Revision sheet · Unit 12 Ratio and proportion

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12.1 Understanding ratio Number

A ratio compares amounts part with part: 3 : 2 means 3 of one for every 2 of the other.
Order matters. 3 : 2 is not the same as 2 : 3.
To simplify, divide every part by their highest common factor — exactly like a fraction.
To find an equivalent ratio, multiply or divide every part by the same number.
Before writing a ratio of two measurements, put them in the same units: 40 cm : 1 m becomes 40 : 100 = 2 : 5.
A fraction compares part with whole. In 2 : 3 the first share is 2/5 of the whole, not 2/3.
The total number of parts is the sum of the ratio: 2 : 3 has 5 parts.
That confusion between 2/3 and 2/5 is the commonest error in the whole unit.

Key words: ratio · simplest form · equivalent ratio · part · proportion

squash 1 : 4
1 part syrup to 4 parts water
so 50 ml syrup needs 200 ml water
boys : girls = 2 : 3
girls : boys = 3 : 2
same class, different ratio
12 : 18
HCF is 6

12.2 Sharing in a ratio Number

Add the ratio numbers to find how many parts there are altogether.
Divide the total by that to find what one part is worth.
Multiply each ratio number by the value of one part.
Three steps, always in that order: add, divide, multiply.
Check by adding the shares — they must give the total back.
The larger number in the ratio always takes the larger share.
Working backwards: if one share is known, divide it by its ratio number to find one part.
If two shares differ by a known amount, that difference is the difference of the ratio numbers, in parts.

Key words: share · total parts · one part · divide in a ratio

share 20 in the ratio 2 : 3
parts: 2 + 3 = 5
one part: 20 ÷ 5 = 4
shares: 8 and 12
8 + 12 = 20  ✓
if you had 8 and 10, the check fails
share 60 in 1 : 2 : 3
parts: 6,  one part: 10

12.3 Direct proportion Number

Two quantities are in direct proportion when doubling one doubles the other.
Then dividing one by the other always gives the same number — the rate.
The unitary method: find the value of one, then multiply.
It is two steps: divide to get one, multiply to get many.
The graph of two directly proportional quantities is a straight line through the origin.
A taxi fare is not directly proportional to distance — the fixed charge lifts the graph off the origin.
In inverse proportion, doubling one halves the other: twice as many workers, half the time.
Ask “does zero of one give zero of the other?” — it is the quickest test for direct proportion.

Key words: direct proportion · unitary method · rate · scale up · inverse proportion

3 pens cost 12  →  4 each
6 pens cost 24  →  4 each
9 pens cost 36  →  4 each
5 pens cost 35
one pen: 35 ÷ 5 = 7
8 pens: 7 × 8 = 56
litres:  2   4   6
cost:   10  20  30