Word Problems

What This Module Covers

This module covers the skill of translating English sentences into mathematical equations — the foundation of every GRE word problem. You'll work through rate, distance, and work problems; mixture problems; profit and cost scenarios; and age problems. Each problem type has its own setup pattern, and once you know the pattern, the algebra becomes mechanical.

Why It Matters on the GRE

Word problems appear in roughly a third of all GRE Quant questions. The math itself is usually straightforward — the difficulty is in the translation step. A student who misreads "5 less than x" as 5−x will get the wrong answer even if they solve the resulting equation perfectly. The GRE deliberately tests whether you can go from words to a correct equation, so that's where this module focuses.

Core Concepts

1. Translation from English to Math

Every word problem is a puzzle written in two languages: English and algebra. Your job is to translate. A handful of phrases come up constantly, and you need to know them cold.

Must Know: "More than" means add. "Less than" means subtract — but the order flips. "Is" means equals. "Of" means multiply. "What" or "how many" is your variable.

The most dangerous phrase on the GRE is "less than." When the problem says "5 less than x," students instinctively write 5−x. That's wrong. "5 less than x" means you take x and subtract 5, giving you x−5. Read it as: start with x, then go 5 less.

Must Know: "5 less than x" = x − 5. "x is 5 less than y" means x = y − 5. Always identify what you're starting from before subtracting.

Example — Translation:

1. Read: "Maria has 8 more coins than twice the number David has."
2. Let d = David's coins.
3. "Twice the number David has" → 2d.
4. "8 more than" → add 8 → Maria = 2d + 8.
5. Done. One variable, clean equation.

2. Rate, Work, and Distance Problems

These three problem types all use the same underlying formula, just arranged differently.

Must Know: Distance = Rate × Time (D = RT). Rearrange as needed: T = D/R and R = D/T.

For work problems, reframe them as rate problems. If a printer finishes a job in 4 hours, its rate is 1/4 of the job per hour. When two workers (or machines) work together, their rates add.

Must Know: Combined work rate = 1/r₁ + 1/r₂. Time to finish together = 1 ÷ (combined rate). Rates add — times do not.

For round-trip or average speed problems, the GRE loves to bait you into averaging the two speeds. Never do that. Average speed = Total distance ÷ Total time, always.

For catch-up problems: if two objects move toward each other, add their speeds to find how fast the gap closes. If they move in the same direction, subtract the slower from the faster.

Example — Distance:

1. Car A leaves at 60 mph. Car B leaves from the same point 1 hour later at 80 mph. How long until B catches A?
2. When B has traveled t hours, A has traveled t+1 hours.
3. Set distances equal: 60(t+1) = 80t.
4. 60t + 60 = 80t → 60 = 20t → t = 3 hours.
5. Verify: B travels 80×3 = 240 miles. A travels 60×4 = 240 miles. ✓

3. Mixture Problems

Mixture problems ask you to combine two substances (solutions, alloys, food items) with different concentrations or properties and find what the result looks like.

Must Know: (concentration₁ × amount₁) + (concentration₂ × amount₂) = concentration_final × total_amount. Label everything before you write the equation.

Example — Mixture:

1. 10 liters of a 20% salt solution is mixed with 30 liters of a 40% salt solution. What is the final concentration?
2. Salt from solution 1: 0.20 × 10 = 2 liters of salt.
3. Salt from solution 2: 0.40 × 30 = 12 liters of salt.
4. Total salt: 2 + 12 = 14 liters. Total volume: 10 + 30 = 40 liters.
5. Final concentration: 14/40 = 0.35 = 35%.

4. Profit, Revenue, and Cost

The GRE tests a small set of business math relationships. Know these three:

Must Know: Profit = Revenue − Cost. Sale price = Original × (1 − discount%). Marked-up price = Original × (1 + markup%). Percent change = (New − Old) / Old × 100%.

Break-even is simply the point where Revenue = Cost, meaning Profit = 0. When the problem asks "which quantity is the base," the base is always the original (pre-change) value for percent problems.

5. Age Problems

Age problems give you two time points and relationships between people's ages. They always require two equations.

Must Know: Let x = current age. "In n years" → x + n. "n years ago" → x − n. Always write both equations before solving — never try to do it in your head.

