GRE Word Problems Translation and Setup

Word problems are the bridge between everyday language and pure algebra, forming a critical part of the GRE's Quantitative Reasoning section. They test not just your math skills, but your ability to model real-world situations. Mastering a systematic approach to translation—converting English sentences into mathematical expressions—transforms these seemingly complex scenarios into straightforward equations you can solve with confidence.

The Translation Mindset: Your Systematic Blueprint

Before tackling specific problem types, you must internalize a universal process. The goal is to move from a vague paragraph to a precise equation. Your three-step blueprint is: Define Variables, Translate Phrases, and Establish Relationships.

First, define variables clearly. Assign letters (e.g., $x$, $y$, $t$) to unknown quantities the question asks you to find. If a problem involves ages, let the current age be $a$. For an unknown quantity, let it be $n$. Write these definitions down.

Second, translate key phrases into algebraic operations. This is the core skill. GRE problems use consistent, testable language:

• "Is," "was," "equals," "results in" → $=$
• "More than," "greater than," "sum of" → $+$
• "Less than," "fewer than," "difference" → $-$
• "Of," "times," "product of" → $\times$
• "Per," "out of," "quotient" → $÷$
• "Increased by" → $+$
• "Decreased by" → $-$

Be meticulous with order. "5 less than x" translates to $x-5$, not $5-x$.

Third, establish the relationship that forms your equation. The problem will contain one or more sentences that link your variables. Your job is to express that link mathematically. For instance, "Jenna is twice as old as Paul" becomes $J=2P$.

Core Problem Type 1: Age Problems

Age problems are classic translation exercises. They rely on the fact that time passes equally for everyone. The key is to express all ages in terms of a single point in time (usually "now" or "a given number of years ago/in the future").

Example Strategy: If Paul is currently $p$ years old and Jenna is currently $j$ years old, then in 5 years, their ages will be $p+5$ and $j+5$. Ten years ago, their ages were $p-10$ and $j-10$. The equation will always connect these expressions.

GRE-Style Example: "Today, Ben is three times as old as Sam. In 5 years, Ben will be twice as old as Sam. How old is Sam today?"

1. Define Variables: Let Sam's current age be $s$. Let Ben's current age be $b$.
2. Translate & Relate (Now): "Ben is three times as old as Sam" → $b=3s$.
3. Translate & Relate (Future): "In 5 years, Ben will be twice as old as Sam." In 5 years, Ben's age is $b+5$, Sam's is $s+5$. The relationship is: $b+5=2(s+5)$.
4. Solve the System: Substitute $b=3s$ into the second equation: $3s+5=2(s+5)$. Solving gives $3s+5=2s+10$, so $s=5$. Sam is currently 5 years old.

Core Problem Type 2: Work-Rate Problems

These problems ask how long it takes individuals or machines working together or sequentially to complete a task. The fundamental concept is the work rate: the portion of the job completed per unit of time. If a painter takes $t$ hours to paint a room alone, their rate is $1/t$ rooms per hour.

The Universal Equation: (Rate of A)$\times$(Time) + (Rate of B)$\times$(Time) = 1 (Total Job). When they work together, you add their rates.

GRE-Style Example: "Pump A can fill a tank in 4 hours. Pump B can fill the same tank in 6 hours. How long will it take to fill the tank if both pumps work together simultaneously?"

1. Define Rates: Pump A's rate: $1/4$ tank/hour. Pump B's rate: $1/6$ tank/hour.
2. Define Variable: Let $t$ be the time in hours to fill the tank together.
3. Establish Relationship: (Combined Rate)$\times$(Time) = 1 Job. Combined rate is $(1/4+1/6)$.

$$(\frac{1}{4}+\frac{1}{6})\times t=1$$

1. Solve: Find a common denominator (12): $(\frac{3}{12}+\frac{2}{12})t=1$ → $(\frac{5}{12})t=1$ → $t=\frac{12}{5}=2.4$ hours.

