PSAT/NMSQT Math Practice Problems: Variables & Expressions
Variables and expressions questions on the PSAT/NMSQT Math section test one core skill above all else: translating an English sentence into a correct algebraic expression. The exam deliberately uses precise but easy-to-mix-up phrasing — "the product of A and the sum of B and C" is not the same as "the sum of the product of A and B, and C" — and choosing the wrong order of operations or grouping is the single most common way students lose points on this topic, even when they understand the underlying math perfectly well.
The 6 problems below cover the full range of how this skill appears on the real exam, from single-step translations to multi-part expressions involving fractions, powers, and real-world word problems. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. The most reliable way through these problems is to translate one phrase at a time, working from the innermost operation outward, and to add parentheses defensively any time a phrase like "decreased by the sum of" or "the square of the sum" appears — that's almost always a signal that grouping matters. For unlimited additional practice, our free PSAT/NMSQT Math QBank has over 2,500 more problems across every PSAT/NMSQT Math topic, each with instant explanations.
1Translating a Verbal Expression
Twice the product of m and n decreased by the square of the sum of m and n. Which of the following is an expression for the statement above?
A) 2mn − (m² + n²)
B) 2mn − (m + n)²
C) (m + n)² − 2mn
D) (m² + n²) − 2mn
Answer Explanation
Break the sentence into pieces. "Twice the product of m and n" is 2mn. "The square of the sum of m and n" is (m + n)² — the sum m + n is squared as a whole, not m² + n².
"Decreased by" means subtract the second quantity from the first: 2mn − (m + n)². The correct answer is B.
2Translating a Verbal Expression
The product of a number x and four decreased by twelve. Which of the following is an expression for the statement above?
A) 4x + 12
B) 4(x + 12)
C) 4(x − 12)
D) 4x − 12
Answer Explanation
"The product of a number x and four" is 4x. "Decreased by twelve" means subtract 12 from that entire product, not from x before multiplying.
That gives 4x − 12. The correct answer is D.
3Translating a Verbal Expression
The quotient of 19 and a number d increased by seven. Which of the following is an expression for the statement above?
A) 19/d + 7
B) d/19 + 7
C) (19 + d)/7
D) (d + 7)/19
Answer Explanation
"The quotient of 19 and a number d" means 19 divided by d, so 19/d — 19 is the dividend, since it's named first in "quotient of 19 and d."
"Increased by seven" adds 7 to that entire quotient: 19/d + 7. The correct answer is A.
4Writing an Expression from a Word Problem
Mario received y text messages each minute for 10 minutes yesterday and received t text messages each minute for 20 minutes today. What is the total number of text messages he received for two days in terms of y and t?
A) 30yt
B) 200yt
C) 20y + 10t
D) 10y + 20t
Answer Explanation
Yesterday's total is the rate times the time: y messages/minute × 10 minutes = 10y. Today's total is t messages/minute × 20 minutes = 20t.
Add the two days together: 10y + 20t. The correct answer is D.
5Translating a Verbal Expression
Which of the following expressions represents the product of 3k and the sum of m and one third of n?
A) 3km + (1/3)n
B) 3k · (1/3)(m + n)
C) 3k(m + (1/3)n)
D) 3k(m + n + 1/3)
Answer Explanation
"One third of n" is (1/3)n. "The sum of m and one third of n" is m + (1/3)n — this whole sum must be grouped together, since it's what gets multiplied by 3k next.
"The product of 3k and" that sum means multiplying the entire grouped sum by 3k: 3k(m + (1/3)n). The correct answer is C.
6Writing an Expression from a Word Problem
The difference between two numbers is eight. If the smaller number is n to the third power, what is the greater number?
A) n³ − 8
B) n³ + 8
C) 8 − n³
D) 8n³
Answer Explanation
The smaller number is n³. Since the greater number minus the smaller number equals the difference (8), the greater number is the smaller number plus that difference.
That gives n³ + 8. The correct answer is B.
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