Number, Year 11: bounds, growth and compound interest

The upper bound of a rounded measurement

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

为什么A 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

提示Make both sides as long as they could possibly be before multiplying.

为什么Each 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

提示Taking away less leaves more.

为什么The 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

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

为什么Upper 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)

提示Round both ends more and more finely and see where they stop agreeing.

为什么To 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

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

为什么The 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

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

为什么Losing 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

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

为什么A 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

提示Ask what percentage of the amount is left after the decrease.

为什么Taking 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

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

为什么A 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

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

为什么Year 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

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

为什么Adding 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

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

为什么Each 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

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

为什么Simple 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

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

为什么Because 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.