The GDP of a country is $13.8 billion, and the total production of one

should be solved like this. i'm not sure how we gonna solve it without calculator.

\(0.5*[13.8*(1.05)^{10}]\) < 3.3 \((1+\frac{x}{100})^{10}\)

\([2.1]^{\frac{1}{10}}*1.05 < (1+\frac{x}{100})\)

\(1.07*1.05< 1+\frac{x}{100}\)

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Hi nick, i have used exactly the same approach for the go ahead. However, i feel the power raised should be 9 and not 10 because the first year there was no increase !!
10th year will have compounded effect of 9 years with increased gdp. What do you think ?

Current GDP = $13.8 billion = ~$14 billion

Current total production of one industry = $3.3 billion = ~$3.5 billion, about 25% of current GDP.

GDP and the total production grow at an annual factor of 1.05 and x, respectively.

In order to increase from ~25% to >50% of the GDP in 10 years, the total production needs to double in 10 years. Thus,

(x/1.05)^9 >=2
--> x/1.05 >=2^(1/9)
--> x/1.05 >=~1.1
--> x >= 1.15

Final answer is (B)

I just approximate that:
1.1^2 = 1.1 * 1.1 = 1.2
1.1^4 = 1.1^2 * 1.1^2 = 1.2*1.2 = 1.4
1.1^8 = 1.1^4 * 1.1^4 = 1.4*1.4 = 1.96
So 1.1^9 must be greater than 2. Therefore, 2^(1/9) =~1.1

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1/2*13.8* (1.05)^9 < 3.3 * (1+t/100)^9

2.3*(1.05)^9<1.1(1+t/100)^9)

Since 2.2 (2*1.1) < 2.3 < 3.3 (3*1.1). So the real growth rate of the industry must be greater than twice of GDP's (5%*2 = 10%) and smaller than thrice of GDP's (5%*3 = 15%).

The question asks for the MINIMUM required annual growth, so we should not take 10%. Looking at the choices, 15% is the smallest bound. IMO B. This approach can be easily screwed if A was 12%, so yeah it's not a general solution to solve this kind of questions.

GDP now = $13.8 billion
GDP 10 years from now = $13.8*1.05^10 = $22.48 billion
Industry 10 years from now = $11.24 billion minimum
Industry now = $3.3 billion
Industry growth in 10 years = 11.24/3.3 = 3.4
(1+x)^10 = 3.4
x = ~13 % ~ 15%

IMO B

similar to Kinshook's approach

Current GDP: 13.8, in ten years time 13.8*(1 + 5%)^10 = 22.4787
so such industry has to present half of it so minimum: 11.23935

Such industry worth: 3.3

so plug in the answer to find out one that's above 11.23935
A) 3.3*(1.1)^10 = 8.559 ( not enough)
B) 3.3 *(1.15)^10 = 13.35 ( above 11.3, so this is it)

Here’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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I think the way to solve this would be using the rule of 72. The rule is that 72/A=Y, where A is the annual rate of return, and Y is the number of years to double. Note that A should not be in % form, so 5% would be just 5.

1. you approximate that the entire GDP using the 72 rule,
72/5=14.4 so in 14.4 years the GPD will double, so you can approximate that in 10 years, it will be
15 years = 26.6
10 years = X
5 years = Y
0 years = 13.3
-->13.3/3=4.4, so,
15 years = 26.6
10 years = 22.1
5 years = 17.7
0 years = 13.3

so in 10 years the GDP will be 22.1, roughly, probably a bit higher given that the growth is exponential and not linear as i calculated it.
If the GDP will be 22.1, the industry has to be at least 11

2. then approximate the industry growth using the 72 rule,
3.3-->11.1 is doubling twice, so we want the ARR to make it double in 5 years
72/X=5, x = 14.4

14.4 growth rate needed --> ans B

 

­Enlighten us Bunuel, please... My test is scheduled on March 20th

­I too followed the same approach as yours, but I am not quite sure if we can solve this problem without using a calculator.