AS DISCUSSED IN CLASS, THERE IS USUALLY MORE THAN ONE WAY TO SOLVE A PROBLEM. THE RESPONSES BELOW ALL HAVE THE RIGHT SOLUTION BUT YOU COULD HAVE USED AN ALTERNATE METHOD OF SOLVING.

Problem # 1

A train travels 107.5 miles in 5 hours.

  1. How many miles per hour does the train go?

Proportional reasoning helps with this problem.

5x = 107.5

X = 21.5 The train goes 21.5 miles per hour.

  1. How far can the train go in 7 hours? Explain.

5x = 752.5

X = 150.5 The train can go 150.5 miles in 7 hours.

Problem # 2

Which box of Cheerios gives you the best price per ounce? Explain.

Small / Medium / Large / Family
8 oz. for $1.09 / 16 oz. for $2.10 / 30 oz. for $3.99 / 60 oz. for $7.75

Using ratios—specifically unit rate—helps with this problem. Since each box is a different size, we’d expect them to all be different costs. If we used a unit rate for each one, we could then compare them all by the price of 1 ounce.

Small / Medium / Large / Family
= $0.13625 per oz. / = $0.13125 per oz. / = $0.133 per oz. / = $0.12916 per oz.

“Family” size gives you the best price per ounce because it is the lowest unit rate of dollars per ounce.

Problem # 3

Danielle wants to paint her daughter’s room pink. She has been told there are two paint mixtures to make pink paint.

Mixture A: Use 2 parts Red and

5 parts white

Mixture B: Use 1 part Red and

4 parts white

Which Mixture will give Danielle the paint that is least red? Explain how you made your decision.

This problem should remind you of the orange juice concentrate problem. A part-to-part ratio or a part-to-whole ratio will help you compare the two mixtures to determine which will be least red.

Part-to-part :

Mix A : Mix B :

Since the ratio of red to white in Mix B is smaller than Mix A, Mix B will be least red.

Part-to-whole :

Mix A : Mix B :

Again, the part-to-whole ratio in Mix B is smaller, making Mix B the least red.

Problem # 4

The values in the table below are proportional. How many miles can you get with 7 gallons of gas?

Gallons / 2 / 5 / 7 / 12
Miles / 46 / 115 / x / 276

Explain your reasoning.

You can go 161 miles on 7 gallons of gas. The unit rate for the situation is 46/2 = 23 miles per gallon, so 23 x 7 = 161 miles. Also, you could have set up a proportion of and picked any coordinate for one side of the proportion and for the other side.

Write an equation relating miles, m to gallons, g.

m = 23g ; the dependent variable (miles) should always go by itself in the equation

Problem # 5

What is the unit rate?

20 km/hr ; find 1 on the x-axis, go up until you hit the line, and then go over to the left to see where you are on the y-axis. That is your unit rate.

Is the relationship between distance and time proportional? How do you know?

The relationship is proportional. I know this because it is a straight line through the origin.

According to the graph, how far can you go in 4.5 hours?

You can go 90 miles in 4.5 hours. (going up from 4.5 to the line, then over to the left to get 90)

Problem # 6

A giant tortoise travels at a steady rate of 15cm every 2 seconds.

a) Complete the table

Time(sec) / 1 / 2 / 3 / 5 / 18
Distance (cm) / 7.5 / 15 / 22.5 / 37.5 / 140

a)What is the unit rate?

7.5 cm per second

b)Is the relationship between time and distance, proportional? Explain.

Yes, the relationship is proportional because there is a constant of proportionality (7.5). Also, all of the ratios are equivalent.

c)What is the constant of proportionality?

The constant of proportionality is the same as the unit rate. It is 7.5.

d)Write an equation to represent the tortoise’s movement.

T= timeD= distanceD = 7.5t

Problem # 7

Hamburgers / 3 / 5 / 6 / 10
Cost ($) / $4.50 / $7.50 / $8 / $15

Frank says the relationship between hamburgers and their cost is proportional. His sister Betty disagrees. She says that the relationship is not proportional. Who is correct? Justify your answer.

Betty is correct, Frank is incorrect. The relationship cannot be proportional because there is no constant of proportionality. All of the ratios have a constant of proportionality of 1.5 however the 6 hamburgers for $8 does not. 8/6 = 1.3333, which is not 1.50.

Problem # 8

a) Ethan writes 1/6 pages in 1/12 minutes. How much time does it take him to write one full page?

Cross simplifying or using scale factor, x=

It takes Ethan ½ a minute to write 1 page.

b) Samuel used 1/5 of an ounce of butter to make 1/15 of a pound of jelly. How many ounces of butter are needed to make one pound of jelly?

= 1/5

X = 3 It takes 3 ounces of butter to make one pound of jelly.