Deck 9: Ratios, Rates, Proportions, and Percents

Question 1

A driveway ramp that is 3.205 feet high is less steep than another ramp that is 3.98 feet high.

Answer 1

False

Question 2

One recipe for eight people requires 5 cups of flour. Al expects only six people; he wants to cut the recipe so that it will serve six.

A) If Al is thinking additively, how much flour will he use? Explain your answer.
B) If you are thinking multiplicatively, how much flour will he use? Explain your answer.
C) In this situation, which way is better, Al's method or your part B method? Explain.

Answer 2

A) If Al is using additive thinking, 8 - 6 = 2, so he should use 5 - 2 = 3 cups of flour.
B) The recipe calls for 5/8 a cup of flour per person, so for six people, Al should use 6 × (5/8), or 3 3/4, cups of flour.
C) The part B method is better because, to taste the same, recipes should use the same proportions of ingredients. Rates allow that.

Question 3

Ruben and Ofilia are painting the walls of a large lecture hall. They mixed 2 gallons of blue paint with 5 gallons of white paint for a total of 7 gallons of paint. They run out of paint. They estimate that they need 1/2 of a gallon to finish.

A) Find the portion of the room that has been painted. Draw a neatly labeled diagram displaying your solution.
B) If they mix 1/4 of a gallon of blue paint with 1 gallon of white paint, will the paint colors match? Explain your answer briefly.
C) Suggest what amounts of blue paint and white paint would finish the room and match the color, perhaps with a little left over for touch-ups.

Answer 3

A) The diagram should show the walls (probably one rectangle) marked into 15 equal pieces since 7 gallons equals 14 half-gallons. They have painted 14/15 of the room with the 7 gallons.
B) No. The original rate was 2:5, which is the same as 2/5:1. But if 1 gallon of white paint is used with 1/4 gallon of blue paint, the ratio is 1/4:1, or 1:4, which is not equivalent to 2/5:1.
C) We need a ratio a:b, where a:b is equivalent to 2:5 and where a + b = 1/2. Using 1/4 gallon blue paint and 5/8 gallon of white paint would provide the correct ratio, and the sum is slightly more than 1/2 but close enough for this situation.

Question 4

María was asked to draw an isosceles triangle (a triangle with two equal sides) as part of a homework assignment. She had a straight edge to draw straight lines but didn't have a ruler to measure the length of her sides. Thus, all she could do was eyeball the correct lengths. She decided to draw more than one triangle so that she could go to school a little early, get a ruler, measure all of her triangles, and select the "best" isosceles triangle to turn in to the teacher.
Suppose María drew the following triangles:

A) Which triangle should she select to turn in to the teacher? Why?
B) Suppose María drew 20 triangles like the ones shown above but of various sizes. Once she measures the lengths of the sides, how should she go about selecting which triangle is the "best" isosceles triangle?

Question 5

There are four water slides at the Six Flags Atlantis water park. The following are the measurements of the four water slides. Which is the steepest slide, A, B, C, or D?

A) length: 100 feet; height: 81 feet
B) length: 60 feet; height: 45 feet
C) length: 80 feet; height: 70 feet
D) length: 10 feet; height: 7 feet

Question 6

A) Three bags of tea are used with 4 cups of water in one teapot, and 10 bags of tea are used with 12 cups of water in another teapot. Which pot will have the stronger tea, or will the two have the same strength? Explain your answer.
B) Five scoops of coffee are used with 8 cups of water in one coffee pot, while 7 scoops of coffee are used with 10 cups of water in another coffee pot. Which pot will have the stronger coffee? Explain your answer.
C) The two problems above were given to a sixth-grade class. Which one do you think was more difficult, and why?

Question 7

Two painters on a large project want to paint different areas the same color. Painter A mixes 3 quarts of red paint with 2 gallons of white paint, and painter B mixes 5 quarts of the same kind of red paint with 4 gallons of white paint. Painter A says the two mixtures will be the same color, and painter B says his mixture will be redder than painter A's. Explain the thinking of each painter. Which one, if either, is correct? Explain your decision.

Question 8

A friend needs a serious operation and is considering two hospitals that have the following records for the surgery: $\begin{array} { l c c } & \text { Successes } & \text { Failures } \\\text { Hospital A } & 20 & 8 \\\text { Hospital B } & 50 & 18\end{array}$

The friend sees that B has more successes, but A has fewer failures and asks you for help in deciding. What will you say?

Question 9

Terry uses 1 cup of Mr. Spiffy in $1 \frac { 1 } { 2 }$
gallons of water to clean the kitchen floor. What percent of the cleaning solution is Mr. Spiffy? (Note: 16 cups = 1 gallon.)

Question 10

Make a sketch and give an explanation to illustrate this situation:
After a bake sale, there were three identical white cakes left. You and your friend split them, but you took only a quarter as much as your friend took. Label your part M (for "me") and your friend's part F. How much of a cake (or how many cakes) did your friend take? (Give the numerical answer that makes sense here and add an explanation if the numerical answer is not clear from your sketch.)

