# Compound Interest - Intuition is wrong

• Apr 12th 2011, 07:44 AM
mathguy80
Compound Interest - Intuition is wrong
Hi All,

I need some help regarding an alternate approach to the following problem.

In 1960 a man earned $2,000 and spent it all. During the next 10 years his salary increased by 5% per annum(compound interest), but inflation caused his expenditure to rise by 4% per annum(compound interest). Find how much he had saved by the end of 1970, giving your answer to two significant figures. My intuition tells me that +5%(income) and -4%(expense) should given a +1% increase compound annually on the initial amount. So I used the formula for the Sum at nth year for compound interest with$\displaystyle r = 1\displaystyle
S = \left(1 + \dfrac{100}{r}\right)\left((1+\dfrac{r}{100})^n - 1\right)P
$But this approach gives an incorrect answer.$\displaystyle S = 21134$However if I break down the problem separately, Using$\displaystyle r = 5$for Income, and$\displaystyle r = 4$for Expenses. I get,$\displaystyle {Total Income} = 26,413$And,$\displaystyle {Total Expenses} = 24,972$And Hence,$\displaystyle Savings = {Total Income} - {Total Expenses} \approx 1400$The answer checks out! So my intuition is wrong in thinking that +5% and -4% would become a G.P. of 1%. I feel I am making an important logical error here. Can you guys explain why this line of thinking is incorrect? Thanks. • Apr 12th 2011, 07:54 AM Carlow52 X(1 + 5%) - Y(1+ 4%) is not = to X-Y(1+1%) which I think it what you have done, or close to it, in effect. You need to follow the BOMDAS rules • Apr 12th 2011, 07:58 AM HallsofIvy Your intuition is wrong because you are trying to apply both the 5% and the 4% to the same base- the same salary. The 5% increase in salary is 5% of last year's salary. The 4% increase in spending is 4% of this year's salary. • Apr 12th 2011, 08:15 AM SpringFan25 i agree with carlow, if you manually calculate the first few net payments its clear that they dont grow at 1% The net payment in any year:$\displaystyle X_n = 2000(1.05^n - 1.04^n) \neq 2000(1.01^n)\$
• Apr 12th 2011, 07:49 PM
mathguy80
Thanks guys! Got it, makes sense now. Working out the net payment in a year helps clarify as well.