Word Problems & Translation • Topic 4 of 4

Setting Up Equations

The hardest part of a multi-quantity word problem is the setup, not the solving. When two unknowns are related, express both in terms of a single variable so you end with one equation — for a mixture of adult and child tickets, let a be the number of adult tickets and (total − a) be the child tickets, rather than juggling two letters. When the relationships genuinely need two variables, write a system and use substitution or elimination. Always attach units and a clear definition to each variable; a labelled "let x = number of nickels" prevents the classic coin-problem error of confusing the count of coins with their monetary value. Check the answer against the sentence, not just the algebra — a value that solves the equation but gives a negative count or a fractional person signals a setup slip. And remember the GRE-specific shortcut: with numeric answer choices, back-solving (testing a choice in the original wording) is frequently faster and less error-prone than building and solving the equation.

✅ Solved examples

1. Tickets cost $8 (adult) and $5 (child). 30 tickets sold for $198 total. How many adult tickets?
Let a = adult tickets, so child = 30 − a. Value: 8a + 5(30 − a) = 198 → 8a + 150 − 5a = 198 → 3a = 48 → a = 16.
2. A jar has nickels and dimes, 20 coins worth $1.35. How many dimes? (nickel = $0.05, dime = $0.10)
Let d = dimes, nickels = 20 − d. Value: 0.10d + 0.05(20 − d) = 1.35 → 0.05d + 1.00 = 1.35 → 0.05d = 0.35 → d = 7.
3. The length of a rectangle is 3 more than twice its width, and the perimeter is 36. Find the width.
Let w = width, length = 2w + 3. Perimeter 2(w + 2w + 3) = 36 → 2(3w + 3) = 36 → 3w + 3 = 18 → w = 5.
4. A 40-liter mixture is 25% acid. How many liters of acid must be added to make it 40% acid?
Acid now = 0.25 × 40 = 10 L. Add x liters of acid: (10 + x)/(40 + x) = 0.40 → 10 + x = 16 + 0.4x → 0.6x = 6 → x = 10 L.

✏️ Practice — try these, take hints as needed

1. $6 and $10 tickets, 25 sold for $186. How many $6 tickets?
Let x = $6 tickets; other = 25 − x.
6x + 10(25 − x) = 186.
250 − 4x = 186.
16
2. Pens cost $2, notebooks $5. 12 items cost $39. How many notebooks?
Let n = notebooks, pens = 12 − n.
5n + 2(12 − n) = 39.
3n + 24 = 39.
5
3. A rectangle's length is 4 more than its width; perimeter is 28. Find the width.
length = w + 4.
2(w + w + 4) = 28.
2(2w + 4) = 28.
5
4. Coins: quarters and dimes, 15 coins worth $2.55. How many quarters? (quarter = $0.25)
Let q = quarters, dimes = 15 − q.
0.25q + 0.10(15 − q) = 2.55.
0.15q + 1.50 = 2.55.
7
5. A 50-liter solution is 20% salt. How many liters of water to dilute it to 10% salt?
Salt = 0.20 × 50 = 10 L, unchanged.
10/(50 + x) = 0.10.
10 = 5 + 0.1x.
50 L

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Interest & Growth Take the chapter test →