What the question looks like
Number manipulation questions are the wordiest block in the Quantitative Skills section. Instead of a series or a set of expressions, each one is a single sentence that describes a number, followed by four choices. The two most common shapes are:
What number is 7 more than one third of 27?
What number, when divided by 4, leaves 3 less than 10?
The first shape asks you to build a number from pieces. The second shape describes what happens to an unknown number and asks you to work backward to find it. Both use small whole numbers, simple fractions, or easy percents, and both can be done in your head or with one line of scratch work.
How to solve it
- Find the anchor number. It is usually the last number in the sentence, like 27 or 10. Start there, not at the beginning of the sentence.
- Apply the operation attached to it. One third of 27 is 9. Three less than 10 is 7. This gives you the inside value.
- Apply the outer operation. Seven more than 9 is 16. If the sentence says when divided by 4, undo that by multiplying: 7 × 4 = 28.
- Reread the sentence with your answer plugged in. Does 28 divided by 4 leave 7, and is 7 three less than 10? Yes, so 28 is right.
- Match your number to a choice. If it is not there, you most likely did the steps in the wrong order.
Key phrases to translate on sight: more than means add, less than means subtract, of means multiply, and when divided by or when multiplied by means undo that operation to find the starting number.
The reversed form, such as one fourth of what number is 15, or 20 percent of what number is 12, asks for the starting number. Undo the operation: if a fourth of the number is 15, the number is 4 times 15, which is 60; if 20 percent of it is 12, the number is 12 divided by 0.2, which is 60. A quick check is that the wrong choices are usually the forward computation, like a fourth of 15, so if your answer is smaller than the number in the stem, reread it.
Two more habits from the answer choices. If a product cannot end the way a choice ends, cross that choice out without finishing the multiplication; a whole number times a whole number cannot produce a decimal. And when you must guess, the pair of choices that are numerically close to each other usually contains the answer, because one of them is the trap for a one-step slip.
Worked examples
Example 1
What number is 8 more than one fourth of 36?
- A.11
- B.17
- C.19
- D.44
Show answer
Answer: B: 17
Start with the anchor: one fourth of 36 is 36 ÷ 4 = 9. Then add 8 to get 17. Choice A, 11, comes from adding first and dividing second, 36 + 8 = 44 and 44 ÷ 4 = 11, which reverses the order the sentence describes. Choice D, 44, is the same mistake with the division left out.
Example 2
What number, when multiplied by 5, gives 3 more than 42?
- A.8
- B.15
- C.45
- D.9
Show answer
Answer: D: 9
Find the target first: 3 more than 42 is 45. The unknown number times 5 equals 45, so divide: 45 ÷ 5 = 9. Choice C, 45, is the target itself and tempts students who stop before the last step. Choice A, 8, is close to right and catches anyone who estimates instead of dividing.
Example 3
What number is 3 less than 20% of 150?
- A.27
- B.33
- C.30
- D.12
Show answer
Answer: A: 27
Twenty percent of 150 is 150 ÷ 5 = 30. Three less than 30 is 27. Choice B, 33, is what you get if you add 3 instead of subtracting, and choice C, 30, is the inside value with the final step skipped. The word less always means the answer will be smaller than the inside value.
Example 4
What number, when divided by 6, leaves 2 less than 7?
- A.5
- B.42
- C.30
- D.54
Show answer
Answer: C: 30
Two less than 7 is 5, so the unknown number divided by 6 must equal 5. Undo the division by multiplying: 5 × 6 = 30. Check it: 30 ÷ 6 = 5, and 5 is 2 less than 7. Choice A, 5, is the inside value and tempts anyone who forgets the final multiply. Choice D, 54, comes from adding 2 to 7 instead of subtracting, then multiplying 9 × 6.
Traps to watch for
- Working left to right. The sentence 8 more than one fourth of 36 is written with the addition first, but the math needs the division first. Always find the anchor number and work outward.
- Stopping one step early. The inside value, like 9 or 30, is almost always one of the wrong choices. Make sure you have applied every operation the sentence names.
- Flipping less than and more than. When you are moving fast, 3 less than 30 can become 33. Say the phrase in your head as subtract 3 from 30.
- Confusing of with more than. One third of 27 means multiply by 1/3. One third more than 27 would mean add a third of 27 to 27. Read the small words carefully.
Pacing
At about 35 seconds each, these questions leave you enough time to reread the sentence once after you compute. If a sentence is confusing you after two readings, plug the choices in one at a time; with only four options, checking them directly is often faster than untangling the wording. And because the HSPT has no guessing penalty, pick something and move on rather than leave it blank.
Practice
Test your speed on number manipulation and the rest of the Quantitative section in our free diagnostic.
Shaky on the skill itself? Start with the lessons on fractions and percents, then come back to the question format.