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Math Help - Compound Interest - Intuition is wrong

  1. #1
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    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

    <br />
S = \left(1 + \dfrac{100}{r}\right)\left((1+\dfrac{r}{100})^n - 1\right)P<br />

    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

    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.
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  2. #2
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    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
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  3. #3
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    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.
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  4. #4
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    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)
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  5. #5
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    Thanks guys! Got it, makes sense now. Working out the net payment in a year helps clarify as well.
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