Ratio and proportion, Year 9: scale factors and multipliers

The scale factor

HintMaps and model kits quote one of these, such as 1 : 50.

The whyA scale factor of 3 makes every length three times as long, and a scale factor of 1/2 halves every length. One number does the whole job, whichever length you pick.

2.5

HintAsk what the old width must be multiplied by to give the new one.

The why20 ÷ 8 = 2.5, so every length on the photo has been multiplied by 2.5. The multiplier is always new ÷ old; a height of 6 cm would become 15 cm.

300 g

HintFind the single multiplier that turns 12 into 8, then use it on the flour.

The why8 ÷ 12 = 2/3, so every quantity is multiplied by 2/3, and 450 × 2/3 = 300. A multiplier works on every ingredient at once, with no need to find the amount for one person first.

smaller

HintThink what multiplying a length by one half does to it.

The whyMultiplying by 0.4 means taking four tenths of a length, which is only part of it. A factor above 1 gives a larger copy, and a factor of exactly 1 changes nothing.

20

HintUndo a multiplication with the multiplier that reverses it.

The whyMultiplying by 1/20 and then by 20 gets you back where you started, because 1/20 × 20 = 1. To reverse any scale factor, multiply by its reciprocal.

1.07

HintStart from the whole amount, 100%, and add the extra on before converting to a decimal.

The why100% + 7% = 107%, and 107 ÷ 100 = 1.07. Single-digit percentages are where the slip happens: 7% is 0.07, not 0.7.

£204

HintWork out what percentage is left after the reduction and write it as a decimal.

The why100% − 15% = 85%, so multiply by 0.85: 240 × 0.85 = 204. One multiplication replaces "find 15%, then subtract it".

1.15 and £92

HintKeep the whole of the starting amount and put the extra part on top of it.

The whyThe original counts as 1 and the increase adds 0.15. Then 80 × 1.15 = 80 + 12 = 92, the same as finding 15% and adding it, but in a single step.

A 30% decrease

HintCompare the multiplier with 1: is some of the amount missing, and how much?

The whyMultiplying by 0.7 keeps 70% of the amount, so 30% has gone. Subtract the multiplier from 1 to find a fall; subtract 1 from the multiplier to find a rise.

2 : 3 : 5

HintBlue already has the same number of parts in each ratio.

The whyBlue is 3 parts in both ratios, so they can be joined directly. When the shared quantity already matches, the three-part ratio is just the two ratios laid end to end.

It must be the same number of parts in both ratios

HintAsk whether one share in the first comparison is as big as one share in the second.

The whyIn 2 : 3 and 4 : 5 the 3 and the 4 can describe one quantity, but measured in different-sized parts. Scaling both ratios until that quantity matches puts everything into one common size of part.

8 and 15, giving a : b : c = 8 : 12 : 15

HintWhatever you multiply one side of a ratio by, do the same to the other side.

The whyThe first ratio is multiplied by 4 and the second by 3. Twelve is the lowest common multiple of 3 and 4, the smallest number of parts that b can be in both ratios.

20

HintFirst make Ben's share the same number of parts in both ratios, then count all the parts.

The whyDoubling 3 : 2 gives 6 : 4, which matches Ben's 4 in 4 : 5, so Amy : Ben : Cara = 6 : 4 : 5. That is 15 parts, each worth £4, so Cara's 5 parts are £20.

1 : 6

HintJoin the two ratios through b first, then leave b out.

The whyDoubling 2 : 3 gives 4 : 6, so a : b : c = 1 : 4 : 6. Dropping the middle term leaves a : c = 1 : 6, which cannot be simplified further.