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Explore: Ratios, unit rates and scale

Ratio and rates · step 2 of 5

A question to start

Earth is about 12 742 km across. Mars is about 6 779 km across. If you made a model of Earth the size of a football, 22 cm across, how big should the model of Mars be? Would a tennis ball do, or a marble, or a grape? To answer, you need to compare sizes with a ratio, and then use the same ratio at a different scale. (Answer below: about 11.7 cm, roughly a grapefruit.)

The idea

Writing a ratio

The "for every" of Play has a short way of writing. "2 orange beads for every 3 blue beads" is

\[\text{orange} : \text{blue} = 2 : 3\]

read "2 to 3". The order matters: blue to orange is \(3 : 2\), a different ratio. A ratio can compare more than two things: a mortar mixed from 1 part cement, 3 parts sand and 1 part water is \(1 : 3 : 1\).

Equivalent ratios and simplest form

Multiplying or dividing every part of a ratio by the same number gives an equivalent ratio, one that describes the same mixture:

\[2 : 3 = 4 : 6 = 10 : 15 = 20 : 30.\]

A ratio is in simplest form when the parts are whole numbers with no common factor except 1. To simplify, divide every part by the greatest common factor (the biggest number that divides them all): \(18 : 24\), both divisible by 6, becomes \(3 : 4\). If a ratio has fractions or decimals, multiply up first: \(1.5 : 2 = 3 : 4\).

Parts and wholes

A ratio \(2 : 3\) compares part to part. The whole has \(2 + 3 = 5\) parts, so orange beads are \(\tfrac25\) of the necklace and blue beads are \(\tfrac35\). Fractions compare part to whole. Mixing these up is the most common ratio error of all, so always ask: "Is this a part compared with a part, or a part compared with the whole?"

Sharing in a ratio

To share an amount in the ratio \(a : b\):

  1. Add the parts: \(a + b\).
  2. Find the size of one part: divide the amount by \(a + b\).
  3. Multiply: the shares are \(a\) parts and \(b\) parts.

This is exactly the "rounds" method from Play, written compactly.

Unit rates and the best buy

A rate compares two different kinds of quantity: dollars per kilogram, kilometres per hour, heartbeats per minute. A unit rate is a rate for one unit of the second quantity: "4 dollars per 1 kilogram". To find it, divide. Unit rates let you compare offers fairly.

Scale

A scale is a ratio between a drawing (or model) and the real thing, usually written with 1 first:

\[\text{map} : \text{real} = 1 : 50\,000\]

means 1 cm on the map is 50 000 cm on the ground. Since 1 km = 1000 m = 100 000 cm, that is 1 cm to 0.5 km. The units must be the same on both sides of a scale ratio, which is why the numbers get so big.

  • Map to real: multiply by the scale number.
  • Real to map: divide by it.

Worked examples

1. Sharing. Share 40 dollars between Kim and Lee in the ratio \(3 : 5\).

  • Parts: \(3 + 5 = 8\).
  • One part: \(40 \div 8 = 5\) dollars.
  • Kim: \(3 \times 5 = 15\) dollars. Lee: \(5 \times 5 = 25\) dollars. Check: \(15 + 25 = 40\), and \(15 : 25 = 3 : 5\).

2. The best buy. A pack of 6 juice boxes costs 4.20 dollars. A pack of 10 costs 6.50 dollars. Which is better value?

  • Small pack: \(4.20 \div 6 = 0.70\) dollars per box.
  • Big pack: \(6.50 \div 10 = 0.65\) dollars per box.
  • The big pack is cheaper per box, by 5 cents (as long as you will drink them all).

3. A map. On a map with scale \(1 : 25\,000\), two towers are 6 cm apart. How far apart are they really?

  • Real distance: \(6 \times 25\,000 = 150\,000\) cm.
  • \(150\,000 \text{ cm} = 1500 \text{ m} = 1.5\) km.

4. Earth and Mars. The ratio of diameters is

\[\text{Earth} : \text{Mars} = 12\,742 : 6\,779.\]

Dividing both by 6 779 gives about \(1.88 : 1\): Earth is a little less than twice as wide as Mars. For a 22 cm football Earth, the Mars model is \(22 \div 1.88 \approx 11.7\) cm across. A grape would be far too small; a grapefruit is about right. On this model, how far apart should the two balls be? At their closest, Earth and Mars are about 55 million km apart. That is about \(55\,000\,000 \div 12\,742 \approx 4300\) Earth-widths, so the Mars ball would be \(4300 \times 22\) cm \(\approx 950\) m away, nearly a kilometre. Space is mostly empty.

Try it

  1. Write \(18 : 24\) in simplest form.
  2. Share 56 in the ratio \(3 : 4\).
  3. A map has scale \(1 : 50\,000\). A road on the map is 6 cm long. How long is the real road, in km?
  4. Which is better value: 500 g of cereal for 3.40 dollars, or 750 g for 4.95 dollars?
  5. A class has 12 girls and 15 boys. Write the ratio of girls to boys in simplest form. What fraction of the class are girls?
  6. A recipe uses flour and sugar in the ratio \(3 : 2\). How much sugar goes with 7.5 cups of flour?
Answers
  1. The greatest common factor is 6: \(18 \div 6 = 3\) and \(24 \div 6 = 4\), so \(3 : 4\).
  2. Parts: \(3 + 4 = 7\). One part: \(56 \div 7 = 8\). Shares: \(3 \times 8 = 24\) and \(4 \times 8 = 32\).
  3. \(6 \times 50\,000 = 300\,000\) cm \(= 3000\) m \(= 3\) km.
  4. Compare price per 100 g. Small: \(3.40 \div 5 = 0.68\) dollars. Large: \(4.95 \div 7.5 = 0.66\) dollars. The 750 g box is slightly better value.
  5. \(12 : 15 = 4 : 5\) (dividing by 3). The class has 27 pupils, so girls are \(\dfrac{12}{27} = \dfrac{4}{9}\) of the class. Notice it is \(\tfrac49\), not \(\tfrac45\).
  6. 7.5 cups of flour is \(7.5 \div 3 = 2.5\) "parts", so sugar is \(2 \times 2.5 = 5\) cups.

Common mistake

Confusing a part-to-part ratio with a fraction. If boys to girls is \(2 : 3\), it is tempting to say "\(\tfrac23\) of the class are boys". But the class has \(2 + 3 = 5\) parts, so boys are \(\tfrac25\) of the class. The ratio compares boys with girls; the fraction compares boys with everybody.

Where it leads