Ratio and proportion
Simplifying ratios
A necklace with 2 red beads and 6 yellow ones has a ratio of 2 : 6. But for every ONE red bead there are three yellow — so 1 : 3 says the same thing more simply.
What a ratio says
A RATIO compares two amounts. If a necklace has 2 red beads and 6 yellow beads, you write red : yellow = 2 : 6 and read it as 'two to six'. The order matters. 2 : 6 and 6 : 2 describe completely different necklaces.
Simplest form
To write a ratio in its SIMPLEST FORM, divide both numbers by their HIGHEST COMMON FACTOR. For 2 : 6 the highest common factor is 2: 2 ÷ 2 : 6 ÷ 2 = 1 : 3 For 8 : 12 the highest common factor is 4: 8 ÷ 4 : 12 ÷ 4 = 2 : 3 Divide by the HIGHEST common factor and you get there in one step. Divide by a smaller one and you will just have to simplify again.
No decimals, no fractions
A ratio in its simplest form must not contain a decimal or a fraction. Zara simplified 4 : 6 to 2 : 3 and stopped. She was right to stop — 1 : 1.5 also compares the same amounts, but a ratio in simplest form has to be whole numbers. So 8 : 12 is 2 : 3, not 1 : 1.5.
Ratios with units
A model car is 6 cm high and the real car is 160 cm high. Write them in the order asked for — model : real — and, because the units match, drop them: 6 : 160 The highest common factor of 6 and 160 is 2, so 3 : 80 If the units DON'T match, change one first: 4 cm : 12 cm works, but 4 cm : 12 m does not until you convert.
Comparing two ratios
Bryn mixes white and blue paint 1 : 2. Alun mixes his 2 : 3. Whose blue is darker? Turn each into a fraction of the whole. Bryn has 3 parts in all, 2 of them blue: 2/3. Alun has 5 parts, 3 blue: 3/5. 2/3 = 10/15 and 3/5 = 9/15, so Bryn's paint has the higher proportion of blue — his is darker.
Worked examples
The necklace: 2 red, 6 yellow
- Write it in the order asked: red : yellow = 2 : 6.
- The highest common factor of 2 and 6 is 2.
- Divide both by 2.
Answer: 1 : 3
Simplifying 8 : 12
- The highest common factor of 8 and 12 is 4.
- 8 ÷ 4 = 2 and 12 ÷ 4 = 3.
Answer: 2 : 3
Model car 6 cm, real car 160 cm
- Order first: model : real = 6 : 160.
- Units match, so drop them.
- The highest common factor of 6 and 160 is 2.
Answer: 3 : 80
Why 8 : 12 is not 1 : 1.5
- 1 : 1.5 does compare the same amounts.
- But simplest form must not contain a decimal or fraction.
Answer: 2 : 3 is the simplest form
The Eiffel Tower and Pierre's model
- The tower is 324 m, the model is 3 m.
- Order asked for is real : model = 324 : 3.
- The highest common factor is 3.
Answer: 108 : 1
Which paint is darker: 1 : 2 or 2 : 3?
- 1 : 2 is 3 parts in all, 2 blue → 2/3 blue.
- 2 : 3 is 5 parts in all, 3 blue → 3/5 blue.
- 2/3 = 10/15 and 3/5 = 9/15.
Answer: The 1 : 2 mix is darker
Where you see this in real life
- Builders mix sand and cement in a ratio to make concrete. Get the ratio wrong and the concrete is too weak.
- A recipe that serves four is really a set of ratios. Keeping them fixed is what lets you cook for eight.
Key words
- Ratio
- a comparison of two amounts, written a : b
- Simplest form
- a ratio divided down as far as it will go, using whole numbers
- Highest common factor
- the largest number that divides into both numbers
- Simplify
- divide both parts of a ratio by a common factor
- Proportion
- how big one part is compared with the whole