100 became 150 in five years. Is that 10% growth per year?

On the left are the starting value, the ending value and the number of years. Top right is the total growth; bottom right divides it by the number of years to give a simple average.

Sales, users, population, output, savings — the calculation is the same whatever the number counts. This article uses 100 → 150.

It grew 50%, so divide by 5 years and get 10%?

100 became 150. It grew by 50, which is 50% of the original 100. The card top right on the board shows 50% too.

Divide that by 5 years: 50% ÷ 5 = 10%. The card bottom right shows 10%.

But this 10% means adding 10% of the original 100 every year — 10 a year. 10 × 5 = 50, which lands exactly on 150. What was added in earlier years never enters the next year's calculation.

At 10% a year, what do you have after five years?

"It grew 10% a year" normally means each year's value is multiplied by 1.1. The board below card on the left starts at 100 and applies 10% five times. Set its count from 1 to 5 to see each year. The card on the right does the same with a rate that comes up later.

• Year 1 — 110
• Year 2 — 121
• Year 3 — 133.1
• Year 4 — 146.41
• Year 5 — 161.051

At 10% a year, five years gives 161.051, not 150. What was added also gets the next year's 10%. So the rate that lands exactly on 150 must be lower than 10%.

Working the rate back from 150

Suppose it grew at the same rate r every year and reached 150 in five years:

100 × (1 + r)⁵ = 150

You want the number that, multiplied in five times, makes 1.5. Working it back:

r = (150 ÷ 100)^1/5 − 1

The ^1/5 means the fifth root: the number that, multiplied by itself five times, gives 1.5. That is about 1.0845, so r is about 8.45%. The board below shows 8.447177%.

Top right is the annual growth rate: (End ÷ Start)^{1 ÷ Years} − 1, times 100 to make it a percentage.

Does 8.45% really get back to 150?

The card bottom right applies this rate to the starting 100 five times. It comes to 150.

To see it year by year, use the card on the right of the second board, At 8.447177%, and change its count from 1 to 5:

• Year 1 — 108.447177
• Year 2 — 117.607902
• Year 3 — 127.54245
• Year 4 — 138.316186
• Year 5 — 149.999999

It ends at 149.999999 because the card's rate is typed to six decimal places — close enough to 150. The 10% card on the left ends at 161.051, so the gap between the two rates grows to about 11 over five years.

An "average" rate is not the average of each year

The average annual growth rate is the one steady rate that, if applied every year, would give the same final value. It does not mean the value actually grew 8.45% each year.

The path could go up and down — 100 → 130 → 110 → 140 → 120 → 150, say. As long as it starts at 100, ends at 150 and takes five years, the average annual growth rate is the same 8.45% or so. What happened in between does not show up in this number.

It is often called CAGR, short for compound annual growth rate.

Try it yourself

Change the start, end and years on the first and third boards.

100 → 150 over 5 years and 100 → 200 over 10 years both have a simple average of 10%. But the annual growth rates are about 8.45% and about 7.18% — quite different. The same 10% simple average does not mean the same year-on-year rate. Of these two, the longer 10-year case sits further below 10%.

200 → 300 gives 8.447177%, the same as 100 → 150, because both grow 1.5 times. With the same multiple and the same number of years, the rate is the same whether you start at 100 or 200.

Read next — Does 5% a year for 10 years mean a 50% increase? The same calculation the other way round: from a known annual rate to the value years later.

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