Number, Year 11: bounds, growth and compound interest

The upper bound of a rounded measurement

HintA two-word name; its partner at the other end is called 'lower'.

The whyA length of 7.4 cm to 1 decimal place lies in 7.35 ≤ L < 7.45, so this value is 7.45 cm. The true length can get as close to it as you like but cannot equal it, because 7.45 would round to 7.5.

29.25 cm²

HintMake both sides as long as they could possibly be before multiplying.

The whyEach side could be up to half a centimetre longer: 6.5 cm and 4.5 cm. The largest possible area is 6.5 × 4.5 = 29.25 cm². The lower bound is 5.5 × 3.5 = 19.25 cm².

The upper bound of p and the lower bound of q

HintTaking away less leaves more.

The whyThe difference is biggest when you start with as much as possible and remove as little as possible: 8.45 − 2.55 = 5.9. For the least possible difference, swap the bounds: 8.35 − 2.65 = 5.7.

14

HintSpeed is greatest when the distance is as long as possible and the time as short as possible.

The whyUpper bound of distance = 105 m; lower bound of time = 7.5 s. Greatest speed = 105 ÷ 7.5 = 14 m/s. In a division, use the upper bound on top and the lower bound underneath to get the largest answer.

2.46 m, to 2 decimal places (3 significant figures)

HintRound both ends more and more finely and see where they stop agreeing.

The whyTo 2 decimal places both bounds give 2.46, so every possible value rounds to 2.46. To 3 decimal places they give 2.461 and 2.465, which differ, so the third decimal place is not certain.

0.99

HintWrite each change as its own multiplier, then combine the two.

The whyThe rise is × 1.1 and the reduction is × 0.9, and 1.1 × 0.9 = 0.99, an overall fall of 1%. For example £200 becomes £220 and then £198. The 10% reduction is taken from the larger amount, so it removes more than the rise added.

£7680

HintEach year's loss is 20% of what the car is worth at the start of that year.

The whyLosing 20% leaves 80%, a multiplier of 0.8. After two years the value is 12 000 × 0.8 × 0.8 = 12 000 × 0.8² = £7680. Year by year: £9600, then £7680.

Depreciation

HintIt is the opposite of 'appreciation', a rise in value.

The whyA car that loses 15% of its value each year is depreciating at 15% a year, so each year its value is multiplied by 0.85. It is a decay problem: the same percentage is lost each year, so the amount lost in pounds gets smaller every year.

0.85

HintAsk what percentage of the amount is left after the decrease.

The whyTaking away 15% leaves 85% of the amount, and 85% as a decimal is 0.85. Three decreases in a row multiply by 0.85³. A multiplier below 1 means decay; one above 1 means growth.

26 620

HintUse a single multiplier for one year's growth, and apply it once for each year.

The whyA 10% increase is a multiplier of 1.1. After 3 years the population is 20 000 × 1.1³ = 20 000 × 1.331 = 26 620. Year by year: 22 000, 24 200, 26 620.

2100; 2205

HintEach year's interest is worked out on whatever is in the account at the start of that year.

The whyYear 1: 2000 × 1.05 = £2100. Year 2 starts from £2100, not £2000: 2100 × 1.05 = £2205. The second year earns £105, which is £5 more than the first, because the first year's £100 of interest earns interest too.

500 × 1.04⁶

HintAdding four per cent is the same as multiplying by one number; do that once for every year.

The whyAdding 4% multiplies an amount by 1.04. Doing that six years running multiplies by 1.04 six times, which is 1.04⁶. On a calculator this gives £632.66 to the nearest penny.

3

HintBuild the balance up one year at a time, multiplying by the same number each year.

The whyEach year multiplies the balance by 1.1: £1100 after one year, £1210 after two and £1331 after three. The balance first passes £1300 at the end of the third year.

£10 more — in the second year, interest is also paid on the first year's £100 of interest

HintFind both totals, then ask what the later 10% is being taken of in each case.

The whySimple interest: £100 each year, £200 in all. Compound interest: £100 in year one, then 10% of £1100 = £110 in year two, £210 in all. The £10 difference is 10% of the first year's £100 of interest.

Compound interest

HintIts name means 'made of parts put together'; the other kind is called simple.

The whyBecause each year's interest joins the balance and then earns interest itself, the amount grows by the same multiplier every year, not by the same number of pounds. Simple interest is always worked out on the original amount only.