Number Manipulation Prompts: Translating 'What Number Is...' Items
Key Takeaways
- Number Manipulation is about 17 of the 52 Quantitative Skills questions, and nearly all of them are one-sentence word prompts that must be translated into a short equation.
- The phrase 'of' signals multiplication and 'is' signals equals, so 'what number is 3 more than 1/2 of 18?' becomes n = (1/2)(18) + 3.
- Prompts built on 'subtracted from' and 'less than' reverse the written order, which is the single most common translation error on the subtest.
- Working from the inside out - resolve the fraction or percent first, then apply the additive phrase - keeps multi-clause prompts to a single pass.
Number Manipulation Prompts: Translating 'What Number Is...' Items
Number Manipulation supplies roughly 17 of the 52 Quantitative Skills questions, and its signature item is a single spoken-sounding sentence:
What number is 3 more than one-half of 18?
There is no hard arithmetic in that sentence. The entire difficulty is translation: turning English word order into correct symbol order under a 35-second clock. Students lose these points to reversed phrases and misplaced operations, not to computation.
The Keyword Map
| English Phrase | Symbol | Worked Fragment |
|---|---|---|
| is, equals, gives | = | what number is 12 becomes n = 12 |
| of | multiply | 1/3 of 27 becomes (1/3)(27) |
| more than, increased by, added to | + | 5 more than 20 becomes 20 + 5 |
| less than, decreased by | - (reversed) | 5 less than 20 becomes 20 - 5 |
| subtracted from | - (reversed) | 5 subtracted from 20 becomes 20 - 5 |
| the product of | multiply | the product of 6 and 4 becomes 6 x 4 |
| the quotient of | divide | the quotient of 20 and 4 becomes 20 / 4 |
| twice, doubled | x 2 | twice 14 becomes 28 |
| the square of | raise to the second power | the square of 7 becomes 49 |
The reversal rule, stated once and memorized forever: less than and subtracted from both put the second number first. "6 less than 20" is 20 minus 6, never 6 minus 20. Roughly one Number Manipulation item per test is built purely on this reversal.
The Inside-Out Method
Multi-clause prompts nest one operation inside another. Resolve the innermost quantity first, then work outward.
Worked Example 1
What number is 4 more than one-third of 27?
- Innermost quantity: one-third of 27 = 9.
- Outer phrase: 4 more than 9 = 13.
Worked Example 2
What number subtracted from 30 leaves 5 more than 12?
- Resolve the right side first: 5 more than 12 = 17.
- Translate the left side, honoring the reversal: 30 minus n = 17.
- Solve: n = 30 minus 17 = 13.
Worked Example 3
What number is 20 percent of one-half of 80?
- One-half of 80 = 40.
- 20 percent of 40 = 0.20 times 40 = 8.
Worked Example 4
Six times a number, decreased by 8, equals 34. What is the number?
- Translate directly: 6n - 8 = 34.
- Add 8: 6n = 42.
- Divide: n = 7.
Percent and Fraction Prompts Without a Calculator
Number Manipulation items lean on benchmark values because no calculator is allowed. Commit these to instant recall.
| Fraction | Percent | Fast Mental Move |
|---|---|---|
| 1/2 | 50% | Halve it |
| 1/3 | 33 1/3% | Divide by 3 |
| 1/4 | 25% | Halve it twice |
| 1/5 | 20% | Divide by 10, double |
| 1/8 | 12.5% | Halve it three times |
| 2/3 | 66 2/3% | Divide by 3, double |
| 3/4 | 75% | Halve it, then add half again |
Two additional shortcuts pay for themselves repeatedly:
- Percent commutativity: "18 percent of 50" equals "50 percent of 18" - which is just 9. When one factor is awkward and the other is friendly, swap them.
- Build from 10 percent: 10 percent of 240 is 24, so 30 percent is 72 and 5 percent is 12. Almost any percent decomposes into tenths and halves of tenths.
Back-Solving From the Choices
Because every HSPT item is multiple choice, the answer is already printed. When translation feels shaky, test the choices against the sentence instead.
What number decreased by 9 equals one-third of 42? Choices: 5, 14, 21, 23
- Resolve the right side: one-third of 42 = 14.
- Test choice 23: 23 minus 9 = 14. Match.
Back-solving costs a little more time than direct translation but is nearly error-proof, so use it whenever a prompt contains two reversal words or a nested fraction you do not trust yourself to order correctly.
Property-Recognition Items
A minority of Number Manipulation questions ask about the structure of an operation rather than its value: which property justifies a rewriting, or whether an operation is commutative. Keep these four straight.
| Property | Statement | Applies To |
|---|---|---|
| Commutative | a + b = b + a; ab = ba | Addition and multiplication only |
| Associative | (a + b) + c = a + (b + c) | Addition and multiplication only |
| Distributive | a(b + c) = ab + ac | Multiplication over addition or subtraction |
| Identity | a + 0 = a; a x 1 = a | Zero for addition, one for multiplication |
Subtraction and division are neither commutative nor associative - 10 minus 4 is not 4 minus 10, and (16 divided by 4) divided by 2 is not 16 divided by (4 divided by 2). Items testing this present a plausible-looking rearrangement and ask whether it is valid.
Common Traps
- Order reversal. Treating "7 less than n" as 7 minus n.
- Percent of the wrong base. In "what number is 25 percent more than 40," the base is 40, so the answer is 50 - not 25 percent of some other quantity.
- Stopping one step early. In "what number is 5 more than twice 9," the answer is 23, not 18. Read to the end of the sentence before computing.
- Fraction of a fraction. "One-half of one-third of 36" is 6, because you take one-third first (12) and then halve it.
What number is 7 more than one-fourth of 32?
What number subtracted from 45 leaves 6 more than 21?
Which rearrangement is NOT justified by a property of arithmetic?
What number is 20 percent of one-half of 90?