GRE Algebra Word Problems Practice Questions with Answers

Approximately 30 to 40 percent of the Quantitative Reasoning section on the GRE involves translating text into mathematical equations. Success on the exam requires more than just knowing formulas; you must be able to decode complex narratives and identify the underlying algebraic structures. By working through GRE Algebra Word Problems Practice Questions with Answers, you can bridge the gap between abstract math and practical problem-solving.

Preparing for this section often feels overwhelming because the test makers intentionally use wordy scenarios to mask simple arithmetic. However, most problems fall into predictable categories such as rate-time-distance, work-rate, mixtures, and age problems. Utilizing a structured GRE Prep strategy allows you to systematically approach these questions, reducing anxiety and increasing accuracy under timed conditions. Let’s explore the core mechanics of these problems and how to solve them efficiently.

Concept Explanation

GRE algebra word problems are mathematical challenges that require you to translate written descriptions into algebraic equations involving variables, constants, and operators. The primary goal is to identify the unknown quantity, assign it a variable (usually $x$), and set up an equation that reflects the relationships described in the text. According to Khan Academy's guide on word problems, specific keywords act as operational cues: "is" or "total" signifies equality ($=$), "more than" or "sum" signifies addition ($+$), and "product" or "of" often signifies multiplication ($\times$).

To solve these effectively, follow a four-step process: First, identify exactly what the question is asking for. Second, define your variables clearly. Third, translate the English sentences into a mathematical equation. Finally, solve the equation and double-check that your answer makes sense in the context of the story. For more complex scenarios, such as those involving multiple unknown variables, you may need to use a system of linear equations. If you find yourself struggling with the speed of translation, practicing with an AI Question Generator can help you see a wider variety of phrasing patterns.

Solved Examples

Example 1: The Age Problem
Ten years ago, Sarah was half as old as she will be in 5 years. How old is Sarah now?

1. Define the variable: Let $s$ be Sarah's current age.
2. Translate the first part: Sarah's age 10 years ago was $s - 10$.
3. Translate the second part: Her age in 5 years will be $s + 5$.
4. Set up the equation: $s - 10 = \frac{1}{2}(s + 5)$.
5. Solve: Multiply both sides by 2 to get $2s - 20 = s + 5$. Subtract $s$ from both sides: $s - 20 = 5$. Add 20: $s = 25$. Sarah is 25 years old.

Example 2: The Rate-Time-Distance Problem
A train travels 300 miles at a constant speed. If the speed had been 10 mph faster, the trip would have taken 1 hour less. What was the original speed?

1. Use the formula $\text{Time} = \frac{ \text{Distance}}{ \text{Speed}}$. Let $r$ be the original rate.
2. Original time: $\frac{300}{r}$. New time: $\frac{300}{r + 10}$.
3. Set up the equation based on the time difference: $\frac{300}{r} - \frac{300}{r + 10} = 1$.
4. Multiply by the common denominator $r(r + 10)$: $300(r + 10) - 300r = r(r + 10)$.
5. Simplify: $300r + 3000 - 300r = r^2 + 10r$, leading to $r^2 + 10r - 3000 = 0$.
6. Factor the quadratic: $(r + 60)(r - 50) = 0$. Since speed must be positive, $r = 50$. The speed was 50 mph.

Example 3: The Mixture Problem
A chemist has 10 liters of a solution that is 10% acid. How many liters of pure acid must be added to make a solution that is 25% acid?

1. Let $x$ be the amount of pure acid added.
2. Initial acid: $0.10 \times 10 = 1$ liter.
3. Total final volume: $10 + x$.
4. Total final acid: $1 + x$.
5. Set up the percentage equation: $\frac{1 + x}{10 + x} = 0.25$.
6. Solve: $1 + x = 0.25(10 + x) \rightarrow 1 + x = 2.5 + 0.25x$. Subtract $0.25x$: $0.75x = 1.5$. Divide: $x = 2$. Add 2 liters.

Practice Questions

1. A rectangular garden has a perimeter of 50 meters. If the length is 5 meters more than the width, what is the area of the garden in square meters?

2. Together, John and Mary have $84. If John gives Mary $12, Mary will have three times as much money as John has left. How much money did John start with?

3. A car travels from Town A to Town B at an average speed of 40 mph and returns at an average speed of 60 mph. What is the average speed for the entire round trip?

