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Rates and unit rates

Ratio, rates and money · level 3, lesson 4 of 6

A question to start

At the grocery store there are two boxes of eggs.

  • A box of 12 eggs costs 4.20 dollars.
  • A tray of 30 eggs costs 9.90 dollars.

Which is the better deal? The big tray costs more money, but it also has more eggs. Comparing 4.20 with 9.90 tells you nothing useful, and neither does comparing 12 with 30.

The trick is to ask a smaller question: how much does one egg cost? Once you can answer that for both boxes, the choice is easy. That one-egg price is called a unit rate, and it is one of the most useful ideas in everyday math.

The idea

What is a rate?

A ratio usually compares two amounts of the same kind: cups of lemon with cups of water. A rate compares two amounts of different kinds, measured in different units.

  • 240 kilometres in 3 hours (distance and time)
  • 6 apples for 3 dollars (apples and money)
  • 18 heartbeats in 15 seconds (beats and time)
  • 45 litres of water in 3 minutes (volume and time)

Because the units are different, a rate always keeps its units. You cannot write "240 : 3" and forget what the numbers mean. We say the rate with the word per, which means "for each": kilometres per hour, dollars per kilogram, beats per minute. A slash is short for "per": km/h means kilometres per hour.

Unit rates

A unit rate is a rate for one of the second thing: one hour, one apple, one minute. To find it, divide.

240 km in 3 hours: in one hour the car goes \(240 \div 3 = 80\) km. The unit rate is 80 km per hour.

Why divide? If the car goes at a steady pace, the 240 km is shared equally among the 3 hours, like sharing 240 sweets among 3 friends. Each hour gets 80 km.

A double number line shows a rate nicely. The top line counts kilometres and the bottom line counts hours. Marks that line up go together.

km h 0 80 160 240 320 0 1 2 3 4

Once you know the unit rate, every other mark is easy: multiply. In 4 hours the car goes \(4 \times 80 = 320\) km.

Two unit rates for every rate

"6 apples for 3 dollars" can be turned into a unit rate in two ways.

  • Dollars per apple: \(3 \div 6 = 0.50\) dollars, so 50 cents for each apple. This is the unit price.
  • Apples per dollar: \(6 \div 3 = 2\) apples for each dollar.

Both are correct. Which one you want depends on your question. "How much do 15 apples cost?" needs dollars per apple: \(15 \times 0.50 = 7.50\) dollars. "How many apples can I buy with 5 dollars?" needs apples per dollar: \(5 \times 2 = 10\) apples.

A good habit: say the units out loud. "Dollars per apple, times apples, gives dollars." The apples "cancel", like the same number on the top and bottom of a fraction.

The unitary method

The unitary method means "go through one". Whenever you know a rate for some amount and want it for another amount:

  1. Divide to find the rate for one.
  2. Multiply to get the amount you want.

If 5 notebooks cost 8 dollars, one notebook costs \(8 \div 5 = 1.60\) dollars, and 12 notebooks cost \(12 \times 1.60 = 19.20\) dollars. It always works, even when the numbers do not fit nicely, which is why it is the method to trust.

The best buy

Now the eggs.

  • Box: \(4.20 \div 12 = 0.35\) dollars, or 35 cents per egg.
  • Tray: \(9.90 \div 30 = 0.33\) dollars, or 33 cents per egg.

The tray is 2 cents cheaper per egg, so it is the better buy, as long as you will use all 30 eggs before they go bad. That last part matters. A unit price tells you which is cheaper per egg; it does not tell you whether you need 30 eggs. Wasted food is money too.

Many Canadian grocery shelf labels show a unit price in small print, such as "per 100 g" or "per kg". That is the store doing this division for you. It is still worth checking that both labels use the same unit (one may be per 100 g and another per kg).

Comparing with a common amount

Sometimes it is easier to compare the price of the same amount instead of one unit. A 500 g box of cereal for 3.40 dollars and a 750 g box for 4.95 dollars: 500 and 750 are both multiples of 250, but 100 g is an even friendlier amount.

  • 500 g is 5 lots of 100 g: \(3.40 \div 5 = 0.68\) dollars per 100 g.
  • 750 g is 7.5 lots of 100 g: \(4.95 \div 7.5 = 0.66\) dollars per 100 g.

The 750 g box is a little cheaper for the same amount of cereal.

Not every price is a rate

A unit rate only makes sense when the amount grows steadily: twice as many apples, twice the cost. Some prices do not work like that. A taxi might charge a fixed fee just to get in, then so much per kilometre. A movie ticket does not cost twice as much because the movie is twice as long. Before you use the unitary method, ask: "If I double one thing, does the other really double?"

