"What Is the Value of (29² + 29)/29?" — GMAT® Worked Solution

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

What is the value of $\dfrac{29^2 + 29}{29}$?

(A) $870$

(B) $841$

(C) $58$

(D) $31$

(E) $30$

Try it before reading on.

Step 1: The Denominator Divides the Whole Numerator

The numerator is a sum, and the denominator divides the entire thing, not part of it. That fact drives every legal route through this problem, because the tempting move (canceling the 29 against the $29^2$ alone) changes the expression's value. The routes that work are the ones that reorganize the sum before anything gets canceled.

Step 2: Split the Fraction

One route will look familiar in reverse: combining two fractions that share a denominator is standard school math, and running the combination backwards is also legal, even though coursework rarely asks anyone to unsimplify. The fraction splits into two:

$\frac{29^2 + 29}{29} = \frac{29^2}{29} + \frac{29}{29}$

Each piece simplifies on its own:

$\frac{29^2}{29} = 29 \qquad \frac{29}{29} = 1$

Adding them gives $29 + 1 = 30$.

Step 3: Or Factor the Numerator

The second route factors the shared 29 out of both terms in the sum:

$29^2 + 29 = 29(29 + 1)$

The fraction becomes

$\frac{29(29 + 1)}{29} = 29 + 1 = 30$

and the 29s cancel cleanly, because now the numerator is a product rather than a sum.

The answer is (E).

Step 4: Where the Wrong Answer Comes From

Choice (C) is $58$, which is $29 + 29$: the result of canceling the denominator's 29 against the $29^2$ while leaving the second 29 untouched. About 4% of test takers pick it.

A denominator can't cancel through addition one term at a time. Canceling is division, and division reaches every term of the sum, which is what the split in Step 2 enforces and what the factoring in Step 3 sidesteps by converting the sum into a product. Choices (A) and (B) are pieces of the computation rather than traps in their own right: $870$ is the numerator added out, and $841$ is $29^2$ by itself.

Why This Problem Matters

The knowledge required here is a middle school fraction, and the question still finds a way to test something else: whether canceling gets applied where it isn't legal. That makes this a permission question as much as a math question, and the exam presents the setup often enough that the split-the-fraction move is worth having ready.

Both approaches are worth knowing, since they shine in different situations. Favoring one is fine, and staying familiar with the other is the flexible position, because a factored numerator unlocks far more than this fraction does.

Ready for the next one? "If x = 1 - 3t and y = 2t - 1, for What Value of t Does x = y?" — GMAT® Worked Solution.

Want the full strategy? Read: GMAT® Execution Habits: The Scratch-Work System for Easy Questions

From Episode 12 of Real GMAT® Problems (The GMAT® Strategy Podcast).