# Math Help - Compound Interest - Intuition is wrong

1. ## 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 $r = 1$

$
S = \left(1 + \dfrac{100}{r}\right)\left((1+\dfrac{r}{100})^n - 1\right)P
$

But this approach gives an incorrect answer. $S = 21134$

However if I break down the problem separately, Using $r = 5$ for Income, and $r = 4$ for Expenses. I get,

${Total Income} = 26,413$

And,

${Total Expenses} = 24,972$

And Hence,

$Savings = {Total Income} - {Total Expenses} \approx 1400$

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.

2. 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

3. 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.

4. 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:
$X_n = 2000(1.05^n - 1.04^n) \neq 2000(1.01^n)$

5. Thanks guys! Got it, makes sense now. Working out the net payment in a year helps clarify as well.