"If 1.5/(0.2 + x) = 5, Then x = ?" — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

If $\dfrac{1.5}{0.2 + x} = 5$, then $x =$

(A) $-3.7$

(B) $0.1$

(C) $0.3$

(D) $0.5$

(E) $2.8$

Try it before reading on.

Step 1: Clear the Denominator

When an equation has a denominator, clearing it is usually the right first move, and the way to do it is multiplying both sides by whatever is in the denominator. The fraction here is $\frac{1.5}{0.2 + x}$, so the multiplier is the whole quantity $0.2 + x$.

The habit worth building has two parts: parentheses around what you're multiplying by, and parentheses around what's being multiplied. Written out:

$1.5 = (5)(0.2 + x)$

The parentheses around the 5 can look like overkill. On the left side, 1.5 stands alone, so nothing can go wrong there. The point of wrapping both sides anyway is that the habit then runs on every equation without a decision, and on a later question, the side where it matters will be the side you didn't feel the need to check.

Step 2: Distribute on the Right

The right side needs the 5 distributed to both terms:

$5(0.2 + x) = 5 \times 0.2 + 5x = 1 + 5x$

That step is where this question sets its trap, which we'll come back to after the solution. The multiplication $5 \times 0.2 = 1$ is one of the fraction equivalences worth having memorized: 0.2 is $\frac{1}{5}$, and five of those fifths make 1.

So the equation reads:

$1.5 = 1 + 5x$

Step 3: Isolate x

Subtract 1 from both sides:

$0.5 = 5x$

Then divide both sides by 5. Three routes, and any of them is fine: decimal long division, seeing that $\frac{0.5}{5} = 0.1$ directly, or converting to a fraction first, $\frac{1}{2} \times \frac{1}{5} = \frac{1}{10} = 0.1$.

$x = 0.1$

The answer is (B).

The Trap: 0.5 Without Distribution

The mistake runs like this. Both sides get multiplied by $0.2 + x$, the left side cancels down to 1.5, and on the right side the 5 multiplies the 0.2 but never reaches the $x$:

$1.5 = 1 + x$

Subtracting 1 gives $x = 0.5$, which is choice (D), and about 7% of test takers pick it. The wrong answer isn't random; it's the correct solution with one distribution missing, which is what makes it easy to produce and easy to believe.

Two things keep that error out of your test. The first is the parentheses habit from Step 1, which makes the distribution nearly impossible to skip, because $1.5 = (5)(0.2 + x)$ visibly has two terms waiting on the right. The second is a rule about practice: making the mistake once while learning is how anyone learns, and repeating it is the pattern worth attacking. An error that shows up once in practice and never again costs nothing. An error that recurs on questions you know how to do is the difference between the quant score you want and the one you settle for.

Why This Problem Matters

On the surface this is a routine algebra question, and that's what makes it useful. The skills it tests, clearing denominators and distributing, are supposed to be automatic by test day, and slips on questions like this one are usually process problems rather than knowledge problems. The fix is a written habit rather than more review.

It's also a question worth solving two ways during review. The route above clears the denominator first; another valid route divides both sides by 5 first, reads $\frac{1.5}{5} = 0.3$, and gets $0.2 + x = 0.3$ immediately. Whichever feels more natural is the one to pre-commit to, and testing both on a question this short is a cheap way to find out which one that is.

Ready for the next one? "If a Positive Integer n Is Divisible by Both 5 and 7..." — GMAT® Worked Solution.

Want the full strategy? Read: GMAT® Percents, Divisibility, and Quadratics: Translate First, Solve Second

From Episode 11 of Real GMAT® Problems (The GMAT® Strategy Podcast).