Talk How can you use the tape diagram to check your answers? Explain. 3. A race car driver completes 5 laps of a race in 3 minutes and 30 seconds. Then the driver continues driving at this rate. How many laps will the driver complete in 17.5 minutes? A. What is the ratio of laps to minutes? Explain. _________________________________ _________________________________ Answer: Given, A racecar driver completes 5 laps of a race in 3 minutes and 30 seconds. Then the driver continues driving at this rate. The ratio of laps to minute is 5: 3.5 B. You can use a double number line to find the unknown quantity. Complete the double number line diagram. Explain how you found the missing numbers of laps and minutes. _________________________________ _________________________________ Answer: As per the ratio seen in part A we can draw the double number line for laps and minutes. C. Another way to find the unknown quantity is to find a factor that generates the equivalent ratio. What number multiplied by 3.5 minutes will result in 17.5 minutes? _____ Multiply both quantities of the first ratio by this factor to find the quantities of the second, equivalent ratio. Answer 5 laps = 3.5 minutes x laps = 17.5 minutes x × 3.5 = 5 × 17.5 x × 3.5 = 87.5 x = 87.5/3.5 x = 25 25 laps in 17.5 mintes D. How many laps will the race car driver complete in 17.5 minutes? _______ Answer: The racecar driver can complete 25 laps in 17.5 minutes. Check Understanding Question 1. A garden center is running a special on houseplants. A selection of any 2 plants costs$7. If a designer buys 22 plants for new homes, how much does the designer spend on plants?
_________________________________
Given,
A garden center is running a special on houseplants.
A selection of any 2 plants costs $7. A designer buys 22 plants for new homes. 22 ÷ 2 = 11 11 ×$7 = $77 Thus the designer spend$22 on plants.

Question 2.
A scale model of the Eiffel Tower uses the scale shown. The Eiffel Tower is 324 meters tall to the tip. What is the height of the model? Given,
Scale model used for the Eiffel Tower :
2 centimeters represents 6 meters
This means that 2 centimeters on the model represent 6 meters of the Eiffel Tower.
To obtain the height of the model in centimeter :
Let the height of the model = h
h of model = actual height of tower
Actual Height of tower = 324 meters
2 = 6
h = 324
6 × h = 324 × 2
6h = 648
h = 648 / 6
h = 108

Question 3.
A hybrid car can drive 53 miles in the city on 1 gallon of gas. How many gallons of gas will it use to drive a total of 371 city miles?
_________________________________
Given,
A hybrid car can drive 53 miles in the city on 1 gallon of gas.
You are trying to find the number of gallons needed for 371 miles you would divide 371 by 53 and you get 7.
371 ÷ 53 = 7 gallons of gas

Question 4.
The table shows the numbers of water bottles and juice boxes sold at two school events. A. What is the ratio of water bottles sold to juice boxes sold at Event 1?
_________________________________
Water bottles sold at event 1 are 24
Juice boxes sold at event 1 are 54.
The ratio of water bottles sold to juice boxes sold at Event 1 is 24:54

B. What is the ratio of water bottles sold to juice boxes sold at Event 2?
_________________________________
Water bottles sold at event 1 are 36
Juice boxes sold at event 1 are 108
The ratio of water bottles sold to juice boxes sold at Event 2 is 36:108

C. Are the ratios of water bottles sold to juice boxes sold equivalent for each event?
_________________________________
Answer: No the ratio of water bottles sold to juice boxes sold equivalent for each event are not equivalent.

Question 5.
Elsa builds her own triangular frames for paintings. She has one frame with a length of 60 centimeters and a height of 45 centimeters. She builds a second frame with an equivalent ratio of length to height. If the length of her second frame is 100 centimeters, what is its height?
Given,
Frame 1:
Length = 60 centimeters
Height = 45 centimeters
Frame 2:
Length = 100 centimeters
Height = x centimeters
Ratio of length to ratio of heights
60 cm : 45 cm = 100 cm : x cm
60 / 45 = 100 / x
cross product
60 × x = 45 × 100
60x = 4500
x = 4500 / 60
x = 75
Therefore, the height of the second frame is 75 centimeters.

