Ratio and proportion, Year 10

24; 920

提示Find 3% of the starting amount once, repeat it for every year, and remember what was there to begin with.

为什么3% of £800 is 0.03 × 800 = £24. Simple interest pays that same £24 every year, because it is always worked out on the original £800. So 5 years earn 5 × 24 = £120 and the account then holds £920.

3

提示Share the interest equally between the years first, then compare one year's worth with the amount invested.

为什么£60 over 4 years is £15 a year. 15 ÷ 500 = 0.03, which is 3%. Simple interest is the same every year, so the total can be split evenly.

VAT (value added tax)

提示Look near the bottom of a shop receipt, where it is often shown as a percentage.

为什么VAT is a percentage put on top of the price before tax. At a rate of 20%, an item priced at £50 before VAT costs 50 × 1.2 = £60.

40

提示Find the money gained, then compare it with what the shop paid, not with what the customer paid.

为什么Profit = 49 − 35 = £14. As a percentage of the cost price: 14 ÷ 35 = 0.4 = 40%.

£3500

提示Only part of her pay is taxed: take off the tax-free slice before you find the percentage.

为什么Taxable income = 30 000 − 12 500 = £17 500. 20% of 17 500 = £3500.

£195

提示One of the two words means money going out of the account and the other means money coming in.

为什么A debit takes money out and a credit pays money in: 240 − 75 + 30 = £195.

30 000 N/m²

提示Ask how many square centimetres cover one square metre: it is not 100.

为什么1 m = 100 cm, so 1 m² = 100 × 100 = 10 000 cm². Each of those square centimetres carries 3 N, so one square metre carries 3 × 10 000 = 30 000 N.

force; area

提示Read the unit aloud as "newtons per square metre" and ask what each part measures.

为什么A compound unit is a formula in disguise. Newtons measure force and square metres measure area, and "per" means divide, so pressure = force ÷ area. In the same way g/cm³ shows that density = mass ÷ volume.

2700 kg/m³

提示Convert the mass part and the volume part separately; a cubic metre holds a million cubic centimetres.

为什么1 m³ = 100³ = 1 000 000 cm³, so one cubic metre of the metal has a mass of 2.7 × 1 000 000 = 2 700 000 g, which is 2700 kg. Overall, changing g/cm³ to kg/m³ multiplies by 1000.

1200

提示Dividing by a number smaller than 1 makes the result bigger than what you started with.

为什么600 ÷ 0.5 = 1200. A whole square metre under the same load would carry 1200 N; the crate covers only half a square metre, so its 600 N is concentrated on that half.

Block B (3 g/cm³, against 2.7 g/cm³)

提示Work out the mass packed into one cubic centimetre of each.

为什么Density = mass ÷ volume. A: 540 ÷ 200 = 2.7 g/cm³. B: 480 ÷ 160 = 3 g/cm³. B has less mass in total, but more in each cubic centimetre.

5 g/cm³

提示Write an expression for how much space the solid takes up, then divide.

为什么The volume of the cube is x × x × x = x³ cm³. Density = mass ÷ volume = 5x³ ÷ x³ = 5 g/cm³. The x³ cancels, so the density is the same whatever the size of the cube.

y = 5x

提示Start from y = kx and use the pair of values to find the missing multiplier.

为什么Direct proportion means y = kx. Putting in the values: 20 = k × 4, so k = 5.

3

提示In this kind of relationship the product of the two quantities never changes.

为什么Inverse proportion means y = k ÷ x, so k = xy = 4 × 6 = 24. When x = 8, y = 24 ÷ 8 = 3. Doubling x halves y.

2; 50

提示Square the x-value before you use it, both times.

为什么18 = k × 3² = 9k, so k = 2 and the equation is y = 2x². When x = 5, y = 2 × 25 = 50.

It becomes a quarter of its old value

提示Write y = k ÷ x² and replace x by 2x; look at the denominator.

为什么y = k ÷ x². Doubling x makes x² four times as large, and dividing by a number four times as large gives a result one quarter the size.

The constant of proportionality

提示It never changes, and its name says which kind of relationship it belongs to.

为什么Every proportion statement becomes an equation by bringing in this one number. Find it from a known pair of values, then use the equation for any other value.