gmat Prep Math Question

[abehrman]

if $1,000 is deposited in a certain bank account and remains in the account along with any accrued interest, the dollar amount of interest, I, earned by deposit in the first n years is given by:

I = 1,000 ((1+r/100)^n -1)

where r percent is the annual interest rate paid by the bank. Is the annual interest rate paid by the bank > 8%?

1) the deposit earns a total of $210 in interest in the first 2 years.

2) (1+r/100)^2 > 1.15

I know 1 is sufficient, why is 2 insufficient? You can solve for r.

[agha79]

becasue there is no way of finding what "r" is

[jitendra.havaldar]

For the 2nd option mentioned above, when such an option is given on the test, do we have to put in some 2-3 values to confirm?
or we simply take it as an insufficient option without giving it a second thought?
why i say this is because of the following 2 scenarios here:

For (1+r/100)^2 > 1.15 :-

1. assume (1+r/100)^2 = 1.16 solving for 'r', we get r = 7(less than 8)

2. assume (1+r/100)^2 = 1.44 solving for 'r', we get r = 12(greater than 8)

so clearly insufficient since we get 2 different answers for the scenarios mentioned.

[kramacha1979]

Ron,

I have 2 questions here

a) The first statement doesn't say that the principal is the same 1000. The initial statement was just an example to show the relation. In that case how can Stmt#1 be Suff

b) How do we go about solving for r in stmt#2easily? . I mean for 1.16 we get r<7 and 1.44 r>8. It's time consuming to find the square root of 1.16 and to prove that Stmt#2 is insuff just by mere plugging values ..
I thought GMAT math calculations aren't tough but need tricks/short cuts

[tejkumar.m]

The first point you made is true.. Not sure abt that.

Second one, shortest way is put 8 in (1+r/100)^2 which will come to 1.16 appr.. now the 2 nd option says it is greater than 1.15.. by this it is possible that r is 8, and hence it is "=8" and it is also possible to be r>8 say 9 or any other number >8 will satisfy == (1.18) which is also > 1.15.. Hope this helps.

[sandeepgupta176]

if $1,000 is deposited in a certain bank account and remains in the account along with any accrued interest, the dollar amount of interest, I, earned by deposit in the first n years is given by:

I = 1,000 ((1+r/100)^n -1)

where r percent is the annual interest rate paid by the bank. Is the annual interest rate paid by the bank > 8%?

1) the deposit earns a total of $210 in interest in the first 2 years.

2) (1+r/100)^2 > 1.15

I know 1 is sufficient, why is 2 insufficient? You can solve for r.

Lets work with statement second !!!

Given the exponent power is 2 that means n=3 (3 years).

Now 1.15 here indicates that in 3 years the investment has increased by more than 15%

That would approximately give around 5% of annual return (although thats not true in
case of compounding interest, but would serve the purpose). Obviously since the return
has been shown as greater than 15% for 3 years that means we can have both less
than 8% per year (like 5% here) or even 10% if its 1.21 and not 1.15 on the right hand
side.

Statement 2 is therefore insufficient !!!

We dnt need to solve for square roots of 1.15 or something, GMAT never requires that.

Hope this helps !!!

[akhp77]

Statement 1:

210 = 1,000 ((1+r/100)^2 -1)
r = 10%

Sufficient

Statement 2:
(1+r/100)^2 > 1.15 = 115/100 = (1+7.23/100)^2

r > 7.23
r may take any value between 7.23 and 8 or >= 8
So, we can't say whether it would be lesser than, greater than, or equal to 8

Insufficient

Ans: A

[StaceyKoprince]

Multiple questions on this one!

abehrman, as some others have mentioned, we cannot actually solve for r with statement 2 alone. First, we can't actually solve for a particular value because statement 2 is an inequality, which means we are only going to be able to solve for a range of values. That still COULD be sufficient, because the question also asks about a range (>8%), so we have to explore a bit further.

jitendra provides one nice way to test - try a couple of different numbers to see whether we can get contradictory results (one result that is greater than 8% and one result that is less than or equal to 8%).

You can also attempt to simplify the given inequality, depending upon whether you feel more comfortable working with real numbers or working with algebra. (Note: most people feel more comfortable - or should feel more comfortable - working with real numbers. Algebra can get a lot messier.)

take the SQRT of both sides:
1+ (r/100) > approx. 1.07
r/100 > 0.07
r > approx 7

(Note: this looks easier than jitendra's try-real-numbers approach, but do you know how to quickly and fairly accurately approximate the square root of a number such as 1.15?)

