Fractions, ratio notation and linear functions
One quantity as a fraction of another
The amount after "of" goes on the bottom.
Put both amounts in the same unit first, then simplify.
- write 45 minutes as a fraction of 2 hours
- 2 hours = 120 minutes
- 45/120
- divide both by 15: 3/8
For a percentage, divide and then multiply by 100: 350 g as a percentage of 2.5 kg is 350 ÷ 2500 × 100 = 14%. Writing 350/2.5 compares grams with kilograms, which gives a nonsense answer.
Ratio notation, in the right order
A ratio lists amounts in the order the words name them.
"There are 4 times as many sheep as goats" means for every 1 goat there are 4 sheep, so sheep : goats = 4 : 1 and goats : sheep = 1 : 4.
With different units, change every part to the same unit before you simplify:
- 1.2 m : 80 cm : 600 mm
- in mm: 1200 : 800 : 600
- divide every part by 200: 6 : 4 : 3
Fractions of a group become a ratio once they share a denominator. If 1/4 walk, 1/3 cycle and the rest come by bus, the bus fraction is 1 − 1/4 − 1/3 = 5/12. Over 12 the fractions are 3/12, 4/12 and 5/12, so walk : cycle : bus = 3 : 4 : 5.
A ratio as fractions and as a linear function
In the ratio 3 : 5 there are 8 parts.
The first amount is 3/5 of the second and 3/8 of the whole. These are different questions: check whether you are comparing with the other part or with the total.
If x and y are always in the ratio 3 : 5, then y ÷ x = 5/3 every time, so
- y : x = 5 : 3
- y = 5x/3
- when x = 12, y = 12 × 5 ÷ 3 = 20
That is direct proportion, y = kx, and its graph is a straight line through the origin. Any point on the line gives the ratio: the point (300, 120) gives 300 : 120 = 5 : 2.
Check with the ratio tool
In the ratio tab, simplify 1200 : 800 : 600. Then try 1.2 : 80 : 600 and see how far off you are if you skip changing the units.
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