Word Problems

Last updated July 07, 2026 · Reviewed by the AnvayaPrep team

Introduction

Word Problems constitute the applied mathematics domain of GRE Quantitative Reasoning, spanning 25 topics that cover every major problem scenario the exam uses to test quantitative reasoning through real-world contexts. The unit covers: distance-rate-time problems, work problems, mixture problems, percent word problems, ratio word problems, age problems, profit and loss, interest problems, counting word problems, data word problems, algebraic word problems, comparison word problems, arithmetic word problems, multi-step problems, and the specific word problem traps the GRE sets for unprepared students. Word problems appear in every GRE question format and represent approximately 30 to 40% of the Quantitative Reasoning section.

The primary challenge of word problems is not mathematical -- students who struggle with word problems typically know the required math. The challenge is translation: converting the verbal description into the right mathematical model. Every GRE word problem involves (1) identifying what is being asked, (2) defining variables clearly, (3) writing equations or expressions from the verbal relationships, and (4) solving for the target quantity. Students who apply this four-step process consistently make far fewer setup errors than those who rush directly from the problem statement to a formula.

Learning Objectives

• Apply the distance-rate-time formula (D = R x T) to single-traveler, multi-traveler, opposite-direction, same-direction, and round-trip scenarios
• Compute combined work rates by adding individual rates expressed as fractions of the job per unit time, and solve for time to complete jointly
• Set up and solve mixture problems by tracking the total amount of one component (the solute) before and after combining, using the relationship: concentration x volume = amount of component
• Translate percent word problem language ("X is what percent of Y," "by what percent did X increase") into equations using the formula Part = Percent x Whole
• Set up algebraic equations for age problems using a common reference time point, defining variables for current ages and expressing past or future ages algebraically
• Apply profit, loss, markup, and discount formulas: Profit = Revenue - Cost; Markup percent = Profit / Cost; Discount percent = Discount / Original Price
• Solve simple and compound interest problems: Simple interest = Principal x Rate x Time; Compound interest = Principal x (1 + Rate/n)^(nt)
• Define variables carefully in all word problems, verifying that the variable represents the correct quantity and that the final answer answers the specific question asked
• Recognize and apply the unit-checking technique: verify that units on both sides of an equation are consistent before solving
• Identify the common GRE word problem traps: solving for the wrong variable, misidentifying the base in percent questions, ignoring units, and setting up mixture concentrations incorrectly

High-Yield Concepts

The Universal Word Problem Process

Every GRE word problem, regardless of category, follows the same four-step solution framework:

1. Read and identify: What quantity does the question ask for? Identify the final target before anything else.
2. Define variables: Assign a variable to the unknown quantity or the most natural quantity to represent the scenario. Write the definition explicitly: "Let x = the number of hours Worker A works."
3. Write equations: Express the relationships stated in the problem as mathematical equations. Most GRE word problems require one to two equations.
4. Solve and verify: Solve for the target quantity. Substitute back into the original problem to verify the answer makes sense.

The most common error -- solving for the wrong variable -- is avoided entirely by Step 1. Write the question's target at the top of your scratch work before setting up any equations.

Distance-Rate-Time Scenarios

All DRT problems use D = R x T in some form. The critical extensions:

Two objects traveling toward each other: Their closing rate is the sum of their speeds. Time to meeting = Total distance / (Speed 1 + Speed 2).

Two objects traveling in the same direction (overtaking): The faster object gains distance at the rate difference. Time to overtaking = Head start distance / (Faster speed - Slower speed).

Round trips: Total distance is twice the one-way distance. The average speed for a round trip is not the average of the two speeds; it is total distance / total time. For a round trip with speeds s1 and s2 over the same distance d: average speed = 2d / (d/s1 + d/s2) = 2s1s2 / (s1 + s2).

Work Problems: Rate Addition

The core insight: if Worker A completes a job in a hours, her rate is 1/a jobs per hour. If Worker B completes the same job in b hours, his rate is 1/b jobs per hour. Working together, their combined rate is 1/a + 1/b jobs per hour.

Time to complete together = 1 / (combined rate) = 1 / (1/a + 1/b) = ab / (a + b).

This formula only applies when both workers are adding to the completion of the job. If one worker fills and another drains (as in a pipe problem), the effective rate is 1/a - 1/b (filling rate minus draining rate), and the formula adjusts accordingly.

Mixture Problems: Component Tracking

Mixture problems track the amount of one component (typically the concentrated substance) across a mixing operation. The governing equation: amount_before = amount_after, where amount = concentration x volume (or percent x quantity).

If you have V1 liters of a solution at concentration C1 and add V2 liters at concentration C2, the resulting concentration is: C_final = (C1 x V1 + C2 x V2) / (V1 + V2).

A special case: adding pure substance (concentration = 1) or pure solvent (concentration = 0) simplifies the numerator. Adding water to dilute is adding concentration 0.

Percent and Ratio Word Problems

Percent word problem language maps to specific equations:

• "X is P% of Y" → X = (P/100) x Y
• "X is what percent of Y" → P = X/Y x 100
• "X increased by P%" → New value = X x (1 + P/100)
• "X decreased by P%" → New value = X x (1 - P/100)
• "Y is P% more than X" → Y = X x (1 + P/100), so X = Y / (1 + P/100)

For ratio word problems, express the ratio using a multiplier variable: if the ratio of A to B is 3:2, let A = 3k and B = 2k. Use the additional constraint (a total or specific value) to solve for k.

Study Strategy

Begin with distance-rate-time and work problems -- these are the highest-frequency word problem types and share the same R x T structure. Mastering this structure first creates a transferable framework.

Study mixture problems after DRT and work, since mixture problems also involve a component-tracking equation that parallels work-rate addition.

Study percent word problems and ratio word problems together, since the translation steps are similar.

Cover the remaining specialized types (age problems, interest problems, profit and loss) individually. These appear less frequently but each has a specific equation structure that is worth learning explicitly.

Finish with the multi-step word problem and word problem traps topics. Multi-step problems require combining two or more of the above frameworks, and the traps topic explicitly identifies the most common setup errors.

Common Mistakes

Solving for the wrong variable. The question asks for time but the setup variable is speed, or the question asks for Worker B's time but the solution produces the combined time. Write the target quantity at the top of your scratch work before defining any variables.

Averaging speeds by taking the arithmetic mean. For round trips or two-leg journeys where speed varies, average speed = total distance / total time. The arithmetic mean of the two speeds is wrong unless the time spent at each speed is equal.

Setting up mixture concentrations backward. The component amount before mixing equals the component amount after mixing: C1xV1 + C2xV2 = C_final x (V1+V2). Students who write the equation backward or omit a volume term get wrong answers.

Using the new value as the base for percent change. The GRE asks "by what percent did X increase?" which means (new - original) / original. Using the new value as the denominator is a trap that produces the wrong answer.

Forgetting to check units. Distance-rate-time problems frequently mix hours and minutes or miles and feet. Convert all units to a consistent system before setting up the equation.

Exam Tips

Draw a diagram or table for every word problem involving motion or multiple agents. For DRT problems: a table with columns for rate, time, and distance for each traveler eliminates most setup errors. For work problems: a table with columns for rate, time, and work fraction completed.

For mixture problems, write the component amount equation explicitly: amount in + amount added = total amount in final mixture. Substituting concentration x volume for each term makes the equation immediately solvable.

For complex multi-step word problems, solve them in stages rather than trying to write one master equation. Get an intermediate result from the first stage and use it as input to the second stage.

Always verify that your final answer is answering the specific question asked. Reread the final sentence of the problem after computing your answer to confirm the match.