M12-11

Question

What is the approximate number of years it takes for a country's population to double if it grows at a rate of 20% per year?

A. 3
B. 4
C. 5
D. 6
E. 7

Bunuel · Answer

Official Solution:

What is the approximate number of years it takes for a country's population to double if it grows at a rate of 20% per year?

A. 3
B. 4
C. 5
D. 6
E. 7

The question basically asks us to find approximate value of \(n\) such that \(1.2^n > 2\)

Express the above as a fraction because powers of 5 and 6 will be easier to calculate than exponentiating 1.2:

\((\frac{6}{5})^n > 2\)

In 1 year the population will increase \(\frac{6}{5} = \frac{36}{25} < 2\) times;

In 2 years the population will increase \((\frac{6}{5})^2 = \frac{36}{25} < 2\) times;

In 3 years the population will increase \((\frac{6}{5})^3 = \frac{216}{125} < 2\) times;

In 4 years the population will increase \((\frac{6}{5})^3 = \frac{1296}{625} > 2\) times.

Hence, the number of years it takes for a country's population to double is between 3 and 4, closer to 4.

An alternate method to solve this problem is by applying the Rule of 72, which states that the number of years it takes for an investment or population to double is approximately equal to 72 divided by the annual growth rate as a percentage. Here is an Official Question talking about this rule In our case, since the population is growing at a rate of 20% per year, we can divide 72 by 20 to get 3.6, indicating that the population will double in approximately 3.6 years.

Answer: B

GMATCoachBen · Answer

I see there is some uncertainty over the problem wording; the intention of the problem is asking:

1.2^n\geq 2 , (solving for n, where n is the lowest possible integer)

I haven't seen this wording on official questions, but on this question, "before" means "how many years will it take for the population to at least double"? If "before" is taken literally as "prior to doubling", then 0 or 1 or 2 would also be acceptable responses, which doesn't make sense.

2 solutions below:

1) A reliable way to approach percent change over a short period of years (essentially, compound interest) is a table, picking the number 100 as a starting point:

Year 0: 100
After 1 year: 120
After 2 years: 120*1.2 = 144
After 3 years: 144*1.2 = 172.8 (a quick way to add 20% is add 2*10% --> add 2*14.4 to 144 --> 172.8 
After 4 years: 172.8*1.2 = 207.36
Here, we can  save time by rounding and approximating -- we just need to know Yes/No whether it's greater than 200 (just like Data Sufficiency, it's best to avoid unnecessary calculations) 
Round to 173 and add 2*10% -->  173 + 2*17.3 > 200?  --> Clearly yes (173 + 34.6 = 207.6)

Answer is  B. 4

2) Using fractions instead of decimals:

In many cases, it's a helpful habit to use fractions:
Translate "increase (grows) by 20%" to "multiply by  \frac{6}{5} "

Glance at the answer choices and try 3 years first:

If you have your cubes memorized,  (\frac{6}{5})^3=\frac{216}{125}<2.  Less than 2, so not the answer.  (2 = \frac{250}{125} )

Next, try 4 years:  \frac{216}{125}*\frac{6}{5}=\frac{1296}{625} > 2.  Greater than 2, so our answer is  B. 4  ( 2 = \frac{1250}{625} )

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

RishiAgarwal231 · Answer

This can easily be solved using rule of 72

Rule of 72: to find out in how many years will a value double when it is compounding at a fixed rate p.a., divide 72 by that value

So, for 20%, 72/20=3.6yrs.

SO, approx. 4yrs. (10 sec solution)

Premkungg · Answer

I like the solution - it’s helpful.

Anish27 · Answer

I like the solution - it’s helpful.

M12-11

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards