Angela borrowed P dollars at 6 percent annual interest compounded

Question

M = P *\frac{(0.005)(1.005)^n}{(1.005)^n - 1}

Angela borrowed P dollars at 6 percent annual interest compounded monthly, and she will repay the loan by making n monthly payments of M dollars each. The formula gives the relationship between P, M and n. Is M > 50?

(1) P = 10,000
(2) n = 48

Bunuel · Accepted Answer

M = P *\frac{(0.005)(1.005)^n}{(1.005)^n - 1}

Angela borrowed P dollars at 6 percent annual interest compounded monthly, and she will repay the loan by making n monthly payments of M dollars each. The formula gives the relationship between P, M and n. Is M > 50?

If we filter the question and remove all extraneous information, the question boils down to whether  P * \frac{(0.005)(1.005)^n}{(1.005)^n - 1} > 50 .

(1) P = 10,000

Substituting this value into the question, we'd get:

Is  10,000 *\frac{(0.005)(1.005)^n}{(1.005)^n - 1} > 50 ?

Is  50 *\frac{(1.005)^n}{(1.005)^n - 1} > 50 ?

Is  \frac{(1.005)^n}{(1.005)^n - 1} > 1 ?

Is  (1.005)^n>(1.005)^n - 1 ?

Is  0> - 1 ?

Since the answer to the above question is YES, then this statement is sufficient.

(2) n = 48

Substituting this value into the question, we'd get:

Is  P *\frac{(0.005)(1.005)^{48}}{(1.005)^{48} - 1 } > 50 ?

P * \frac{(0.005)(1.005)^{48}}{(1.005)^{48} - 1}  can be more than (or equal to) 50 as well as less than 50, depending on the value of P. Hence, this statement is not sufficient.

Answer: A.

MartyMurray · Answer

M = P *\frac{(0.005)(1.005)^n}{(1.005)^n - 1}

Angela borrowed P dollars at 6 percent annual interest compounded monthly, and she will repay the loan by making n monthly payments of M dollars each. The formula gives the relationship between P, M and n. Is M > 50?

(1) P = 10,000

This is a tricky question. Since there are two variables, P and n, on the right side of the equation, it's tempting to believe that we need both to determine M. However this is a GMAT question. So, probably, determining the answer is not that simple.

So, let's consider this statement carefully.

Notice that the greater the value of n, the greater is the value of  (1.005)^n . Accordingly, the greater the value of n, the smaller is the relative difference between  (1.005)^n  and  (1.005)^n - 1 . So, the greater the value of n, the smaller is the fraction  \frac{(1.005)^n}{(1.005)^n - 1} .

Thus, the greater the value of n, the smaller will be the value of  \frac{(0.005)(1.005)^n}{(1.005)^n - 1} .

So, for a given amount of money, the greater the number of months Angela has to repay it, in other words, the greater the value of n, the smaller M will be. So, to find the smallest possible M for P = 10,000, let's maximize n by saying that n = infinity.

If n = infinity, then both  (1.005)^n  and  (1.005)^n - 1  equal infinity as well.

So, if n = infinity, then  (1.005)^n = (1.005)^n - 1 .

So, if n = infinity, then

\frac{(1.005)^n}{(1.005)^n - 1} = 1

and

\frac{(0.005)(1.005)^n}{(1.005)^n - 1} = 0.005 × 1 = 0.005 .

In that case, if  P = 10,000 , then  M = 10,000 *\frac{(0.005)(1.005)^n}{(1.005)^n - 1} = 10,000 × 0.005 = 50 .

So, if P = 10,000, then we know that, if n = infinity, the monthly payment is $50.

However, since we never reach infinity, M must always be greater than 50.

Sufficient.

(2) n = 48

Without knowing the amount of money to be repaid, there's no way we can determine whether M > 50. After all, P could be $10, in which case M < 50 regardless of how many months Angela has to repay the money, and if P = 1,000,000, then if Angela has 48 months to repay the money, M > 50.

Insufficient.

Correct Answer  A

Takeaway:  Sometimes the best way to determine what happens in a minimum or maximum scenario is to let a variable be infinity.

rebinh1982 · Answer

Can some please explain this question in an easier way? Thanks in advance

sayan640 · Answer

KarishmaB  Would you like to discuss this question ?

sayan640 · Answer

Bunuel  How do you infer that the highlighted part can be more than 50 or less than 50 , depending on the value of P ? WHats the logic behind this ?

Angela borrowed P dollars at 6 percent annual interest compounded monthly, and she will repay the loan by making n monthly payments of M dollars each. The formula gives the relationship between P, M and n. Is M > 50?

If we filter the question and remove all extraneous information, the question boils down to whether  P * \frac{(0.005)(1.005)^n}{(1.005)^n - 1} > 50 .

(1) P = 10,000

Substituting this value into the question, we'd get:

Is  10,000 *\frac{(0.005)(1.005)^n}{(1.005)^n - 1} > 50 ?

Is  50 *\frac{(1.005)^n}{(1.005)^n - 1} > 50 ?

Is  \frac{(1.005)^n}{(1.005)^n - 1} > 1 ?

Is  (1.005)^n>(1.005)^n - 1 ?

Is  0> - 1 ?

Since the answer to the above question is YES, then this statement is sufficient.

(2) n = 48

Substituting this value into the question, we'd get:

Is  P *\frac{(0.005)(1.005)^{48}}{(1.005)^{48} - 1 } > 50 ?

P * \frac{(0.005)(1.005)^{48}}{(1.005)^{48} - 1}  can be more than (or equal to) 50 as well as less than 50, depending on the value of P.  Hence, this statement is not sufficient.

Answer: A.

Bunuel · Answer

The logic is very simple actually. Try substituting a very small value, close to 0, for P and a very large value, say 10^100, for P.

KarishmaB · Answer

First impression : Easy C which means it is likely sufficient using a single statement. Now the question is - which statement alone is sufficient?

Note the question stem asks for not the value of M but whether M > 50. So less info could be sufficient to answer it.

M = P *\frac{(0.005)(1.005)^n}{(1.005)^n - 1}

Whether the monthly payment is more than 50 or less than 50 must depend on the amount borrowed. What if Angela borrowed only $10? Then her monthly payment cannot be greater than $50. Any reasonable amount will lead to M > 50. So we certainly need statement 1. Statement 2 alone cannot be sufficient.

Now we need to evaluate whether statement 1 alone is sufficient or we need statement 2 too. Do we need the value of n if we know that P = 10,000?

If n = 1 (we repay in one go), we get M = P * 1.005 in the given formula. Since P = 10,000, this value is certainly greater than 50.
If n is a very large number, then (1.005)^n is approximately the same as (1.005)^n - 1 and hence we get M = P * .005
Since P = 10,000, we get M = 50. Since the denominator (1.005)^n - 1 is slightly smaller than (1.005)^n, we know that M will be slightly greater than 50 in this case.

Hence no matter what the value of n (from 1 to a very very large value), M will always be greater than 50. So statement 1 is sufficient alone.

Answer (A)

jpear · Answer

Not sure if my logic would hold true in all problems but since this is compounded monthly I just did the simple annual interest at a 6% rate.

for 1)

10,000 * 6% = 600 / 12 = 50

so if it is compounded more frequently it would be above $50.

for 2)

without knowing the P you cant determine this

Angela borrowed P dollars at 6 percent annual interest compounded

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