Question

Cohen took out a loan which gathers compound interest each year

At the end of 2004 cohen owed £637.88
At the end of 2009 cohen owed £863.75
How much interest did the loan gather over the next 3 years
give your answer in pounds to the nearest 1p

Answer 1

£172.29

Explanation:

You want the compound interest a loan accumulates after 3 more years when the amount owed in 2004 was £637.88 and in 2009 was £863.75.

Account value

The formula for an amount subject to annually compounded interest is ...

A = P(1 +r)^t

where P is the principal amount, r is the interest rate, and t is the number of years.

Solution

We can find the value of (1+r) from ...

863.75 = 637.88(1 +r)^(2009 -2004)

(1 +r) = (863.75/637.88)^(1/5)

Then the amount owed after 3 more years will be ...

A = 863.75(1 +r)^3

This amount includes the initial value of 863.75, so the increase is ...

863.75(1 +r)^3 -863.75 = 863.75((1 +r)^3 -1)

= 863.75(((863.75/637.88)^(1/5))^3 -1) ≈ 172.29

The loan gathered an additional £172.29 in interest over the next 3 years.

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Additional comment

We don't really need to know the initial loan value or the number of years prior to 2004 for which interest was accumulated. As a byproduct of the intermediate result (1+r) we computed, we learned the interest rate is about 6.25%. Since we used the exact ratio of account values, this number, too, is irrelevant.

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