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.