Ratio and proportion, Year 9: scale factors and multipliers

The scale factor

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

为什么A 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

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

为什么20 ÷ 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

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

为什么8 ÷ 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

提示Think what multiplying a length by one half does to it.

为什么Multiplying 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

提示Undo a multiplication with the multiplier that reverses it.

为什么Multiplying 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

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

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

£204

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

为什么100% − 15% = 85%, so multiply by 0.85: 240 × 0.85 = 204. One multiplication replaces "find 15%, then subtract it".

1.15 and £92

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

为什么The 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

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

为什么Multiplying 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

提示Blue already has the same number of parts in each ratio.

为什么Blue 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

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

为什么In 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

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

为什么The 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

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

为什么Doubling 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

提示Join the two ratios through b first, then leave b out.

为什么Doubling 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.