Proportional Reasoning with Rates — Print
A rate compares two different kinds of quantity. Turn it into a unit rate and you can compare anything.
What the practice questions cover
These questions use 1 idea from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Better buy — The option with the lower unit price.
Two jars of honey sit on the shelf. The small one is 400 g for $3.20. The big one is 750 g for $5.40.
Which is the better buy? The big jar costs more — but you also get more of it.
Comparing the prices tells you nothing. Comparing the weights tells you nothing either.
So make them comparable. $3.20 for 400 g is $0.008 per gram. $5.40 for 750 g is $0.0072 per gram.
Now they can be compared, because both numbers answer the same question: what does one gram cost?
That is a unit rate — a rate with a 1 on the bottom. Almost every "which is better?" question becomes easy the moment you divide down to one.
What a rate really says
A ratio compares two amounts. When the two amounts have different units, we call it a rate.
If a car travels 150 miles in 3 hours, the rate is 150 miles3 hours. That fraction is the rate. It is true, but it is hard to compare to other cars.
So we shrink the bottom to 1. Divide both the top and the bottom by 3: 150 ÷ 33 ÷ 3 = 501. The car goes 50 miles every 1 hour. That is the unit rate.
The one move
To find a unit rate, divide the top number by the bottom number. Nothing else. ab → a ÷ b per one.
Why unit rates win
You cannot compare $7.50 for 3 pounds against $11.60 for 4 pounds just by looking. The numbers are all different sizes.
But 7.50 ÷ 3 = $2.50 per pound, and 11.60 ÷ 4 = $2.90 per pound. Now it is obvious. Unit rates put two deals on the same footing.
Take it a step at a time
Ask yourself out loud: "per one what?" That question tells you which number goes on the bottom. Cost per pound → pounds on the bottom. Miles per hour → hours on the bottom.
Rates with fractions
Sometimes the numbers themselves are fractions: a snail crawls 12 foot in 14 hour. The rule does not change — still divide.
12 ÷ 14 = 12 × 41 = 2. The snail moves 2 feet per hour.
Dividing by a fraction smaller than 1 makes the answer bigger. That is not a mistake — it is because a quarter hour is a short time, so in a whole hour the snail gets much further.
The double number line
Draw two parallel lines: one for each quantity. Line them up so matching values sit above each other. Now you can step along by equal jumps and read off any value you need.
This is the picture behind every rate problem. If 4 tickets cost $18, mark 4 under $18, then halve both to get 2 tickets for $9, then halve again for 1 ticket at $4.50.
Worked examples
Better buy
A 12-ounce box of cereal costs $3.96. An 18-ounce box costs $5.58. Which is the better buy?
- Decide what "per one" means. We want cost per ounce, so ounces go on the bottom.
- Small box: 3.96 ÷ 12 = 0.33 33 cents per ounce.
- Large box: 5.58 ÷ 18 = 0.31 31 cents per ounce.
- Compare: 0.31 < 0.33 The large box costs less for each ounce.
The 18-ounce box is the better buy, by 2 cents per ounce.
Rate with fractions
Maya paints 34 of a wall in 23 of an hour. How much can she paint in one hour?
- Write the rate as a division: 34 ÷ 23 Wall per hour, so hours go on the bottom.
- Flip and multiply: 34 × 32 Dividing by 23 is the same as multiplying by 32.
- 3 × 34 × 2 = 98 Multiply straight across.
- 98 = 118 More than one wall — makes sense, since 23 hour is less than an hour.
118 walls per hour.
Working backwards from a unit rate
A printer runs at 14 pages per minute. How long to print 196 pages?
- The unit rate is 14 pages per 1 minute. Every minute adds 14 pages.
- Set up: 14 × (minutes) = 196 Pages per minute times minutes gives total pages.
- 196 ÷ 14 = 14 Undo the multiplication by dividing.
14 minutes.
Words to know
- Ratio
- A comparison of two quantities, like 3 cups flour to 2 cups water.
- Rate
- A ratio of two different units, like miles per hour or dollars per pound.
- Unit rate
- A rate with denominator 1 — how much for exactly one.
- Constant of proportionality
- The unit rate, written k, in y = kx.
Mistakes to watch for
Dividing the wrong way round. "Miles per hour" means miles ÷ hours, because you want to know how many miles fit into one hour. The word after "per" always goes on the bottom.