Question 6.
A court reporter can type 215 words per minute. How many minutes will it take the reporter to type a document that is 1,505 words long?
_________________________________
Given,
A court reporter can type 215 words per minute.
1505 ÷ 215 = 7
Thus it takes 7 minutes for the reporter to type a document that is 1,505 words long.

Question 7.
The ratio of dahlias to sunflowers is the same in all of the flower arrangements at a banquet.
A. Use Structure Complete the double number line to show the ratios of dahlias to sunflowers in all the flower arrangements at the banquet.   The ratio is 4:3

B. How many dahlias will there be when there are 33 sunflowers? _______
The ratio of dahlias to sunflowers is 4:3
There are 33 sunflowers
Let the number of dahlias is x.
4:3 = x:33
4/3 = x/33
4 × 33 = x × 3
132 = 3x
x = 132/3
x = 44
Thus there are 44 dahlias.

C. What is the ratio of dahlias to all flowers when there are 21 total flowers? ______
Answer: If there are 21 flowers the number of dahlias is 28.

D. What is the ratio of sunflowers to all flowers in a bouquet with 70 total flowers? _____

Question 8.
Nadine shops at three different grocery stores. She uses the ads to determine where to buy certain items. The table shows the cost of a brand of laundry detergent at each store. A. What is the cost per ounce of detergent at each grocery store?
Fresh Grocers: ______
Jim’s Corner Store: _____
City Market: _______
By seeing the table we can see that
Fresh Grocers: $5.76 Jim’s Corner Store:$ 3.38
City Market: $5.94 B. Where should Nadine shop to get the lowest price per ounce? __________________ Answer: 1 pound = 16 ounces First we have to find the cost of detergent per pound. Fresh Grocers:$5.76
48 ounces = 3 pounds
3 pounds = $5.76 1 pound = x 3 × x = 5.76 x = 5.76/3 =$1.92
1 pound = $1.92 Jim’s Corner Store:$ 3.38
26 ounces = $3.38 26 oz = 1.62 pounds 1.62 = 3.38 1 = x x × 1.62 = 3.38 x = 3.38/1.62 = 2.08 1 pound =$2.08
City Market: $5.94 54 oz =$5.94
54 oz = 3.3 pound
3.3 = $5.94 1 = x x × 3.3 = 5.94 x = 5.94/3.3 = 1.8 1 pound =$1.8
City market is the lowest price to buy the detergent powder.

Question 9.
The tape diagram shows the ratio of hamburgers to veggie burgers served at a family reunion. A. What is the ratio of veggie burgers served to hamburgers served?
__________________
Number of hamburgers served = 12
Number of veggie burgers served  = 5
So, the ratio of veggie burgers served to hamburgers served is 5 : 12

B. If 30 veggie burgers were served, how many hamburgers were served?
__________________
The ratio of veggie burgers served to hamburgers served is 5 : 12
5 × 30 : 12 × 30
150 : 360
Thus if 30 veggie burgers were served 360 hamburgers will served.

Question 10.
Reason A sheet of 28 stickers has star and heart stickers. There are 16 heart stickers.

A. How many heart stickers does a pack of 140 stickers have? Write equivalent ratios to show your reasoning.
__________________
A sheet of 28 stickers has star and heart stickers. There are 16 heart stickers.
28 – 16 = 12 star stickers
The ratio is 16 : 12 or 4 : 3
140 = 80 + 60
So, for 140 stickers there will be 80 heart stickers.
80:60 = 4:3

B. How many star stickers does a pack of 140 stickers have? Write equivalent ratios to show your reasoning.
__________________
A sheet of 28 stickers has star and heart stickers. There are 16 heart stickers.
28 – 16 = 12 star stickers
The ratio is 16 : 12 or 4 : 3
140 = 80 + 60
So, for 140 stickers there will be 60 star stickers.
80:60 = 4:3

C. What is another way you could have found the number of star stickers in a pack of 140 stickers?
__________________
4:3 = 80:x
4/3 = 80/x
x × 4 = 80 × 3
x × 4 = 240
x = 240/4
x = 60
So, the number of star stickers in a pack of 140 stickers is 60.

