Proportional Reasoning with Rates

A rate compares two different kinds of quantity. Turn it into a unit rate and you can compare anything.

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.

which jar is the better buy? 400 g $3.20 750 g $5.40 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.

A double number line linking tickets to dollars 01234 $0.00$4.50$9.00$13.50$18.00 tickets cost

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?

  1. Decide what "per one" means. We want cost per ounce, so ounces go on the bottom.
  2. Small box: 3.96 ÷ 12 = 0.33 33 cents per ounce.
  3. Large box: 5.58 ÷ 18 = 0.31 31 cents per ounce.
  4. 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?

  1. Write the rate as a division: 34 ÷ 23 Wall per hour, so hours go on the bottom.
  2. Flip and multiply: 34 × 32 Dividing by 23 is the same as multiplying by 32.
  3. 3 × 34 × 2 = 98 Multiply straight across.
  4. 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?

  1. The unit rate is 14 pages per 1 minute. Every minute adds 14 pages.
  2. Set up: 14 × (minutes) = 196 Pages per minute times minutes gives total pages.
  3. 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.

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