What the question looks like
In this block of the Quantitative Skills section, each question gives you three expressions instead of three pictures. They are labeled (A), (B), and (C) and stacked one above the other, like this:
(A) 3 × 8 (B) 2 to the 4th power plus 8 (C) 30 − 6
Then come four statements about how the values relate, and your job is to pick the one that is true. The statements look like:
- (A) is greater than (B) and equal to (C).
- (A), (B), and (C) are all equal.
- (B) is less than (C).
- (C) is greater than (A) and (B).
Two of the three expressions are often equal, and the wrong choices are written so that a single arithmetic slip lands you on one of them.
How to solve it
- Compute (A), (B), and (C) one at a time and write each value in the margin. Do not try to hold three numbers in your head.
- Watch order of operations. Exponents and roots first, then multiply and divide, then add and subtract. An expression like 2 to the 4th power plus 8 means 16 + 8 = 24, not 2 to the 12th power.
- Put all three values in the same form. If one is a fraction and another is a decimal, convert to decimals or to a common denominator before comparing.
- Write the order from smallest to largest, noting any ties. Then read the four choices and keep only the one that matches your order exactly.
- If no choice matches, recompute the value you were least sure about. A mismatch almost always means one expression was evaluated wrong.
Worked examples
Example 1
(A) 4 × 7 (B) 2 to the 5th power minus 4 (C) 50 − 22. Which statement is true?
- A.(A) is greater than (B).
- B.(A), (B), and (C) are all equal.
- C.(C) is less than (B).
- D.(B) is greater than (A) and (C).
Show answer
Answer: B: (A), (B), and (C) are all equal.
Compute each one: (A) is 4 × 7 = 28, (B) is 32 − 4 = 28, and (C) is 50 − 22 = 28. All three equal 28, so choice B is true. Choice D catches anyone who reads 2 to the 5th power as 2 × 5 = 10 and then gets 6, or who treats the exponent as 25 and gets 21; either way the values no longer match.
Example 2
(A) 3/4 of 40 (B) 25% of 120 (C) 0.5 × 64. Which statement is true?
- A.(A) is equal to (B) and less than (C).
- B.(A) is greater than (B).
- C.(C) is less than (A).
- D.(A), (B), and (C) are all equal.
Show answer
Answer: A: (A) is equal to (B) and less than (C).
Work each one out: (A) is 40 ÷ 4 = 10, then 10 × 3 = 30. (B) is 120 ÷ 4 = 30. (C) is half of 64, which is 32. So (A) equals (B), and both are less than (C), matching choice A. Choice D tempts students who see three answers near 30 and assume they all tie without finishing the last multiplication.
Example 3
(A) 6² (B) √49 + 30 (C) 5 × 7. Which statement is true?
- A.(A) is greater than (B) and (C).
- B.(C) is greater than (A).
- C.(B) is greater than (A), and (A) is greater than (C).
- D.(A) and (C) are equal.
Show answer
Answer: C: (B) is greater than (A), and (A) is greater than (C).
Evaluate: (A) is 6 × 6 = 36, (B) is 7 + 30 = 37, and (C) is 35. Ordered from largest to smallest that is (B), then (A), then (C), so choice C is true. Choice A is what you get if you read 6² as 6 × 2 = 12 for (A) and make a similar slip on (B); check that a small exponent means repeated multiplication, not multiplication by the exponent.
Example 4
(A) 2/3 (B) 5/8 (C) 0.65. Which statement is true?
- A.(B) is greater than (C).
- B.(A) and (C) are equal.
- C.(C) is greater than (A).
- D.(A) is greater than (C), and (C) is greater than (B).
Show answer
Answer: D: (A) is greater than (C), and (C) is greater than (B).
Convert everything to decimals: (A) 2/3 is about 0.667, (B) 5/8 is exactly 0.625, and (C) is 0.65. The order from largest to smallest is (A), (C), (B), so choice D is true. Choice C is the trap for students who round 2/3 down to 0.6 in their heads; two thirds is more than 0.65, not less.
Shortcuts that skip the arithmetic
The method above always works. These shortcuts get you the same answer faster, and on a section that gives you about 35 seconds a question, faster is the point.
- Compare to one half. A fraction is more than one half when the numerator is more than half the denominator. 3/7 is below one half because 3 is less than 3.5; 5/9 is above because 5 is more than 4.5. If two fractions land on different sides of one half, you are done without touching a common denominator.
- Cross-multiply two fractions. To compare 3/7 and 4/9, multiply each numerator by the other denominator: 3 times 9 is 27 and 4 times 7 is 28. The larger product belongs to the larger fraction, so 4/9 is bigger. Write each product above its own fraction so you do not swap them.
- Same numerator, bigger denominator loses. 5/8, 5/9, and 5/11 are already in order from largest to smallest; the more pieces a whole is cut into, the smaller each piece.
- Slide the decimal for percents. 0.35 is 35 percent and 3/8 is 0.375, so a comparison of 0.35, 3/8, and 34 percent is really 35, 37.5, and 34. Converting everything to a percent or a decimal takes a few seconds and removes every trap.
- Take 10 percent first. 10 percent of a number is the number with the decimal moved one place. 20 percent is double that, 5 percent is half of it, and 15 percent is 10 percent plus half again. 15 percent of 60 is 6 plus 3, which is 9.
- Plug in a small number for a variable. When the quantities are expressions in x, try x equals 2 or 3 and compare the results. Avoid 0 and 1, which make expressions tie that are not equal in general. If the stem says x is greater than 2, choose 3 and, if two quantities come out close, check 10 as well.
- Read scientific notation as a plain number. 3.2 times 10 to the third is 3,200. Write the plain number under each quantity; once the powers of ten are gone, the comparison is ordinary.
- Do not fully compute a distributive expression. 3(4 + 5) and 3 times 4 plus 3 times 5 are the same number by the distributive property. Recognizing that saves the arithmetic and avoids the classic trap, 3 times 4 plus 5, which forgets to distribute.
Traps to watch for
- Exponent confusion. 2 to the 4th power is 16, not 8. 6² is 36, not 12. If a choice matches the wrong reading, it was put there on purpose.
- Skipping order of operations. In an expression like 3 + 4 × 5, multiply first to get 23. Reading left to right gives 35, which will usually appear among the wrong choices.
- Comparing mixed forms. A fraction, a percent, and a decimal cannot be compared by looking at them. Convert first, even if it costs five seconds.
- Answering before computing (C). Many students find (A) and (B) equal and grab the first choice that says so. The third expression can still break the tie or change the order.
Pacing
Plan on about 35 seconds each, which is enough to compute three simple expressions if you write them down. If one expression is taking too long, estimate it, pick the choice that fits your estimate, and move on. There is no guessing penalty on the HSPT, so an answered question is always better than a skipped one.
Practice
Work a timed mix of comparison questions and the rest of the Quantitative section in our free diagnostic.
Shaky on the skill itself? Start with the lessons on fractions, decimals, percents, and exponents and roots, then come back to the question format.