Question 11.
Kyle recorded the results of his basketball free throw practice for three different sessions in the table shown. Any attempt not missed is a free throw he made.

A. Are the ratios of misses to attempts in the table equivalent ratios? Explain how you know. __________________
Yes the ratios are equivalent
3:10 can be written as 3 × 2 : 2 × 10 = 6 : 20
3:10 can be written as 3 × 3 : 3 × 10 = 9 : 30
Thus the table shows the equivalent ratios.

B. If the ratio remains the same, how many free throws would Kyle make if he misses 15 free throws in his next practice session?
__________________
As we can see misses is multiple of 3
The ratio is 3:10
For 15 misses
3 × 5 : 10 × 5 = 15 : 50
50 free throws would Kyle make if he misses 15 free throws in his next practice session.

Question 12.
Geography On a small island, 63 people live on 21 acres of land.

A. If the ratio of people to acres is about the same on each acre, how many people would you expect to live on 1 acre of land?
__________________
Given,
On a small island, 63 people live on 21 acres of land.
63 people = 21 acre
x people = 1 acre
x × 21 = 63
x = 63/21
x = 3
3 people live on 1 acre of land
Thus the ratio of people to the acre is 3 : 1.

B. If another 42 people move to the island and the ratio of people to acres is equivalent to the original ratio, how many acres will the 105 people live on?
__________________
If 63 people live on 21, acres, that means that 63/21 = 3,
so every 1 acre is 3 people.
So we can divide 105/3 and we get 35, which means 105 people will live on 35 acres.

Question 13.
The length-to-width ratios of three rectangles are equivalent. One of the rectangles has a width of 2.5 centimeters and a length of 15 centimeters.
A. The second rectangle has a length of 9 centimeters. What is its width?
__________________
Given,
The length-to-width ratios of three rectangles are equivalent.
One of the rectangles has a width of 2.5 centimeters.
So, if the length is 9 cm then the width is 2.5 cm.

B. The third rectangle has a width of 6 centimeters. What is its length?
__________________
The third rectangle has a width of 6 centimeters.
One of the rectangles has a length of 15 centimeters.
So, the length is 15 cm

C. Reason Write the length and width of two more rectangles whose length-to-width ratios are equivalent to that of the three rectangles. Explain how you found the length and width of each of your rectangles.
__________________
__________________
__________________
__________________

Lesson 5.5 More Practice/Homework

Solve Ratio and Rate Problems Using Proportional Reasoning

For Problems 1—4, use the following information.

Piet Mondrian, a Dutch artist, painted images of rectangles and squares. The image shown is an example of art similar to Mondrian’s art. Question 1.
What is the ratio of orange rectangles to green rectangles?
_____________________
Answer: The ratio of orange rectangles to green rectangles is 2:3.

Question 2.
What is the ratio of green rectangles to white rectangles?
_____________________
Answer: The ratio of green rectangles to white rectangles is 3:2

Question 3.
What is the ratio of orange rectangles to white rectangles?
_____________________
Answer: The ratio of orange rectangles to white rectangles is 2:2 or 1:1

Question 4.
Describe in words two ratios from the image that are equivalent.
_____________________
Answer: The ratio of orange rectangles to white rectangles is equivalent say 2:2 = 1:1

Question 5.
Reason Carmen can make 9 bracelets each day. She needs to make at least 120 bracelets to sell at a craft fair. She estimates she will need more than 13 days to make enough bracelets. Is she correct? Explain.
_____________________
Carmen can make 9 bracelets each day. She needs to make at least 120 bracelets to sell at a craft fair.
120 ÷ 9 = 13.33 > 13 days
Thus she needs more than 13 days to make enough bracelets.

