DS

[prachi18oct]

Please explain few approaches to solve this problem.
During exam, I couldn't think of any short approach.

[Ian Stewart]

The formula in the question makes this look complicated, but really we're just making an investment, and applying interest twice (with compounding). There's no need to even look at the formula.

If we know the exact size of the investment, and exactly how much interest we earn in two years, we can work out the interest rate, so Statement 1 is sufficient.

For Statement 2, we multiply the value of our investment by (1 + r/100) every time we want to add r% interest. So if we multiply by (1 + r/100)^2, we're adding r% interest twice. In words, Statement 2 just says "in two years, the investment earns more than 15% in interest". Certainly our interest rate could be very high, so the answer to the question could be yes - the annual interest rate might be larger than 8%.

We want to know if it's possible that the interest rate is less than 8%. You might think about what would happen with an annual interest rate exactly equal to 7.5% (exactly half of 15%). If we were applying simple interest, the value of the investment would increase by exactly 15% in two years. But if we compound, as we're doing here, we'll always earn more than if we don't compound. So at 7.5% annual interest, the investment in this question would earn more than 15% in two years. That means, if Statement 2 is true, our annual interest rate could also be 7.5%, and the answer to the question can be 'no'. So Statement 2 is not sufficient, and the answer is A.

[GMATGuruNY]

If 1,000 is deposited in a certain bank account and remains in the account and any accumulated interest, I, earned by the deposit in the first n years is given by the formula I = 1,000((1 + r/100)^n - 1), where r percent is the annual interest rate paid by the band, is the annual interest rate paid by the bank greater than 8 percent?

1) The deposit earns a total of $210 in interest in the first 2 years
2) (1 + r/100)² > 1.15

OA A

Question: Is r > 8?

Statement 1: The deposit earns a total of $210 in interest in the first 2 years.
Plug I=210 and n=2 into the given equation:
210 = 1,000 ((1+r/100)^2 - 1)
We can solve for r.
SUFFICIENT.

Statement 2: (1+r/100)² > 1.15
Test r=8.
(1 + 8/100)² > 115/100
(108/100)² > 115/100
(108*108)/(100*100) > 115/100
(108*108)/100 > 115

Ballpark the left side:
(11,600)/100 > 115
116 > 115.

Since r=8 works, we know that r>8 also works.
INSUFFICIENT.

The correct answer is A.