Bunuel wrote:
The GDP of a country is $13.8 billion, and the total production of one of its industries is $3.3 billion. If the GDP were to grow by 5% per year in the future, which of the following would be the MINIMUM required annual growth in this industry that would it represent more than half of the GDP in ten years?
A. 10%
B. 15%
C. 20%
D. 25%
E. 30%
Are You Up For the Challenge: 700 Level QuestionsHere’s how I thought about it:
3.3/13.4 ~= 1/4
We need this to become 2/4 in ten years.
If both the numerator and denominator grow at 5% per year for ten years, the ratio would remain the same. For the ratio to double, we need to also multiply the numerator by 2.
To get that factor of two in terms of an interest rate, we can use the rule of 72. To double something in ten years requires a rate of approx 72/10 = 7.2%.
So if the numerator were multiplied by (1.05)^10 * (1.072)^10, then the ratio would double.
Now the question is what is (1.05)^10 * (1.072)^10 in terms of a single annual interest rate?
We know from algebra rules that (a^c)*(b^c) is NOT (a + b)^c.
However, we can still sum a and b as a rough approximation. For example, if something grew by 10% and then 10%, the overall growth would be close to 20%; (1.1)^2 = (11/10)^2 = 121/100= 1.21 > than our approx.
In this case, adding the rates would mean an annual rate of ~ 12.2%.
Now since 1.05*1.072 > 1 + 0.05 + 0.072, our estimate is lower than actual*, and this lower estimate will be compounded 10 times. So the actual required rate is probably a little more than 12.2% and we can probably safely choose B.
*You can prove this to yourself pretty quickly.
1.1*1.1 = 1.21 > 1.2 = 1 + 0.1 + 0.1
1.2*1.2 = 1.44 > 1.4 = 1 + 0.2 + 0.2
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