Skip to content

Writing a ratio

Ratio, rates and money · level 3, lesson 1 of 6

A question to start

A bottle of orange squash says on the label: "Mix 1 part squash with 4 parts water."

Mia and Leo read the label. Mia says: "So the drink is one quarter squash." Leo says: "No, it is one fifth squash."

They cannot both be right. Who is?

Here is a hint. Pour 1 cup of squash and 4 cups of water into a jug. How many cups of drink are in the jug?

By the end of this lesson you will know who is right, and you will know the exact mistake the other one made. It is the most common mistake people make with ratios, and grown-ups make it too.

The idea

Two ways to compare

There are two ways to compare two amounts.

Say a necklace has 6 orange beads and 4 blue beads.

  • Comparing by subtracting: there are \(6 - 4 = 2\) more orange beads than blue ones.
  • Comparing by multiplying: for every 3 orange beads there are 2 blue beads.

The first way tells you a difference. The second way tells you a ratio. A ratio describes how the amounts are mixed, and that is what matters for a recipe, a paint colour, or a drink. If you make a bigger necklace with the same pattern, say 30 orange and 20 blue, the difference changes (now 10) but the ratio does not (still 3 orange for every 2 blue). The ratio is the part that stays the same when you make more.

The colon

"3 orange for every 2 blue" has a short way of writing:

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

The dots are called a colon, and you read \(3 : 2\) as "3 to 2". The two numbers are the parts (or terms) of the ratio.

3 orange 2 blue :

You can also write a ratio in words ("3 to 2"), or as "3 for every 2". All mean the same thing.

Order matters

The order of the numbers must match the order of the words. Orange to blue is \(3 : 2\). Blue to orange is \(2 : 3\). These are different ratios: one says "more orange than blue", the other says "more blue than orange". So always write the words first, then the numbers in the same order:

  • cats : dogs \(= 5 : 1\) means 5 cats for every 1 dog;
  • dogs : cats \(= 1 : 5\) means the same thing, written the other way round.

If you see a ratio with no words, you cannot tell what it means. "\(5 : 1\)" on its own could be cats to dogs or dogs to cats. Writing the words is not extra fuss; it is part of the answer.

Part to part

A ratio like orange : blue \(= 3 : 2\) compares one part with another part. Neither number is the whole necklace. This is called a part-to-part ratio.

Part to whole

Sometimes we want to compare one part with the whole thing. In the 3 : 2 pattern, each "group" of the pattern has \(3 + 2 = 5\) beads. So:

  • orange : whole \(= 3 : 5\),
  • blue : whole \(= 2 : 5\).

These are part-to-whole ratios. And a part-to-whole ratio is the same as a fraction: orange beads are \(\tfrac35\) of the necklace, blue beads are \(\tfrac25\).

orange: 3 parts blue: 2 parts whole: 3 + 2 = 5 parts

Here is why the fraction has 5 at the bottom and not 2. A fraction like \(\tfrac35\) means "3 out of every 5". Out of every 5 beads in the pattern, 3 are orange. The bottom number counts everything, so it must include the blue beads too.

The rule to remember:

To go from a part-to-part ratio \(a : b\) to a fraction of the whole, add the parts to get the whole. The first part is \(\dfrac{a}{a + b}\) of the whole, and the second is \(\dfrac{b}{a + b}\).

Now you can settle the squash argument. Squash : water \(= 1 : 4\) is part to part. The whole drink has \(1 + 4 = 5\) parts, so the drink is \(\tfrac15\) squash. Leo is right. Mia treated the part-to-part ratio \(1 : 4\) as if it were the fraction \(\tfrac14\). Her mistake would be correct only if the label said "1 part squash out of 4 parts of drink".

Ratios with more than two parts

A ratio can compare three or more things. A fruit salad with 2 cups of melon, 3 cups of grapes and 1 cup of berries has

\[\text{melon} : \text{grapes} : \text{berries} = 2 : 3 : 1.\]

The whole salad has \(2 + 3 + 1 = 6\) cups, so berries are \(\tfrac16\) of the salad and grapes are \(\tfrac36\), which is half.

A ratio does not tell you how many

Knowing that a bag of marbles has red : green \(= 3 : 2\) does not tell you how many marbles are in the bag. It could be 3 red and 2 green, or 6 and 4, or 300 and 200. The ratio tells you the mix, not the size. To know the actual numbers you need one more fact, such as the total or one of the parts. That is what the next two lessons are about.

