Solving equations questions on the PSAT/NMSQT Math section test a wide range of the same core skill: isolating an unknown through a careful, ordered sequence of algebraic steps. This shows up in three main forms — solving a literal equation for one variable in terms of the others, solving a straightforward numeric equation for a specific value, and translating a word problem into an equation before solving it. A large share of these questions also reward a shortcut: many ask for the value of a whole expression (like −n + 3, or 33 + 3a) rather than the variable itself, and setting up the algebra to solve for that expression directly is almost always faster than solving for the bare variable first.
The 22 problems below cover this full range, including six literal-equation problems where you solve for one variable in terms of several others, and sixteen numeric and word-problem equations of increasing complexity. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. For literal equations especially, treat every other variable as if it were just a number — the same steps (combining like terms, moving terms across the equals sign, dividing to isolate the target variable) apply exactly the same way. 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.
If 2x + 3y = 18, which of the following gives y in terms of x?
Isolate the y-term by subtracting 2x from both sides: 3y = 18 − 2x.
Divide every term by 3: y = 6 − (2/3)x. The correct answer is B.
If P = 2l + 2w, which of the following gives w in terms of P and l?
Isolate the term with w by subtracting 2l from both sides: 2w = P − 2l.
Divide every term by 2: w = P/2 − l. The correct answer is C.
If c = a/(a+b), which of the following gives a in terms of b and c?
Multiply both sides by (a+b) to clear the denominator: c(a+b) = a, so ca + cb = a.
Move all a-terms to one side: cb = a − ca, so cb = a(1−c). Divide by (1−c): a = cb/(1−c) = bc/(1−c). The correct answer is A.
If (ab−1)/3 = c, which of the following gives b in terms of the other variables?
Multiply both sides by 3: ab − 1 = 3c.
Add 1 to both sides: ab = 3c + 1. Divide by a: b = (3c+1)/a. The correct answer is A.
If gh − f = g − h, which of the following gives g in terms of the other variables?
Move all g-terms to one side and everything else to the other: add h to both sides and subtract g from both sides: gh − g = f − h.
Factor out g on the left: g(h−1) = f−h. Divide both sides by (h−1): g = (f−h)/(h−1). The correct answer is D.
If n = a + (k−1)d, which of the following gives k in terms of the other variables?
Subtract a from both sides: n − a = (k−1)d.
Divide both sides by d: (n−a)/d = k − 1. Add 1 to both sides: k = (n−a)/d + 1 = (n−a+d)/d. The correct answer is D.
If 7n + 3 = 2n − 12, what is the value of −n + 3?
Solve for n first: subtract 2n from both sides and subtract 3 from both sides: 5n = −15, so n = −3.
Substitute into the target expression: −(−3) + 3 = 3 + 3 = 6.
If 7(h−5) − 3h = (3/2)h, what is the value of (1/7)h?
Distribute and simplify the left side: 7h − 35 − 3h = 4h − 35, so the equation becomes 4h − 35 = (3/2)h.
Subtract (3/2)h from both sides: (5/2)h = 35. Multiply both sides by 2/5: h = 14. Then (1/7)h = 14/7 = 2.
r/3 + s/11 = 39/33. Given the above equation, if s = 2, what is the value of r?
Substitute s = 2: r/3 + 2/11 = 39/33. Simplify 39/33 to 13/11 (dividing by 3), so r/3 + 2/11 = 13/11.
Subtract 2/11 from both sides: r/3 = 11/11 = 1. Multiply both sides by 3: r = 3.
If (9−2k)/3 = k−2, what is the value of k?
Multiply both sides by 3 to clear the denominator: 9 − 2k = 3(k−2) = 3k − 6.
Add 2k and 6 to both sides: 15 = 5k. Divide by 5: k = 3.
A $48 shirt costs $22 more than one half the cost of a pair of pants. How much does the pair of pants cost?
Translate into an equation, letting p be the cost of the pants: 48 = (1/2)p + 22.
Subtract 22 from both sides: 26 = (1/2)p. Multiply both sides by 2: p = 52.
Twice a number n, increased by 11 is the same as six times the number decreased by 9. What is the value of n?
Translate into an equation: 2n + 11 = 6n − 9.
Add 9 to both sides and subtract 2n from both sides: 20 = 4n. Divide by 4: n = 5.
One half of a number increased by 3 is five less than two thirds of the number. What is the number?
Let x be the number. Translate into an equation: (1/2)x + 3 = (2/3)x − 5.
Add 5 and subtract (1/2)x from both sides: 8 = (2/3 − 1/2)x = (1/6)x. Multiply both sides by 6: x = 48.
Four times the greatest of three consecutive odd integers exceeds three times the least by 31. What is the greatest of the three consecutive odd integers?
Let the least integer be n, so the three consecutive odd integers are n, n+2, and n+4. Translate into an equation: 4(n+4) = 3n + 31.
Distribute: 4n + 16 = 3n + 31. Subtract 3n and 16 from both sides: n = 15. The greatest integer is n + 4 = 19.
−11 + x = 9. Given the above equation, what is the value of 20 − (11−x)?
Solve for x first: add 11 to both sides: x = 20.
Substitute into the target expression: 20 − (11−20) = 20 − (−9) = 20 + 9 = 29.
If 33 − a = a + 27 − 5a, what is the value of 33 + 3a?
Simplify the right side: a + 27 − 5a = −4a + 27, so the equation becomes 33 − a = −4a + 27.
Add 4a to both sides and subtract 27: 6 = −3a, so a = −2. Substitute into the target expression: 33 + 3(−2) = 33 − 6 = 27.
If (1/2)x + 3 = 3/4 − x, what is the value of x?
Add x to both sides: (3/2)x + 3 = 3/4.
Subtract 3 from both sides: (3/2)x = 3/4 − 3 = −9/4. Multiply both sides by 2/3: x = −3/2.
If x − (3−2x) + (4−5x) = −7, what is the value of x?
Distribute the negative signs and combine like terms: x − 3 + 2x + 4 − 5x = (x+2x−5x) + (−3+4) = −2x + 1.
So −2x + 1 = −7. Subtract 1 from both sides: −2x = −8. Divide by −2: x = 4.
If three quarters of a number decreased by twenty is equal to eighty two, what is that number?
Translate into an equation, letting x be the number: (3/4)x − 20 = 82.
Add 20 to both sides: (3/4)x = 102. Multiply both sides by 4/3: x = 136.
Two and three fifths of a number equals −26. What is the number?
"Two and three fifths" as a mixed number is 13/5. Translate into an equation, letting x be the number: (13/5)x = −26.
Multiply both sides by 5/13: x = −26 × (5/13) = −10. The correct answer is B.
There are one hundred forty-two students in a high school band. These students represent two ninths of the total students in the high school. How many students attend the school?
Translate into an equation, letting T be the total number of students: (2/9)T = 142.
Multiply both sides by 9/2: T = 142 × (9/2) = 639. The correct answer is C.
820c + 380r = 4,360. The above equation models the amount of calories in a snack of c cups of cashews and r cups of raisins. The amount of calories per cup of cashews is 820 and the amount of calories per cup of raisins is 380. According to the equation, how many cups of raisins are used, if 3 cups of cashews are used to make the snack?
Substitute c = 3 into the equation: 820(3) + 380r = 4,360, so 2,460 + 380r = 4,360.
Subtract 2,460 from both sides: 380r = 1,900. Divide by 380: r = 5. The correct answer is C.
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