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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Official Answer

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.

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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."

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GMAT Data Sufficiency (DS) Questions

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