Word problems show up across Arithmetic and Algebra content areas and account for a significant portion of GRE Quant. The underlying formulas are not complicated. The challenge is converting a paragraph of English into a clean equation without losing anything in translation.
Most word problem errors happen at the setup stage, not the calculation stage. Students who get these wrong usually set up the wrong equation with the right arithmetic. The fix is not more formula memorization. It is a disciplined translation process.
The Translation Process
Before writing any equation, do three things:
- Identify what the question is actually asking for. Underline it.
- Assign a variable to every unknown. Write them down explicitly. Do not keep them in your head.
- Translate sentence by sentence, not word by word.
The trap is rushing to an equation because the problem looks familiar. Word problems are designed to use slightly different phrasing than the version you practiced. If you read too fast and pattern-match instead of translating, you will set up an equation for a similar but different problem.
One practical check: once you have your equation, read the original problem again and confirm your equation says the same thing the words say. This takes 10 seconds and catches most setup errors.
Rate Problems: D = rt
Every rate problem is a version of Distance = Rate x Time.
For a single traveler: if someone drives at 60 mph for 2.5 hours, they cover 150 miles.
The traps appear when the problem involves two travelers or two segments:
Average speed is not the average of two speeds. If you drive 60 mph for 1 hour and 40 mph for 1 hour, your average speed is (60+40)/2 = 50 mph. That works only because the time intervals are equal. If you drive 60 mph for 1 hour and 40 mph for 2 hours, the average is not 50 mph.
The correct formula for average speed: Total Distance / Total Time.
In the second example: you cover 60 miles in the first hour and 80 miles in the next two hours. Total distance = 140 miles. Total time = 3 hours. Average speed = 140/3 ≈ 46.7 mph.
The GRE frequently gives you a problem where a car travels from A to B at one speed and returns at another speed, then asks for average speed for the round trip. The answer is never the arithmetic average of the two speeds. Use total distance divided by total time.
For opposite-direction and same-direction problems:
- Two objects moving toward each other: their speeds add. Combined closing rate = r1 + r2.
- Two objects moving in the same direction: the faster one gains at rate r1 - r2.
Work Problems: Combined Rate Formula
If person A can complete a job alone in x hours and person B can complete the same job alone in y hours, working together they complete it in:
Combined time = (xy) / (x + y)
The logic: A's rate is 1/x jobs per hour. B's rate is 1/y jobs per hour. Together their rate is 1/x + 1/y = (x + y)/(xy) jobs per hour. To find the time to complete 1 job, take the reciprocal: xy/(x+y).
Example: A takes 4 hours, B takes 6 hours. Together: (4 x 6)/(4 + 6) = 24/10 = 2.4 hours.
The common trap is adding the times directly: 4 + 6 = 10, then concluding the job takes 10 hours together. That is backwards. Two workers are always faster than one, so the combined time must be less than either individual time.
For problems where the work is partially done: if A works alone for 2 hours and then B joins, calculate how much of the job A completed in those 2 hours (at rate 1/x per hour), subtract that from 1, and then solve for how long A and B need together to finish the remainder.
Mixture Problems: Weighted Averages
Mixture problems ask you to combine two substances of different concentrations, prices, or properties to hit a target blend.
The approach: set up a weighted average equation.
If you mix m liters of a 20% salt solution with n liters of a 50% salt solution to get a 30% salt solution, the equation is:
0.20m + 0.50n = 0.30(m + n)
Expand: 0.20m + 0.50n = 0.30m + 0.30n Simplify: 0.20n = 0.10m Therefore: m/n = 2/1
You need twice as much of the 20% solution as the 50% solution.
The same framework works for price mixtures, alloy problems, and any scenario where you are combining two things with different per-unit values.
One useful shortcut for ratio questions: the ratio of the two quantities is inversely proportional to their distances from the target. If one solution is 10 percentage points away from the target and another is 20 percentage points away, you need twice as much of the first (the one closer to the target).
Sets and Venn Diagrams: Inclusion-Exclusion
For two overlapping sets, the core formula is:
Total = Set A + Set B - Both + Neither
This is the inclusion-exclusion principle. If you add Set A and Set B separately, you count the "Both" group twice. Subtracting it once corrects for the double-count. Then add back any elements that belong to neither set.
Example: In a group of 100 students, 60 study French, 45 study Spanish, and 25 study both. How many study neither?
100 = 60 + 45 - 25 + Neither 100 = 80 + Neither Neither = 20
For three overlapping sets, the formula extends:
Total = A + B + C - (A and B) - (A and C) - (B and C) + (A and B and C) + Neither
Three-set Venn diagrams appear occasionally on the GRE. Draw the diagram. Fill in the innermost region (all three) first, then work outward to the pairwise overlaps, then to the remaining portions of each circle.
The most common trap on set problems is forgetting to add "Neither" back in. The formula accounts for it, but students frequently write Total = A + B - Both and forget that some elements belong to no set at all.
Percent Change Problems
Percent change = (New - Old) / Old x 100.
The base is always the original value. This is consistent across word problems and DI questions.
Traps on percent change word problems:
Successive percent changes do not add. If a price increases by 20% and then decreases by 20%, you do not end up where you started. Start at 100. After 20% increase: 120. After 20% decrease: 120 x 0.80 = 96. You end up at 96, not 100.
Percent of vs. percent more than. "A is 150% of B" means A = 1.5B. "A is 150% more than B" means A = B + 1.5B = 2.5B. These are different statements. Read carefully.
Profit and loss: Profit = Selling Price - Cost. Profit percentage = Profit / Cost x 100. The base is cost, not selling price.
Age Problems
Age problems ask about relationships between ages now and ages at a different point in time.
The setup: define everyone's current age with variables. If the problem references ages in the past or future, add or subtract a constant.
Example: Sarah is twice as old as Tom. In 6 years, Sarah will be 1.5 times Tom's age. How old are they now?
Let Tom's current age = t. Sarah's current age = 2t.
In 6 years: Sarah = 2t + 6, Tom = t + 6.
Equation: 2t + 6 = 1.5(t + 6) 2t + 6 = 1.5t + 9 0.5t = 3 t = 6
Tom is 6, Sarah is 12.
The trap: students often write Tom's future age as t + 6 but forget to add 6 to Sarah's age as well. Everyone ages by the same amount in the same time period.
Building the Translation Habit
The pattern across all word problem types is the same: translate carefully, assign variables explicitly, and verify your setup before you solve.
The most efficient way to practice this is to work problems without a calculator first, then verify with arithmetic. This forces you to set up clean equations rather than approximating your way to an answer.
At a pace of roughly 90-120 seconds per word problem, you should be able to complete the translation and setup in the first 30-45 seconds and the actual calculation in the remaining time. If you find yourself spending most of your time on arithmetic, the setup is probably messier than it needs to be.
The GRE lessons in Module 5 cover each word problem type with worked examples: rate, work, mixture, sets, percent change, and age. Once you have the setup mechanics down for each type, take them into the practice builder to work through mixed word problem sets under timed conditions.
Word problems reward patience at the setup stage. Students who rush the translation and spend 90 seconds on the algebra are working in the wrong order. Slow down at the start, read precisely, write down your variables, and the math will be straightforward.