"Which of the Following Ratios Is Most Nearly Equal to the Ratio 1 + √5 to 2?..." — GMAT® Worked Solution

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

Which of the following ratios is most nearly equal to the ratio $1 + \sqrt{5}$ to $2$?

(A) 8 to 5

(B) 6 to 5

(C) 5 to 4

(D) 2 to 1

(E) 1 to 1

Try it before reading on.

Step 1: Memorize the Key Square Root

The square root of 5 is not an integer. It's a long, non-terminating decimal that you can't compute without a calculator. So memorize it:

$\sqrt{5} \approx 2.2$

This is the level of precision you need for almost every GMAT® estimation problem. When a problem says "most nearly equal," it's inviting you to estimate, and 2.2 is close enough to distinguish between the answer choices.

The three square roots worth memorizing before test day: $\sqrt{2} \approx 1.4$, $\sqrt{3} \approx 1.7$, $\sqrt{5} \approx 2.2$.

Step 2: Set Up the Ratio as a Fraction

Express the given ratio as a fraction so you can compute its decimal value:

$\frac{1 + \sqrt{5}}{2} \approx \frac{1 + 2.2}{2} = \frac{3.2}{2} = 1.6$

The ratio $1 + \sqrt{5}$ to $2$ is approximately 1.6.

Step 3: Convert Each Answer Choice to a Decimal

Now convert each answer ratio to a decimal and compare:

(A) $\dfrac{8}{5} = 1.6$

(B) $\dfrac{6}{5} = 1.2$

(C) $\dfrac{5}{4} = 1.25$

(D) $\dfrac{2}{1} = 2$

(E) $\dfrac{1}{1} = 1$

The answer is (A). The ratio 8 to 5 equals 1.6, which matches our estimate exactly.

The Shortcut: Memorized Decimal Equivalents

You can save time on Step 3 if you've memorized the decimal equivalents of common fractions. Knowing that $\frac{3}{5} = 0.6$ means you can instantly evaluate $\frac{8}{5}$ as $1 + \frac{3}{5} = 1.6$ without long division.

The fractions worth memorizing: thirds, fourths, fifths, eighths, ninths, and elevenths. Knowing these cold speeds up ratio comparisons and helps you recognize equivalent forms across different problems. The investment is small (a few flashcard sessions) and the payoff compounds across the quant section.

Why This Problem Matters

About 28% of test takers miss this problem, making it a medium-difficulty question. That's a significant jump from the first two problems in this episode, and the difficulty comes from combining several concepts that are each manageable on their own: ratios, square roots, estimation, and decimal conversion.

The problem doesn't test any single hard concept. It tests whether you can string together several easy ones without making a careless error at any step. That's the pattern for most medium-difficulty GMAT® quant questions: they combine 3-4 elementary concepts in a way that creates friction.

The fix is having systems for each sub-skill. If you have $\sqrt{5}$ memorized, you don't lose time or confidence on that step. If you have decimal equivalents memorized, you don't make a long-division error on the comparison step. Each piece is easy. The system makes the whole thing easy.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Algebra, Compound Interest, and Estimation

From Episode 22 of Real GMAT® Problems (The GMAT® Strategy Podcast).