Core Problem Type 3: Distance-Rate-Time Problems

Governed by the essential formula Distance = Rate $\times$ Time ($D=RT$), these problems often involve two moving objects. The translation challenge is setting up equations for each object's travel and finding the relationship between their distances (e.g., they meet when the sum of their distances equals the total distance between them).

Exam Tip: Pay close attention to units. If rate is in miles per hour, time must be in hours. The GRE often presents a "trap answer" for those who use minutes without converting.

GRE-Style Example: "Two trains start 300 miles apart and travel toward each other. Train A travels at 70 mph, and Train B travels at 50 mph. How long will it take before they meet?"

1. Define Variable: Let $t$ be the time in hours until they meet.
2. Translate for Each: Train A's distance: $D_A=70t$. Train B's distance: $D_B=50t$.
3. Establish Relationship: When they meet, the sum of the distances they've traveled equals the initial separation: $D_A+D_B=300$.
4. Solve: $70t+50t=300$ → $120t=300$ → $t=300/120=2.5$ hours.

Core Problem Type 4: Profit, Discount, and Mixture Scenarios

These test your understanding of percentages and weighted averages.

• Profit/Loss: Selling Price = Cost Price + Profit (or - Loss). Profit is often a percentage of the cost: Profit = (Percent Profit/100)$\times$(Cost).
• Discount/Markup: Sale Price = Original Price - Discount. Discount = (Percent Discount/100)$\times$(Original Price).
• Mixtures: The core concept is that the total amount of a substance (e.g., sugar, alcohol, cost) in the mixture is the sum of the amounts from each component. This leads to equations like: (Conc. A)$\times$(Vol. A) + (Conc. B)$\times$(Vol. B) = (Final Conc.)$\times$(Total Vol.).

GRE-Style Example (Mixture): "How many liters of a 20% salt solution must be added to 10 liters of a 5% salt solution to produce a 10% salt solution?"

1. Define Variable: Let $x$ be the liters of 20% solution added.
2. Translate Amounts of Salt: Salt from 20% solution: $0.20x$. Salt from 5% solution: $0.05\times 10=0.5$. Total salt in final mix: $0.10(x+10)$.
3. Establish Relationship: Salt in = Salt out (in final mix).

$$0.20x+0.5=0.10(x+10)$$

1. Solve: $0.2x+0.5=0.1x+1$ → $0.1x=0.5$ → $x=5$ liters.

Common Pitfalls

1. Misordering Subtraction and Division: The most frequent error. Remember, "5 less than x" is $x-5$. "The ratio of x to y" is $x/y$, not $y/x$. Always identify the subject of the sentence.
2. Inconsistent Units in Rate Problems: Using a rate in "miles per hour" with a time given in "minutes" will lead to a wrong answer. Your first step with any DRT or work-rate problem should be to ensure all units are aligned. The GRE's trap answers often include the result of this exact mistake.
3. Forgetting to Distribute in Age/Time Problems: A statement like "In 4 years, Maria will be twice as old as she was 8 years ago" translates to $M+4=2(M-8)$. A common error is to write $M+4=2M-8$, forgetting to multiply the 2 by both the $M$ and the $-8$. Always use parentheses when setting up the future/past expression.
4. Solving for the Wrong Variable: After meticulously setting up and solving an equation, you might find $x=10$, but the question asks for "how old he will be in 5 years." Always take the final step of re-reading the question to ensure you answer what is being asked. The solved-for variable is often an intermediate step.

Summary

• Adopt a Systematic Process: Always follow the Define-Translate-Relate blueprint to deconstruct any word problem.
• Master Key Translations: Memorize the algebraic equivalents for common English phrases like "less than," "product of," and "per." Precision here is non-negotiable.
• Know the Core Formulas: Internalize $D=RT$ and the work-rate equation (Combined Rate)$\times$Time = 1 Job. Understand that profit and mixture problems are built on percentage and weighted average principles.
• Check Units and the Final Question: Consistently convert units to be the same before solving, and always verify that your final answer corresponds to what the problem is asking for, not just the variable you solved for.
• Practice Deliberate Translation: The path to speed and accuracy is seeing the English-to-math translation as the primary task. Once the equation is correctly set up, the algebra is usually simple.