Kate invested x dollars, where X 2 1,000, for one year in a new

Question

Kate invested x dollars, where x ≥ 1,000, for one year in a new account that earned interest at an annual rate of r percent, compounded semiannually. Bob invested y dollars for one year in a new account that earned interest at an annual rate of 5 percent, compounded annually. If there were no other transactions to these accounts and if the sum consisting of the amount invested and the interest earned for the year was the same for both accounts, what was the value of r?

(1) y = 1.05x
(2) y = x + 100

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A

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gmatophobia · Accepted Answer

The value of Katie's investment after one year = x(1+\frac{r}{200})^2 The value of Bob's investment after one year = y(1+\frac{5}{100}) if the sum consisting of the amount invested and the interest earned for the year was the same for both accounts x(1+\frac{r}{200})^2 = y(1+\frac{5}{100}) Statement 1 (1) y = 1.05x x(1+\frac{r}{200})^2 = 1.05x(1+\frac{5}{100}) As x eq 0 , we can divide both sides of the equation by x (1+\frac{r}{200})^2 = 1.05(1+\frac{5}{100}) We have one unknown, in the equation. Hence, we can solve the equation to obtain the value of r . The statement alone is sufficient and we can eliminate B, C, and E. Statement 2 (2) y = x + 100 x(1+\frac{r}{200})^2 = (x+100)(1+\frac{5}{100}) We have two unknown variables in the equation x and r . Hence, we can't find a unique value of r from this equation alone. The statement alone is not sufficient to obtain the value of r . Eliminate D. Option A GMAT-Club-Forum-4lbudj7m.png

Asan92 · Answer

gmatophobia

After evaluating Statement 1, we are left with the quadratic equation with just one variable "r". However, how do we know, without solving, that the solution of "r" will not be two positive figures, which situation will imply that Statement 1 is not sufficient?

CapnSal · Answer

When you equate the two equations, you will see that the right hand side has  1.05^2 . Upon taking the square root, since the left hand side is square, only a single value of r will be possible

RSaurastri · Answer

Without solving you can tell if you’ve got a relation like in statement 1, the x’s will get cancelled in the equation and you’ll have one variable, but with statement 2, it won’t get cancelled

Usernamevisible · Answer

Yes, this is a very useful GMAT DS pattern.

When the original equation is homogeneous in x and y (both variables appear as multiplicative factors), a ratio statement often lets you cancel x or y and get a unique value.

Example:

x(1 + r/200)^2 = 1.05y

Statement (1):

y = 1.05x

Substitute:

x(1 + r/200)^2 = 1.05(1.05x)

The x cancels completely.

You are left with only r.

Therefore Statement (1) is very likely sufficient.

---

Contrast with Statement (2):

y = x + 100

Substitute:

x(1 + r/200)^2 = 1.05(x + 100)

Now x does NOT cancel because of the +100.

You still have both x and r.

Not sufficient.

---

GMAT shortcut:

If the main equation looks like

Ax = By

and a statement gives

y = kx

expect x to cancel and a unique value to emerge.

If instead the statement gives

y = x + c

or

y = x - c

the variables usually do not cancel, so sufficiency is much less likely.

This is not a theorem, but it is an excellent first-check pattern for DS. In many GMAT questions, "ratio relationship → cancellation → sufficient" and "difference relationship → no cancellation → not sufficient."

RobertVericPrep · Answer

Start off by writing two equations, one for Kate and one for Bob. Then we’ll ask, “What is the question behind the question? What do we really need to know in order to answer this question?”

We'll actually start with the value of Bob's investment after one year because that's a little simpler. The value of Bob's investment = y * (1 + 5/100), which is the same as y * (1.05)

The formula to find the value of Kate's investment is a little more interesting because her interest is compounded semiannually. How do you handle that? It means after six months you don't multiply x by (1 + r/100); you multiply x by (1 + r/200). You cut the annual interest rate in half, but THEN six months later you multiply by (1 + r/200) again. In other words, the value of Kate's investment after one year is equal to: x * (1 + r/200)^2

That's how you handle interest that is compounded semiannually.

Next, we're told that the amount each person invested and the interest earned for the year was the same for both accounts. That means the total value of Kate's investment after one year was equal to the total value of Bob's investment after one year. Let's say that the value of each person's investment after one year = Total.

Our equations are:

Bob: Total = y * (1.05) 
Kate: Total = x * (1 + r/200)^2

The question asks, “What was the value of r?” We'll set those equations equal to each other:

x * (1 + r/200)^2 = y * (1.05)

To find the value of r, first divide both sides of the equation by x:

(1 + r/200)^2 = (y/x) * 1.05

We could simplify this further to isolate r itself, but the point is that if you want to find r, the QUESTION BEHIND THE QUESTION — what we really need to know to answer this question — is simply: “What is y/x?”

That's exactly what we need to know in order to answer this question.

Statement 1:

y = 1.05x

Can we find out what y/x is? Yes, divide both sides by x and you get y/x = 1.05.

SUFFICIENT

Statement 2:

y = x + 100

Can we find out what y/x is? No. We can't isolate a single value for the ratio without knowing the actual values of x or y.

INSUFFICIENT

Answer A

Kate invested x dollars, where X 2 1,000, for one year in a new

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