Example — Age:

1. Today, Alex is 3 times as old as Ben. In 4 years, Alex will be twice as old as Ben. Find their current ages.
2. Equation 1 (now): A = 3B.
3. Equation 2 (in 4 years): A + 4 = 2(B + 4).
4. Substitute A = 3B: 3B + 4 = 2B + 8 → B = 4, A = 12.
5. Verify: In 4 years, Alex = 16, Ben = 8. 16 = 2 × 8. ✓

6. Setting Up Variables

Use the fewest variables possible. If the problem says "John has twice as many apples as Mary," you only need one variable: let Mary = m, John = 2m. Assigning separate variables to related quantities forces you to write more equations than necessary and increases the chance of an error.

Must Know: When two quantities are explicitly related by a ratio or multiple, express them both in terms of one variable. Save two-variable setups for problems where no such relationship is given.

Common Traps

• "Less than" reversal: "5 less than x" is x − 5, not 5 − x. The GRE puts this in answer choices deliberately. Slow down on any subtraction phrase and ask: what am I starting from?
• Averaging speeds: If you drive 30 mph one way and 60 mph the other, the average speed is NOT 45 mph. Compute total distance ÷ total time. This trap appears on nearly every GRE test.
• Wrong base for percentages: "The price increased by 20% from last year" means the base is last year's price. Using this year's price as the base gives a wrong answer. Always identify the original before computing a percent change.
• Adding times instead of rates: In work problems, rates add — not times. If printer A takes 4 hours and printer B takes 6 hours, the combined time is NOT 4 + 6 = 10 hours (and definitely not 5 hours). Add 1/4 + 1/6 and take the reciprocal.
• Using too many variables: Introducing x and y when you could use x and 2x doubles the algebra and the opportunity to make errors. Push back against the habit of assigning a new letter to every unknown.

GRE Strategy

• Translate one clause at a time. Don't try to read the whole problem and write the equation all at once. Cover up everything after the first sentence, translate it, then move on.
• Label your variables explicitly. Write "let x = Mary's current age" on your scratch paper. When you arrive at x = 12, you need to know what 12 means — especially if the question asks for something different.
• Verify with the original words, not your equations. After solving, re-read the problem sentence by sentence and confirm your answer satisfies each condition. Equations can have errors; the words don't lie.
• On rate problems, draw a table. Three columns: Rate | Time | Distance (or Work). Fill in what you know, then write the equation from what's missing.
• When stuck, try Plug In. If the answer choices are numbers, plug them into the problem and see which one works. On a timed test, a 90-second plug-in is often faster than a 3-minute algebra grind.

Worked Example

Question: Printer A can print 100 pages in 4 hours. Printer B can print 100 pages in 6 hours. Working together, how long does it take them to print 100 pages?

(A) 2.0 hours (B) 2.4 hours (C) 2.5 hours (D) 3.0 hours (E) 5.0 hours

Solution:

Step 1 — Find each printer's rate. Printer A: 1/4 of the job per hour (it completes 1 full job in 4 hours). Printer B: 1/6 of the job per hour.

Step 2 — Add the rates. Combined rate = 1/4 + 1/6. Find a common denominator (12): 3/12 + 2/12 = 5/12 of the job per hour.

Step 3 — Find the time. Time = 1 job ÷ (5/12 jobs per hour) = 12/5 = 2.4 hours.

Answer: (B) 2.4 hours.

Step 4 — Verify. In 2.4 hours, Printer A prints 2.4 × (1/4) = 0.6 of the job. Printer B prints 2.4 × (1/6) = 0.4 of the job. Together: 0.6 + 0.4 = 1.0 job. ✓

Why the other choices are wrong:

• (A) 2.0 hours — Too fast. This would require a combined rate of 1/2 per hour. You'd only get there if both printers were faster than they actually are.
• (C) 2.5 hours — A tempting rounding error. 12/5 = 2.4, not 2.5. Always compute the fraction fully before converting.
• (D) 3.0 hours — This is the average of 4 and 6 divided by 2 (i.e., 10/2 = 5... actually it's closer to just picking a middle number). Either way, averaging times is never valid for work problems.
• (E) 5.0 hours — This is 4 + 6 divided by 2, or the simple average of the two times. The GRE puts this here specifically to catch students who average instead of adding rates. Together, two printers must always be faster than either one alone. Any answer ≥ 4 hours should be immediately eliminated.