This is a good question that tests your understanding of Compound interest as an application of percentage increase. Solving this question in less than 2 min is about how well you apply the conceptual understanding and logic rather than calculation-intensive approaches.
Let's take an example: 1000$ is invested at a rate of r% for n years at C. I annually.
The amount, A=\(1,000((1+r/100)^n\)
The first step is to understand what this formula is doing. It is very simple. Here, 1000 $ is increased at a rate of r% every year.
Since it was deposited for n years, 1000$ will be increased by r %, n times successively. So it is a pure
application of successive percentage increase.
Let me make it more clear for you.,
if r = 8% and n = 2 years.
A = \(1,000((1+8/100)^2\) = \(1000 * \frac{108}{100} * \frac{108}{100}\)
Multiplying 1000 by 108/100 means you are increasing 1000 by 8%.
So here, 1000$ is successively increased by 8%,2 times. I hope you all got a hang of how compound interest works.
Interest amount = Final amount after successive increase - Initial amount deposited.
I=\(1,000(1+r/100)^n- 1000\) = \(1,000((1+r/100)^n-1).\)
Now it's time to analyze the question stem.
Question stem: Is the annual interest rate paid by the bank greater than 8 percent? so is
r > 8%?
There is no info given in the question regarding the no of years you have deposited. Also, it will not affect the interest rate charged by the bank.
But, if you analyze both the statements given we will know that the data is given for 2 years. So let's try to re-frame the question stem w.rt 2 years.
If 1000$ is deposited for 2 years at 8 % CI annually, then the interest would = 80 + 80 + 6.4 = 166.4
Explaining my calculation in detail below:
Total interest = First-year interest + Second-year interest.
First year interest = 8%of initial amount = 8 % of 1000 = 80$
Second year interest = 8%of initial amount + 8% of ( first year interest) = 8 % of 1000 + 8 % 80 = 80 + 6.4
So total interest in 2 years =
80 + 80 + 6.4 = 166.4Instead of using intensive calculation formula to find Compound interest, I prefer to stick to this logical approach. Now, it's your choice.
If r =8%, we found that interest for 2 years is 166.4$
If r is greater than 8 %, the interest amount for 2 years also should be greater than 166.4$
Hence, we can re-frame the question stem
is r > 8%? as is Interest for 2 years > 166.4$?So here we need to check if the interest for 2 years is greater than 166.4 $ or not.
St1:
The deposit earns a total of $210 in interest in the first two years.You will get a definite
Yes as it is given that the deposit earned a total of $210 in interest in the first two years. Therefore,
Statement 1 alone is sufficient to answer the question.St2: \((1+r/100)^2 >1.15\)
\((1+r/100)^2 \) will give you an
idea about the overall percentage increase for 2 years. Analyzing in detail : \(A=1,000(1+r/100)^2\)
If \((1+r/100)^2\) = 1.15
A = 1000 * 1.15 = 1000 * 115/100 = 1150
Multiplying 1000 by 1.15 means 1000 is increased by 15 % in 2 years i.e 1000 + 150 = 1150.
So here, \((1+r/100)^2 > 1.15\) means
the amount after 2 years > 1150 Or we can say that the overall percentage change happened in 2 years is greater than 15 %
Since Interest = Final amount - initial deposit, We can conclude that
total interest for 2 years is greater than 150.If you could recollect how we re-framed the question stem in terms of the total interest in St 1, Can we use it here as well?
is r > 8%? as is Interest for 2 years > 166.4$?
Greater than 150 means it could be between 150 and 166.4 or greater than 166.4. So you will get a
NO as well as
YES to the question stem.
Therefore,
statement 2 alone is not sufficient.
There is also an alternative approach you can think about in st 2, in terms of the overall percentage change.
Let's try to re-frame the question stem in terms of overall percentage increase for 2 years.
If you are increasing an amount by 8 % successively 2 times, the overall percentage increase would be 8 + 8 + 8*8/100 = 16.64 %
I hope all of you are aware of this formula for overall percentage change.
If not, please have a look here.
If a value is increased by a % then again increased by b %, the overall percentage change = a + b + ab/100.
Note: This formula is applicable only for 2 changes. Since the value is increased by a%, we are putting + a. So if it's a decrease, use the '-' sign instead of '+'.
For eg. If a value is decreased by a % then increased by b %, then overall percentage change = -a + b + (-a)(b)/100 = -a +b -ab/100
Coming back to the question stem, can we re-frame is r > 8% ? as is the overall percentage increase for 2 years > 16.64 %?
In the st 2, (1+r/100)^2 >1.15 means the overall percentage increase for 2 years is greater than 15 %. This will not give you a definite answer to the question stem.
Greater than 15 % means it can be greater than 16.64 % or it can also be between 15 and 16.64 %. So, it can be a yes or a no.
Hence, St 2 alone is not sufficient.
Option A is the correct answer.
I hope this explanation helps to have a better understanding of compound interest and how to apply concepts and logic instead of calculation-intensive approaches.
Thanks,
Clifin J Francis,
GMAT QUANT SME