Comparing totals instead of unit rates. A bigger box costing more does not make it a worse deal — it just makes it bigger. Only the price per gram, per sheet or per litre can settle it.
Forgetting that dividing by a number less than 1 makes the answer larger. Half a tank in a quarter of an hour is 2 tanks per hour, not 18 — dividing by 14 multiplies by 4.
Answering with the rate when the question wanted the total. "$2.50 per pound" is not the cost of the shopping. Read the question again once you have the rate.
Unit pricing on supermarket shelf labels was forced on shops by law, not offered by them.
The US state of Massachusetts required it in 1971 after studies showed shoppers could not reliably pick the cheaper option without it.
The law was doing exactly what this unit does: dividing everything down to one, so the numbers can be compared at a glance.
Problem 1
A car travels 250 miles in 5 hours. Find the speed in miles per hour.
Worked answer
50
250 ÷ 5 = 50
The car covers 50 miles every hour.
Problem 2
Six pounds of apples cost $12.60. What is the cost per pound?
Worked answer
2.10
12.60 ÷ 6 = 2.10
Each pound costs $2.10.
Problem 3
Leo types 480 words in 8 minutes. How many words per minute?
Worked answer
60
480 ÷ 8 = 60
60 words each minute.
Problem 4
Five notebooks cost $7.25. What does one notebook cost?
Worked answer
1.45
7.25 ÷ 5 = 1.45
One notebook costs $1.45.
Problem 5
A car drives 360 miles using 12 gallons of gas. Find the miles per gallon.
Worked answer
30
360 ÷ 12 = 30
30 miles for each gallon.
Problem 6
A 15-ounce jar costs $4.35. A 24-ounce jar costs $6.72. Which is the better buy?
- The 15-ounce jar
- The 24-ounce jar
- They cost the same per ounce
Worked answer
The 24-ounce jar
15-ounce: 4.35 ÷ 15 = 0.29 per ounce
24-ounce: 6.72 ÷ 24 = 0.28 per ounce
0.28 < 0.29, so the larger jar is cheaper per ounce.
Problem 7
A snail crawls 34 mile in 12 hour. What is its speed in miles per hour?
Worked answer
1.5
34 ÷ 12 = 34 × 21
= 64 = 32 = 1.5
In a full hour it travels 1.5 miles.
Problem 8
Nina paints 23 of a room in 35 of an hour. How many rooms per hour?
Worked answer
10/9
23 ÷ 35 = 23 × 53
= 109
A little more than one room per hour.
Problem 9
A printer runs at 22 pages per minute. How many minutes to print 341 pages?
Worked answer
15.5
341 ÷ 22 = 15.5
It takes 15.5 minutes.
Problem 10
A recipe uses 34 cup of sugar for 2 dozen cookies. How much sugar per dozen?
Worked answer
3/8
34 ÷ 2 = 34 × 12
= 38 cup per dozen.
Problem 11
A heart beats 72 times a minute. How many beats in 1 hour and 30 minutes?
Worked answer
6480
1 hr 30 min = 90 minutes
72 × 90 = 6480 beats
Problem 12
A car travels at 45 miles per hour. How far does it go in 2 hours and 20 minutes?
Worked answer
105
Time = 213 = 73 hours
45 × 73 = 15 × 7 = 105 miles
Problem 13
A faucet fills 58 of a tank in 15 minutes. How many minutes to fill the whole tank?
Worked answer
24
One eighth takes 15 ÷ 5 = 3 minutes
Eight eighths take 3 × 8 = 24 minutes
Or: 15 ÷ 58 = 15 × 85 = 24.
Problem 14
A shop sells pens 3 for $4.80 or 5 for $7.75. How many cents cheaper per pen is the better deal?
Worked answer
5
4.80 ÷ 3 = $1.60 each
7.75 ÷ 5 = $1.55 each
1.60 - 1.55 = $0.05, which is 5 cents.
Problem 15
Kai drives 60 miles at 30 mph, then 60 miles at 60 mph. What is his average speed for the whole trip?
Worked answer
40
First leg: 60 ÷ 30 = 2 hours
Second leg: 60 ÷ 60 = 1 hour
Total: 120 miles in 3 hours
120 ÷ 3 = 40 mph — less than 45, because he spent longer going slowly.
Problem 16
A machine makes 7 parts every 4 minutes. How many parts in 3 hours?
Worked answer
315
3 hours = 180 minutes
180 ÷ 4 = 45 blocks
45 × 7 = 315 parts