Question 6.
A catering company mixes taco seasoning and sour cream to make a vegetable dip. The table shows the amounts of taco seasoning for various amounts of sour cream. A. Complete the table. B. How much taco seasoning will the catering company use if it makes a dip with 15 cups of sour cream? __________________
The ratio of Taco Seasoning to Sour Cream is 2 : 3.
If it makes a dip with 15 cups of sour cream then taco seasoning is 16.

Question 7.
Math on the Spot On the map, the distance between the trailhead and the campground is 4 inches. What is the actual distance? ________ The ratio of inch to mile is 1 : 3.
1 in = 3 miles
4 in = 4 × 3 = 12 miles
Thus the actual distance is 12 miles.

Question 8.
The speed of an object is the ratio of its distance traveled to time. If the speed of a bicycle is 4.5 meters per second, what is the distance traveled in 9 seconds? __________
Given,
The speed of an object is the ratio of its distance traveled to time.
4.5 meters = 1 sec
x = 9 sec
x × 1 = 4.5 × 9
x = 40.5 meters
Therefore the distance traveled in 9 seconds is 40.5 meters.

Test Prep

Question 9.
The ratio of students who prefer pineapple to students who prefer kiwi is 12 to 5. Which pair of equivalent ratios could be used to find how many students prefer kiwi if there are 357 total students?
A. $$\frac{12}{5}$$ = $$\frac{357}{?}$$
B. $$\frac{5}{12}$$ = $$\frac{?}{357}$$
C. $$\frac{12}{17}$$ = $$\frac{?}{357}$$
D. $$\frac{5}{17}$$ = $$\frac{?}{357}$$
Given,
The ratio of students who prefer pineapple to students who prefer kiwi is 12 to 5.
12:5 = 357
Let the number of students prefer kiwi be x.
That means $$\frac{12}{5}$$ = $$\frac{357}{x}$$
Option A is the correct answer.

Question 10.
The table shows the ratio of boxes of cornbread mix to cornbread muffins. If you have only half a box of mix, how many muffins could you make? 1 cornbread mix = 10 cornbread muffins
The ratio of boxes of cornbread mix to cornbread muffins is 1 : 10.
Half of box of mix = 0.5 or 1/2
1/2 × 10 = 5
Therefore If you have only half a box of mix you could make 5 cornbread muffins.

Question 11.
Jackson answered every 7 out of 9 questions correctly on a Spanish quiz. If the quiz had 63 questions, how many questions did Jackson answer correctly?
Given,
Jackson answered every 7 out of 9 questions correctly on a Spanish quiz.
If the quiz had 63 questions.
9 × 7 = 63
7 × 7 = 49
63 – 49 = 14
Thus Jackson answered 49 correctly.

Question 12.
The double number line shows the cost of different quantities of cheddar cheese at a grocery store. One large wheel of cheddar cheese weighs a total of 15 pounds. How much would it cost to buy one large wheel? The cost of one large wheel with 1 pound is $1.8 The cost of one large wheel with 15 pound is x x × 1 = 1.8 × 15 x = 27 Thus the cost of one large wheel with 15 pound is$27.

Spiral Review

Question 13.
A blue ribbon has a length of 1.694 yards. A red ribbon is 1.228 yards long. How much longer is the blue ribbon than the red ribbon?
_________________________________
Given,
A blue ribbon has a length of 1.694 yards.
A red ribbon is 1.228 yards long.
1.694 – 1.228 = 0.466 yards
The blue ribbon is 0.466 yards longer than the red ribbon.

Question 14.
What is the greatest common factor of 54 and 72? ______  