If x dollars are placed in a savings account that earns y% annual inte

Question

If x dollars are placed in a savings account that earns y% annual interest compounded quarterly. What is the balance at the end of the year?

(1) If the x dollars had been in the account for 1 year longer, the amount earned in interest would have been $60 greater.
(2) If 2x dollars had been in the account for 18 months, the interest would have been $75 greater.

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nick1816 · Answer

Let P is the balance at the end of the year

P= x(1+\frac{y}{400})^4

We need to figure out both x and y in order to figure out balance at the end of the year.

Statement 1-

x(1+\frac{y}{400})^8- x(1+\frac{y}{400})^4= 60

we have 2 variables and 1 equation

Insufficient

Statement 2-

2x(1+\frac{y}{400})^{18/3}- x(1+\frac{y}{400})^4= 75

we have 2 variables and 1 equation

Insufficient

Combining 2 statements, we have 2 variables and 2 equations; hence, we can figure out x and y.

Sufficient

ask1 · Answer

nick1816  Can you explain how the 18/3 came in solving the 2nd statement?

IanStewart · Answer

There are a few issues with a problem set up this way:

- if you actually solve the problem, using both statements, you discover that the amount invested is some infinite decimal, close to $10.46 (and the annual interest rate turns out to be roughly 124%). You can't invest $10.46209... in any kind of account, so the numbers don't make sense;

- if you write out the equations that result from each Statement, and consider both equations together, you might think "I have two equations in two unknowns, so I can solve". But they're not linear equations. When you have two non-linear equations in two unknowns, a lot of things can happen, and you have to do some work to figure out if you have zero solutions, one solution, or many solutions. In fact, as you'd expect from equations resembling quadratics, you do have two solutions here if you solve purely algebraically, and it's only because one solution is negative that we can answer the question.

- one of the worst DS strategies a higher-level test taker can use on the GMAT is "counting equations, counting unknowns". A test taker who simply counts equations and unknowns, and guesses information is sufficient when there are at least as many equations as unknowns, will get roughly 2/3 of medium and hard questions wrong on the actual GMAT (anyone can confirm that just by trying that strategy on DS questions in the last half of any OG). But a question designed like this one rewards the test taker who uses this gimmicky 'strategy' with no mathematical basis (that test taker will pick C quite quickly) and punishes the test taker who recognizes that math is more complicated than that, and that you need to do a lot more than count equations when equations are non-linear. Real GMAT questions never punish test takers who know math well and reward those who don't, so the structure of the problem is unrealistic.

Anyway, there's very possibly an easier way to do this, but if we think of the interest rate y as a decimal instead of as a percentage (so we don't have to divide by 100), then the investment is worth:

x (1 + y/4)^4

after one year. We can call that "S". It's also convenient to let k = (1 + y/4)^2. Then Statement 1 tells us

S*k^2 - S = 60
S(k^2 - 1) = 60

and Statement 2 tells us

2S*k - S = 75
S(2k - 1) = 75

and now dividing the first equation by the second, we find

(k^2 - 1)/(2k - 1) = 60/75

and cross-multiplying and rearranging, 5k^2 - 8k - 1 = 0. That's a quadratic with two solutions, but because the product of the roots is negative, only one solution is positive. So there is only one positive value for k, and with the value of k we can find S from the equations above.

But it's not a realistic question for a few reasons.

ask1 · Answer

IanStewart

Very grateful for this detailed insight. Im better equipped to tackle such problems now.
But may I know how you have factored the 18 months part in the second statement? I dont see it reflected in your expression of K.

Because 18 months would mean 4 for every 3 months for 18 months. That equals 6 and not 4.

IanStewart · Answer

Yes, I probably should have provided a bit more detail about that. If you invest $x at a y% annual interest rate, compounded quarterly, then after one year the investment is worth

$ x (1 + y/4)^4

where I'm treating y as a decimal, not as a percentage, so I don't need to divide by 100. If, as Statement 2 describes, you invest $2x for 18 months at a y% annual interest rate compounded quarterly, you're investing at a y/4 percent interest rate for 6 quarters, so the investment will be worth

$ 2x (1 + y/4)^6

Notice this is equal to

(2)   (1 + y/4)^2

and because in my solution above I let S = (x)(1 + y/4)^4, and I let k = (1 + y/4)^2, this is equal to

2*S*k

I find these kinds of substitutions useful, because the equations look like a bit of a mess if you leave them in terms of x and y. I hope that answers your question!

ask1 · Answer

Excellent! It's all clear now

Thanks again!

nick1816 · Answer

IanStewart   y^4 =k  will give 1 positive root, 1 negative and 2 imaginary. 
Hence, counting number of equations and variables gonna work in this case, as we're interested in only +ve values. Tho i shoulda explain it further.

Your approach is better tho as always.

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