Real GMAT® Problems Ep. 15: Roots and Ratios

What This Episode Covers

Three real GMAT® problems from the 11th edition of the Official Guide for GMAT® Review, walking through exponent rules, ratio translation, and converting percentages to ratios. Isaac covers why you can't distribute exponents over addition (and why that mistake is so tempting), how to translate "three times as many A as B" into algebraic variables, and why the habit of writing what's given and what's asked is the single biggest defense against missing questions you know how to do.

The problems increase in organizational complexity — the math itself isn't hard, but the habits that prevent careless errors are what separate a solid score from a disappointing one.

Problems Covered

(√7 + √7)² — A computation problem testing PEMDAS and exponent rules. The key move is combining like terms inside the parentheses first (√7 + √7 = 2√7), then squaring. About 5% of test takers pick (E) 14 by incorrectly distributing the exponent across addition — √7² + √7² = 7 + 7 = 14. That's mathematically illegal and the episode explains why with a numerical example.
Ratio of people 21 or under to total population — A word problem involving ratio translation. "Three times as many people aged 21 or under as there are people over 21" means if X is the over-21 group, 3X is the under-21 group, and the total is 4X. The ratio of under-21 to total is 3:4. About 11% of test takers pick (C) 1:4 — the inverse ratio (over-21 to total) — by not reading carefully what's being asked.
Kelly and Chris packing boxes — ratio from percentages — Chris packed 60% of boxes, Kelly packed the rest. What's the ratio of Kelly's boxes to Chris's boxes? The answer is 2:3 (40:60 reduced). Isaac walks through both the intuitive shortcut and the methodical substitution approach, and discusses when each is appropriate based on your personal error rate.

Key Takeaways

You can't distribute exponents over addition or subtraction. (√7 + √7)² is not the same as √7² + √7². The PEMDAS order exists for a reason — parentheses before exponents. Combine like terms inside the parentheses first, then apply the exponent.

"Three times as many A as B" means A = 3B. For every B, there are three A's. Create a variable for the smaller group (B = X) and express the larger group in terms of it (A = 3X). This pattern shows up constantly in GMAT® word problems.

Write what's given and what's asked before you start solving. On Problem 2, 11% of test takers solved for the wrong ratio — over-21 to total instead of under-21 to total. Writing both quantities down before you calculate eliminates this class of error entirely.

Your error rate should determine your method. If you never make careless errors, the intuitive shortcut (40% and 60% → 2:3) is fine. If you sometimes miss questions you know how to do — which is most people — the methodical approach (define variables, substitute, solve) costs a few extra seconds but dramatically reduces mistakes.

Missing easy questions hurts more than missing hard ones. The GMAT® scoring algorithm is adaptive. Dropping a question you know how to do has an outsized negative effect. The extra seconds spent writing scratch work are the cheapest insurance you can buy.

Building foundations is faster than skipping them. Isaac shares his own experience of trying to speed through GMAT® prep without solidifying math basics — his score went down by 20 points over six months. Doing it right the first time is the fastest path, even when it feels slower in the moment.

Episodes Referenced

Transcript

Read the full transcript

Welcome to the GMAT Strategy Podcast. You're here because you believe there's a better way to study for the GMAT and so do we.

We created the GMAT Strategy to maximize your results and minimize your efforts so you can get to the fun parts about business score and life as quickly as possible.

My name is Isaac Pullia and I've been teaching GMAT classes and tutoring privately for the GMAT for almost a decade and I've achieved a 99th percentile score on the GMAT and helped thousands of students get into the business schools of their choice.

If this show is bringing new value, please share it with your friends and family who are studying so that together we can make this process as easy and as painless as it can possibly be.

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Problem 1: (√7 + √7)²

Welcome to Real GMAT Problems episode 15. The problems we're going to go through are from the 11th edition of the Official Guide for GMAT Review. You can web search them if you'd like to follow along visually. The 11th edition is out of print — that's partly why we're going through these questions, so there's no overlap with the current Official Guide.

Problem 49: (√7 + √7)²

(A) 98 (B) 49 (C) 28 (D) 21 (E) 14

This is really a question about computation. PEMDAS governs all computation, so let's work with that.

First, what's inside the parentheses? We have like terms — √7 and √7. Just like x + x = 2x, √7 + √7 = 2√7.

Now we've got (2√7)², which means 2√7 × 2√7. We can rearrange: 2 × 2 × √7 × √7 = 4 × 7 = 28.

The answer is (C) 28.

About 5% of people pick (E) 14. Here's how that happens: they try to "distribute" the exponent to each term — √7² + √7² = 7 + 7 = 14. That's not mathematically valid.

You cannot distribute exponents over addition and subtraction. Think of (5 + 5)². PEMDAS says parentheses first: 5 + 5 = 10, then 10² = 100. But if you distributed: 5² + 5² = 25 + 25 = 50. That's not 100.

The lesson: if your PEMDAS and exponent foundations are shaky, this question is surprisingly easy to miss. Those skills scale to medium and hard problems.

Problem 2: Ratio of People 21 or Under to Total Population

Problem 50: In a certain population, there are three times as many people aged 21 or under as there are people over 21. The ratio of those 21 or under to the total population is:

(A) 1 to 2 (B) 1 to 3 (C) 1 to 4 (D) 2 to 3 (E) 3 to 4

The key to this problem is the pattern "three times as many A as B." When you see that, it means for every B, there are three A's. So you can create a variable for B and express A as 3B.

Let X = people over 21. Then 3X = people 21 or under. Total population = 3X + X = 4X.

The ratio of those 21 or under to total = 3X to 4X = 3:4.

The answer is (E) 3 to 4.

About 11% of people pick (C) 1:4. That's the ratio of people OVER 21 to the total population — they solved for the wrong thing. This is why the habit of writing what's given and what's asked is so important. It's not a math skills issue — it's an organizational and reasoning issue.

Problem 3: Kelly and Chris Packing Boxes

Problem 52: Kelly and Chris packed several boxes with books. If Chris packed 60% of the total number of boxes, what was the ratio of the number of boxes Kelly packed to the number of boxes Chris packed?

(A) 1 to 6 (B) 1 to 4 (C) 2 to 5 (D) 3 to 5 (E) 2 to 3

First, note that "Kelly and Chris packed several boxes" means nobody else packed any — Chris + Kelly = total.

If Chris packed 60%, Kelly packed 40%. The ratio of Kelly to Chris = 40:60 = 2:3.

The answer is (E) 2 to 3.

Isaac walks through the methodical approach too: define C + K = T, C = 60/100 × T, substitute and solve for K = 40/100 × T, then set up K/C = (40/100 × T)/(60/100 × T), cancel, reduce to 2/3.

The methodical approach takes more steps but dramatically reduces errors. If you're someone who misses questions you know how to do — even occasionally — the extra scratch work is worth it.

The bigger lesson: don't let early success reinforce bad habits. Getting problems right by doing them in your head can create the belief that you don't need to write anything down. That works until it doesn't — and on test day, the stakes are too high to find out the hard way.

If you want more help and you're not seeing the results you want with your current study plan, we offer a free call where we'll assess your situation and set you up with a plan. Head to TheGMATStrategy.com for more.

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