Question 11

Make a sketch and give an explanation to illustrate this situation:
Karen and Sara both jog. Sara jogged 2 1/2 miles, which is 2/3 as far as Karen jogged. How many miles did Karen jog?

Question 12

If the Browns were to save an additional $14,000, they would have 1 2/5 times as much money as the amount the Joneses have in their savings account. The Joneses have $65,000. How much do the Browns currently have in their savings account?

Question 13

A group of 37 people goes to a holiday camp for 35 days. They need to buy enough sugar for the trip. They read that the average consumption of sugar is 2.2 kilograms per week for 10 people. How much sugar do they need?

Question 14

A pollster noticed that, for every 40 men who were in favor of X, there were 28 women who were in favor of X. According to these figures, if 280 men were in favor of X, how many women were in favor of X?

A) 400
B) 196
C) 168
D) 40

Question 15

A machine can make 700 bolts in 40 minutes. At that rate, how many bolts can the machine make in one hour?

A) 900
B) 1050
C) 2800
D) 28,000

Question 16

Miguel runs 200 meters in 40 seconds; Paul runs 150 meters in half a minute. Who runs faster?

A) Miguel
B) Paul
C) They run at the same speed.
D) More information is needed.

Question 17

John drives 7 3/10 miles to campus each day, while Vaneta drives only 4 miles to campus. John drives how many times as far as Vaneta?

Question 18

Your answer book says that the probability of drawing a B ball from box 1 (below) is the same as for box 2.
Box 1 BBW
Box 2 BBBBBBWWW

A child says, "They say 2 to 1 gives the same probability as 6 to 3 does, but I don't see it. I think box 2 has a lot more chances of giving a B ball."
How would you try to convince the child that 2:1 is indeed equal to 6:3, using the balls in the boxes (not just symbolically with numbers)?

Question 19

A child says that the two situations below would give the same "chocolateyness" since "each way has one more spoonful of chocolate sprinkles."
Situation 1 3 spoonfuls of chocolate sprinkles on 2 scoops of vanilla ice cream
Situation 2 4 spoonfuls of chocolate sprinkles on 3 scoops of vanilla ice cream
Make a drawing (not just calculations) that would help the child see that the "chocolateyness" in the two situations would be different. (Use words as needed.)

Question 20

A store sells 48 pecan pies for every 15 banana pies it sells. At this rate, if the store sells 60 banana pies, how many pecan pies will it sell?

A) 12
B) 93
C) 120
D) 192

Question 21

A pollster noticed that, for every 8 men who were in favor of a ballot measure, there were 5 women who were in favor of the measure. According to these figures, if 1200 women were in favor of the measure, how many men were in favor of the measure?

A) 750
B) 1197
C) 1920
D) 2100

Question 22

Clarissa runs 240 meters in 45 seconds; Doña runs 160 meters in half a minute. Who runs faster?

A) Clarissa
B) Doña
C) They run at the same speed.
D) More information is needed.

Question 23

Use a three-column table to solve this problem. Label the columns Picture Frames, Cost, and Notes.
If a box of 36 picture frames (all the same kind) costs $86.40, how much would five frames cost? (Assume no sale price, discount for volume, sales tax, etc.)

Question 24

Felicia can run a mile in 4.7 minutes. Why is it not reasonable to treat a question about the time it takes her to run 8 miles as if it were a problem solved by proportions?

Question 25

Fifteen percent of a given amount is the same as 10% of the amount plus 10% of twice the amount.

Question 26

What does "the sales tax rate is 7 3/4%" mean?

Question 27

You receive 12 points out of 20 on the first quiz, 15 points out of 20 on the second quiz, and 60 points out of 75 on the first midterm.

A) On which quiz or test did you perform best? Show your work.
B) How would you find your average score?
C) Estimate what percent that average represents.

Question 28

Kathy owed her dad $80 and then paid off $24 from her tip earnings.

A) What percent of the original debt had she paid?
B) What percent was still owed?
C) What was her new debt after the payment?
The following evening, she paid him an additional $20 from her tip earnings.
D) What percent of her new debt was paid off?
E) What was her debt after the last payment?
F) What percent of her original debt is now paid off?

Question 29

If this box represents 75% of something, illustrate 125% of the same thing.

Question 30

The median cost of housing rose 25% in one city in a particular year. The median price was then $360,000. What was the median cost of housing at the beginning of the year?

Question 31

Gala apples are on sale today at $1.30 per pound. This is a discount of 30%. What was the cost before the sale?

Question 32

The NASDAQ closed at 1590 today, down by 0.09%. What was the NASDAQ number at yesterday's closing?

Question 33

Estimate each of the following and explain how you arrived at your estimate.

A) 79% of $119
B) 0.5% of 89 kilograms
C) 31% of 21,343 voters
D) 121% of $29