Same units

When the two amounts are measured, they must be in the same unit before you write the ratio. A plank 2 m long and a stick 50 cm long are not in the ratio \(2 : 50\). Change the metres to centimetres first: \(200 : 50\). Then the ratio means "200 centimetres for every 50 centimetres", and it is fair to compare. Because both parts have the same unit, the units cancel and a ratio is just numbers.

Worked examples

1. Reading a picture. A tray has 4 muffins and 6 cookies.

  • Muffins to cookies: \(4 : 6\).
  • Cookies to muffins: \(6 : 4\).
  • Muffins to everything: the whole tray has \(4 + 6 = 10\) treats, so \(4 : 10\), which means muffins are \(\tfrac{4}{10}\) of the tray.

(In the next lesson you will see that \(4 : 6\) can be written more simply as \(2 : 3\).)

2. A class. A class has 12 girls and 15 boys.

  • Girls : boys \(= 12 : 15\).
  • The class has \(12 + 15 = 27\) children, so girls are \(\tfrac{12}{27}\) of the class.
  • It is not \(\tfrac{12}{15}\). That fraction compares girls with boys, not girls with the whole class.

3. From a fraction back to a ratio. In a garden, \(\tfrac38\) of the flowers are red and the rest are yellow. What is the ratio of red to yellow?

  • Out of every 8 flowers, 3 are red.
  • So \(8 - 3 = 5\) are yellow.
  • Red : yellow \(= 3 : 5\).

Going this way you subtract the part from the whole to find the other part.

4. Three parts. A paint shop mixes a colour from 5 tins of white, 2 tins of blue and 1 tin of yellow.

  • White : blue : yellow \(= 5 : 2 : 1\).
  • Total: \(5 + 2 + 1 = 8\) tins. Blue is \(\tfrac28\) of the paint, which is \(\tfrac14\).

5. Mixed units. A recipe uses 1 kg of flour and 250 g of butter. Write flour : butter.

  • Change kilograms to grams: 1 kg \(= 1000\) g.
  • Flour : butter \(= 1000 : 250\), which is 4 times as much flour as butter, or \(4 : 1\).

Try it

  1. A box holds 7 pencils and 3 pens. Write the ratio of pencils to pens, and of pens to pencils.
  2. In question 1, what fraction of the box is pens?
  3. A drink is made with juice : water \(= 2 : 3\). What fraction of the drink is juice?
  4. A farm has 5 cows for every 2 horses. Write the ratio of horses to cows.
  5. In a bag of sweets, \(\tfrac27\) are lemon and the rest are cherry. Write lemon : cherry.
  6. A trail mix has nuts : raisins : seeds \(= 4 : 3 : 1\). What fraction of the mix is raisins?
  7. A rope is 3 m long and a string is 75 cm long. Write the ratio of rope to string using the same unit.
  8. Sam says: "In my class the ratio of left-handed to right-handed pupils is \(1 : 6\), so \(\tfrac16\) of the class is left-handed." Explain what is wrong, and give the right fraction.
Answers
  1. Pencils : pens \(= 7 : 3\). Pens : pencils \(= 3 : 7\).
  2. The box has \(7 + 3 = 10\) things, and 3 are pens: \(\tfrac{3}{10}\).
  3. The whole drink has \(2 + 3 = 5\) parts, and 2 are juice: \(\tfrac25\).
  4. Horses : cows \(= 2 : 5\) (the same fact written the other way round).
  5. Out of every 7 sweets, 2 are lemon, so \(7 - 2 = 5\) are cherry. Lemon : cherry \(= 2 : 5\).
  6. Total parts: \(4 + 3 + 1 = 8\). Raisins are \(\tfrac38\) of the mix.
  7. 3 m \(= 300\) cm, so rope : string \(= 300 : 75\). (That is 4 times as long, so it simplifies to \(4 : 1\), as you will see in the next lesson.)
  8. The ratio \(1 : 6\) is part to part: left-handers compared with right-handers. Out of every \(1 + 6 = 7\) pupils, 1 is left-handed, so the right fraction is \(\tfrac17\), not \(\tfrac16\).

Common mistake

Turning a part-to-part ratio straight into a fraction. If boys : girls \(= 2 : 3\), 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. Before you write a fraction, ask yourself: "Is my bottom number the whole thing?"

Writing the numbers in the wrong order. "There are 2 teachers for every 25 children" is teachers : children \(= 2 : 25\), not \(25 : 2\). Write the words, then the numbers under them.

Where it leads