Small rates, long years
- Start£100After year one£110After year two£121
A basket of shopping cost £100. That year prices rose 10%, so it cost £110. The next year prices rose 10% again. This time the 10% is worked out on £110, not on £100. So it adds £11, and the basket costs £121.
- Ten plus ten20%Ten on top of ten21%
Each year's rise lands on top of last year's price. The word for this is compounding. It just means the rise is worked out on the new number, not the old one. That is why 10% and then 10% came to 21%, not 20%.
- Start£100After year one£110
A basket that cost £100. Prices rise 10% a year for two years. By how much have they risen in total?
21%
Right. The second year's 10% is worked out on £110, not on £100, so it adds £11. The rises stack on each other rather than adding up.20%
Not yet. Ten plus ten. That would be right if the second rise were worked out on the original £100. It is worked out on £110.It depends what you buy.
Not yet. It would, for a real household. Here the figure is given: 10% a year on the whole basket, so the only question is how the two years combine. - Start£100After year one£110After year two£121
Prices rise 10% a year for three years. What does the £100 basket cost at the end?
£Add 10% of £121, not 10% of £100.
Right. £110, then £121, then £133.10. Each year's rise is worked out on last year's price, not the original one. That is all compounding is. - £0£124£247Year 0Year 40
A £100 basket at 2% a year. Two per cent a year sounds like nothing. But the rise stacks every year, and it never stops. Over forty years, about a working life, a £100 basket at 2% a year ends up costing £221. It more than doubled.
A £100 basket, forty years, and one inflation rate. Change the rate.
£0£83£167Year 0Year 40Inflation rate: 1% a year
After forty years the basket is £149. Slow, and it never stops.
£0£124£247Year 0Year 40Inflation rate: 2% a year
After forty years the basket is £221. It more than doubled, at a rate that sounds like nothing.
£0£269£538Year 0Year 40Inflation rate: 4% a year
After forty years the basket is £480. Twice the rate gave more than three times the rise.
£0£839£1,677Year 0Year 40Inflation rate: 7% a year
After forty years the basket is nearly £1,500. Fifteen times the start, from seven per cent a year.
Doubling the rate from 2% to 4% did what to the rise after forty years?
More than tripled it — from about £121 extra to about £380 extra.
Right. The rises stack, so twice the rate is far more than twice the rise. The gap between 2% and 4% is bigger than the gap between 0% and 2%, and it keeps growing.Doubled it, from £121 to about £242.
Not yet. That is what doubling the rate would do if each year's rise were on the original £100. It is on last year's price, and last year's price is bigger at 4%.Added 80% over the forty years.
Not yet. Two extra per cent times forty years. That adds the rates up. The rises stack instead, so adding them gives too small an answer, and the gap grows every year.Move the control.
- At 2% a year35 yearsAt 5% a year14 yearsAt 10% a year7 years
Years until prices double. There is a quick way to see this in your head. Divide 70 by the rate. That is about how many years until prices double. At 2% a year it is 35 years. At 5% it is 14. At 10% it is 7. Then they double again from there. People call this the rule of 70.
- Inflation7% a yearYears until prices double?
Seventy divided by the rate. A country has 7% inflation. Using the rule of 70, how many years until prices double?
yearsSeventy divided by the rate.
Right. Seventy divided by seven. Prices double every ten years at 7%. A house, a pint of beer, a bus fare: whatever it costs now, it costs twice that in ten years. At 7% a year a £100 basket doubles to £200 in ten years. Slide the marker to what it costs after TWENTY years.
Slide the marker to where you think the answer is, then press Lock. Nothing is scored.
your guess · £100the answer · £387- 1Start£100
- 2After ten years£200
Your guess: —
Doubling again is times two, not plus a hundred. The second ten years start from £200, so they end near £400, not £300. Every ten years it doubles from wherever it is.- £1,000The pension, every month, for life3.5% less each yearThe shopping it buys
The number on the statement never changes. The shopping does. A pension pays a fixed £1,000 a month for life, with no rises ever. Inflation runs at 3.5%. Roughly when does it buy half of what it does today?
Seventy divided by the rate.
Never. It is still £1,000.
Not yet. The pounds never change. What those pounds buy shrinks a little every year. The pensioner lives on what the pounds buy, not on the number.After about twenty years.
Right. Seventy divided by 3.5. The pension pays £1,000 every month for twenty years, and by the end that £1,000 buys half the shopping it did.After about thirty years.
Not yet. That is 100 divided by 3.5, which adds the rate up year by year. The rises stack, so halving comes sooner than that.
Lesson complete
Divide 70 by the rate. That is about how many years until prices double.