So, if r is greater than about 7, then r could be less than or equal to 8 and it could also be greater than 8.

kramacha, the information given in the question stem applies to the entire problem. When statement 1 says "THE deposit" it is referring to the only deposit that has been mentioned in the problem - the $1,000 deposit in the question stem.

[joehurundas]

To prove sufficiency of (2), we can test r=8, and r > 8.
first, r=8 --> (1.08)^2 = 1.166 > 1.15...we respond "No, r <> 8"
for r>8 we expect (1+ r/100)^2 > 1.15...we respond "YES, r > 8"
Insufficient
Hope my suggestion helps.

[RonPurewal]

To prove sufficiency of (2), we can test r=8, and r > 8.
first, r=8 --> (1.08)^2 = 1.166 > 1.15...we respond "No, r <> 8"
for r>8 we expect (1+ r/100)^2 > 1.15...we respond "YES, r > 8"
Insufficient
Hope my suggestion helps.

i'm not really following this solution.
first, i don't know what "<>" means (looks like a typographical error, but i can't be sure). please explain; thanks.

second, i think that you might have the right idea, but a couple of steps are missing.
here's the whole explanation:
when you plug in r = 8, you get 1.1664.
this means that 1.15 would correspond to an r-value of LESS than 8.
since huge values (much greater than 1.15) can clearly correspond to r-values that are greater than 8 -- in fact, they can correspond to r-values as large as desired -- it follows that we can get r-values that are either less than or greater than 8.
so, insufficient.

[wxandr]

If $1000 is deposited in a certain bank account and remains in the account along with any accumulated interest, the dollar amount of interest, I, is given by:

I = 1000[(1 + r/100)^t - 1]

where r is the annual interest rate, and t is measured in years.

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 two years.
(2) (1 + r/100)^2 > 1.15

(1) 210 = 1000[(1 + r/100)^2 - 1]
     0.21 = (1 + r/100)^2 - 1
     1.21 = (1 + r/100)^2
     (1.21)^1/2 = (1 + r/100)
     1.1 =  1 + r/100
     0.1 = r/100
     10 = r

(2) 1 + r/50 + (r^2)/10000 > 1.15
      (r^2)/10000 + r/50 > 0.15
      r^2 + 200r - 1500 > 0

r >  [-200 +/- sqrt(46,000)]/2
r >  [-200 +/- 214.47]/2
r > 7.238

A note: the computations for (2) are performed
by solving the quadratic formula for positive roots of r.
They confirmed test logic that the lower boundary for
r might not be great enough to determine whether the
interest rate paid by the bank is greater than 8 percent.

In contrast to the range that satisfies the
inequality in (2), the equation in (1) provides an exact value for r.

[RonPurewal]

In contrast to the range that satisfies the
inequality in (2), the equation in (1) provides an exact value for r.

this is true.
notice one consequence of this fact in particular -- namely, once you notice that the process is going to yield a unique solution, it's a complete waste of your time to continue solving it!

in other words, after writing about half of the steps that you wrote above for statement 1, you should have stopped and written "this is going to give only one solution ... sufficient".

don't waste your time! there's already plenty of time pressure on this test; no reason to add any more.

[prashant.ranjan]

For statement (2) following method can be useful:

(1 + r/100)^2 > 1.15
Now we know that 1.1^2 = 1.21.
So sqrt(1.15) falls between 1 and 1.1 (No need to compute the actual sqrt of 1.15)

1< (1 + r/100) < 1.1
0 < r/100 < 0.1
0< r < 10
So it seems r is less than 10 but it may be less than 8 or greater than 8. So (2) is insufficient.

Thanks
Prashant

[RonPurewal]

For statement (2) following method can be useful:

(1 + r/100)^2 > 1.15
Now we know that 1.1^2 = 1.21.
So sqrt(1.15) falls between 1 and 1.1 (No need to compute the actual sqrt of 1.15)

au contraire, you do need to estimate this square root -- at least, you have to figure out whether it is greater or less than 1.08.
if the square root is greater than 1.08, then the statement is sufficient. if it's less, the statement is insufficient.

1< (1 + r/100) < 1.1
0 < r/100 < 0.1
0< r < 10
So it seems r is less than 10 but it may be less than 8 or greater than 8. So (2) is insufficient.

Thanks
Prashant

no, that doesn't work, although you get lucky and accidentally get the right answer with these particular numbers.
i don't know where you are getting either side of this inequality, actually; the correct inequality is (1 + r/100) > √1.15, which has basically no relation to what you've written here.

[ghong14]

It seems that the key to statement to is recognizing that the square root of 1.15 is 1.07. Any suggestions on what is the easiest way to know that other than calculating. Because I can estimate that the square root of 1.15 is between 1 and 2 but down to .07. Not sure how to do that.