Worked examples

1. A car trip. A car travels 240 km in 3 hours at a steady speed. How far in 5 hours?

  • Unit rate: \(240 \div 3 = 80\) km per hour.
  • In 5 hours: \(5 \times 80 = 400\) km.

2. Apples. 6 apples cost 3 dollars. How much for 15 apples? How many for 5 dollars?

  • Dollars per apple: \(3 \div 6 = 0.50\). Fifteen apples: \(15 \times 0.50 = 7.50\) dollars.
  • Apples per dollar: \(6 \div 3 = 2\). Five dollars: \(5 \times 2 = 10\) apples.

3. A heartbeat. You count 18 heartbeats in 15 seconds. What is your heart rate in beats per minute?

  • One minute is 60 seconds, which is \(60 \div 15 = 4\) lots of 15 seconds.
  • Beats per minute: \(18 \times 4 = 72\).
  • (Through one second: \(18 \div 15 = 1.2\) beats per second, then \(1.2 \times 60 = 72\). Same answer.)

4. Typing. Kai types 210 words in 5 minutes. How long will 630 words take?

  • Unit rate: \(210 \div 5 = 42\) words per minute.
  • Time: \(630 \div 42 = 15\) minutes.
  • Quick check: 630 is 3 times 210, so it takes 3 times as long, \(3 \times 5 = 15\) minutes. ✓

5. Two pack sizes. Batteries come 4 for 5.96 dollars or 10 for 13.90 dollars. Which is cheaper per battery?

  • 4-pack: \(5.96 \div 4 = 1.49\) dollars each.
  • 10-pack: \(13.90 \div 10 = 1.39\) dollars each.
  • The 10-pack is 10 cents cheaper per battery.

Try it

  1. A train goes 150 km in 2 hours at a steady speed. What is its speed in km per hour?
  2. 8 pens cost 6 dollars. What does one pen cost? What do 20 pens cost?
  3. A tap fills 45 litres in 3 minutes. How long does it take to fill a 120-litre tank?
  4. Rice costs 3.80 dollars for 2 kg. How many kilograms can you buy for 9.50 dollars?
  5. Which is the better buy: 1 litre of juice for 2.29 dollars, or 1.75 litres for 3.85 dollars?
  6. A car uses 7 litres of fuel for every 100 km. How much fuel does it use for 350 km?
  7. Runner A does 3 laps in 12 minutes. Runner B does 5 laps in 21 minutes. Who runs faster?
  8. A bag of 6 oranges costs 2.70 dollars and loose oranges cost 0.50 dollars each. Which is cheaper per orange, and by how much?
Answers
  1. \(150 \div 2 = 75\) km per hour.
  2. One pen: \(6 \div 8 = 0.75\) dollars (75 cents). Twenty pens: \(20 \times 0.75 = 15\) dollars.
  3. Unit rate: \(45 \div 3 = 15\) litres per minute. Time: \(120 \div 15 = 8\) minutes.
  4. Price per kg: \(3.80 \div 2 = 1.90\) dollars. Kilograms: \(9.50 \div 1.90 = 5\) kg.
  5. Small: 2.29 dollars per litre. Big: \(3.85 \div 1.75 = 2.20\) dollars per litre. The 1.75 litre bottle is cheaper per litre, by 9 cents.
  6. 350 km is 3.5 lots of 100 km, so \(3.5 \times 7 = 24.5\) litres.
  7. A: \(12 \div 3 = 4\) minutes per lap. B: \(21 \div 5 = 4.2\) minutes per lap. A takes less time per lap, so A is faster. (In laps per minute: A \(3 \div 12 = 0.25\), B \(5 \div 21 \approx 0.238\). A is still faster.)
  8. Bag: \(2.70 \div 6 = 0.45\) dollars each. The bag is cheaper by \(0.50 - 0.45 = 0.05\) dollars, 5 cents per orange.

Common mistake

Dividing the wrong way round. For "8 pens cost 6 dollars", \(8 \div 6 \approx 1.33\) is pens per dollar, not dollars per pen. Both numbers are real rates, but only one answers "how much does one pen cost?". Say the units: "dollars divided by pens gives dollars per pen", so it is \(6 \div 8\).

Thinking the bigger pack is always cheaper. Often it is, but not always. Shops sometimes put a small size on sale, or price a big size higher per unit. Always work out the unit price.

Where it leads