4. Working alone, Pump A can fill a tank in 4 hours, and Pump B can fill the same tank in 6 hours. If both pumps work together, how many hours will it take to fill the tank?

5. The sum of three consecutive integers is 72. What is the value of the largest integer?

6. A store sells shirts for $20 each and pants for $30 each. If a customer buys a total of 12 items and spends $310, how many pants did they buy?

7. A father is currently four times as old as his son. In 20 years, the father will be twice as old as his son. How old is the son now?

8. A boat travels 24 miles upstream in 3 hours and 24 miles downstream in 2 hours. What is the speed of the current in miles per hour?

9. If 5 workers can build 2 houses in 30 days, how many days would it take 10 workers to build 4 houses, assuming all workers work at the same rate?

10. A merchant mixes 5 lbs of coffee worth $8 per lb with 3 lbs of coffee worth $12 per lb. What is the cost per lb of the mixture?

Answers & Explanations

1. Answer: 150. Let width be $w$. Length is $w + 5$. Perimeter $2(w + w + 5) = 50$. $4w + 10 = 50 \rightarrow 4w = 40 \rightarrow w = 10$. Length = 15. Area = $10 \times 15 = 150$.
2. Answer: $33. Let John have $J$. Mary has $84 - J$. After the gift, John has $J - 12$, Mary has $(84 - J) + 12 = 96 - J$. Equation: $96 - J = 3(J - 12)$. $96 - J = 3J - 36$. $132 = 4J$. $J = 33$.
3. Answer: 48 mph. Assume distance is 120 miles (LCM of 40 and 60). Time to B: $120/40 = 3$ hrs. Time back: $120/60 = 2$ hrs. Total distance = 240. Total time = 5. Average speed = $240/5 = 48$.
4. Answer: 2.4 hours. Combined rate = $\frac{1}{4} + \frac{1}{6} = \frac{3+2}{12} = \frac{5}{12}$ tanks per hour. Time = $\frac{12}{5} = 2.4$ hours.
5. Answer: 25. Let integers be $n, n+1, n+2$. $3n + 3 = 72 \rightarrow 3n = 69 \rightarrow n = 23$. Largest is $23+2=25$.
6. Answer: 7. Let $p$ be pants, $s$ be shirts. $p + s = 12$ and $30p + 20s = 310$. Substitute $s = 12 - p$: $30p + 20(12 - p) = 310$. $30p + 240 - 20p = 310$. $10p = 70$. $p = 7$.
7. Answer: 10. Let son be $s$, father is $4s$. In 20 years: $4s + 20 = 2(s + 20)$. $4s + 20 = 2s + 40$. $2s = 20$. $s = 10$.
8. Answer: 2 mph. Upstream speed $(b-c) = 24/3 = 8$. Downstream speed $(b+c) = 24/2 = 12$. Subtract: $(b+c) - (b-c) = 12 - 8$. $2c = 4$. $c = 2$.
9. Answer: 30 days. Doubling workers doubles productivity (4 houses in 30 days). Since the work (4 houses) is also doubled, the time remains the same.
10. Answer: $9.50. Total cost = $(5 \times 8) + (3 \times 12) = 40 + 36 = 76$. Total weight = 8 lbs. Price = $76/8 = 9.5$.

1. If a number \( x \) is increased by 20% and the result is 72, what is the value of \( x \)?

• A50
• B60
• C62
• D64

Frequently Asked Questions

What is the most common mistake in GRE algebra word problems?

The most frequent error is misidentifying the variable or failing to translate "more than" or "less than" correctly, often swapping the terms in subtraction. Always re-read the final sentence to ensure you are solving for the correct unknown.

How do I handle problems with multiple rates?

For combined work problems, add the individual rates (work per unit of time) rather than the times themselves. You can also use the harmonic mean for average speed problems when distances are equal.

Should I use the answer choices to solve word problems?

Back-solving, or plugging in answer choices, is a highly effective strategy for GRE word problems when the algebra feels too complex. Start with choice C to determine if you need a larger or smaller value.

How can I improve my speed on these questions?

Speed improves through pattern recognition, which is best developed by using tools like the AI Exam Simulator to practice under timed pressure. Learning to quickly translate "is" to "=" and "of" to "*" saves vital seconds.

Is it necessary to memorize many formulas?

While some formulas like $d = rt$ are essential, most algebra word problems rely on logical translation and basic equation solving. Focus on understanding the relationship between quantities rather than rote memorization.