Quantitative Reasoning: Arithmetic and Algebra

What you'll learn

Apply core arithmetic operations -- percentages, ratios, exponents -- accurately under time pressure
Translate a word problem into an algebraic equation carefully
Recognize common algebra traps, including sign errors when distributing a negative across parentheses

Explanation

Quantitative Reasoning content spans four broad areas across its two sections: arithmetic, algebra, geometry, and data analysis. Arithmetic and algebra together are the most frequently tested combination, covering percentages, ratios and proportions, exponents and roots, and translating word problems into solvable equations.

One of the most common error sources is confusing percentage points with percent change. Going from 40% to 50% is a 10 percentage-point increase, but it is a 25% relative increase, because the change (10) is measured against the original value (40): 10/40 = 0.25. Treating these two ideas as interchangeable produces a wrong answer even when every arithmetic step is otherwise correct.

Consider this original worked example: a store raises an item's price by 20%, then later lowers the new price by 20%. Is the final price higher, lower, or the same as the original? Multiplying the original price by 1.20 (the increase) and then by 0.80 (the decrease) gives 1.20 x 0.80 = 0.96 -- the final price is 4% lower than the original, not the same, because the second 20% is taken from a larger number than the first 20% was.

A separate, purely algebraic trap is failing to distribute a negative sign correctly across every term inside parentheses: -(x - 5) correctly simplifies to -x + 5, not -x - 5. Under time pressure, this sign error is easy to make and easy to avoid once you know to watch for it specifically.

Common mistakes

Confusing 'percentage points' with 'percent change' -- for example, treating a rise from 40% to 50% as a 50% increase instead of a 10-percentage-point (25% relative) increase.
Forgetting to distribute a negative sign across every term inside parentheses, writing -(x - 5) as -x - 5 instead of the correct -x + 5.
Solving only part of a multi-step word problem, such as an intermediate value, and reporting it instead of the specific quantity the question actually asks for.

Knowledge check

1. A value rises from 40% to 50%. Which statement correctly distinguishes percentage points from percent change?

This is both a 50% increase and a 50 percentage-point increaseThis is a 10 percentage-point increase, which corresponds to a 25% relative increaseThis is a 10% increase and a 40 percentage-point increasePercentage points and percent change always describe the exact same quantity

2. An item's price increases by 20%, and the new price then decreases by 20%. Compared with the original price, the final price is:

Exactly the same as the original price4% lower than the original price4% higher than the original price40% lower than the original price

3. What is the correct simplification of -(x - 5)?

-x - 5-x + 5x - 5x + 5

4. When finishing a multi-step word problem, what is the most reliable final check?

Confirm that your final answer is the specific quantity the question actually asks for, not just an intermediate value you computed along the wayConfirm your final answer is a whole numberConfirm your final answer is positiveConfirm you used every number given anywhere in the problem

Takeaway

Separate percentage points from percent change, watch for sign-distribution errors, and always confirm your final answer is the exact quantity the question asks for.

Summary

Arithmetic and algebra questions reward careful translation of word problems into equations and awareness of common traps: percentage-point versus percent-change confusion, and negative-sign distribution errors.

References

ETS GRE Quantitative Reasoning

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