How to Solve Word Problems in Math (Any Level, Any Subject)

TL;DR

• Word problems are just regular math problems wearing a disguise — your job is to translate English into equations
• Use the RUESC method: Read, Underline, Equation, Solve, Check
• Most students fail at the "translate" step, not the math itself — once you set up the equation correctly, the rest is just computation
• Practice spotting keyword patterns: "total" usually means add, "difference" means subtract, "per" or "each" means multiply or divide

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"A train leaves Chicago heading west at 60 mph while another train leaves Denver heading east at..."

And your brain just shut down. Welcome to every student's least favorite part of math class.

Here's the thing about word problems: the math inside them is usually not that hard. The hard part is figuring out what the math IS. You're doing translation work — English to algebra — and nobody really teaches you how to translate. They teach you algebra. They teach you English. But not the bridge between them.

Let's build that bridge.

The RUESC Method: 5 Steps for Any Word Problem

This framework works whether you're in pre-algebra or differential equations. The complexity of the math changes, but the approach doesn't.

Step 1: READ the entire problem (twice)

Read it once to get the big picture. What's the scenario? What's being asked?

Read it again slowly to catch details. Word problems love to bury important information in the middle of a sentence.

Don't start solving anything yet. Just read.

Step 2: UNDERLINE the key information

Go through the problem and mark:

• Numbers and measurements (quantities, rates, percentages)
• What you're solving for (the question being asked)
• Relationships (words that tell you how quantities connect)
• Units (hours, miles, dollars, liters — these guide your setup)

Here's an example:

"Maria earns $15 per hour at her part-time job and $12 per hour tutoring. Last week she worked a total of 30 hours and earned $402. How many hours did she spend at each job?"

Now you can see the structure.

Step 3: Set up an EQUATION

This is the critical step. Translate your underlined information into math:

• Assign variables to what you don't know
• Write equations based on the relationships described

For Maria's problem:

• Let x = hours at part-time job, y = hours tutoring
• Equation 1 (total hours): x + y = 30
• Equation 2 (total earnings): 15x + 12y = 402

Step 4: SOLVE

Now you're doing regular math. For this system of equations:

• From Equation 1: y = 30 - x
• Substitute into Equation 2: 15x + 12(30 - x) = 402
• 15x + 360 - 12x = 402
• 3x = 42
• x = 14, so y = 16

Maria worked 14 hours at her part-time job and 16 hours tutoring.

Step 5: CHECK

Plug your answers back into the original problem (not just your equations):

• 14 + 16 = 30 hours ✓
• 15(14) + 12(16) = 210 + 192 = 402 ✓
• Does the answer make sense? Both values are positive and less than 30. ✓

The check step catches more mistakes than you'd think. Always do it.

The Keyword Cheat Sheet

Certain English words almost always correspond to specific math operations. This isn't 100% foolproof, but it works the vast majority of the time:

Addition Keywords

• "Total," "sum," "combined," "altogether," "increased by," "more than," "plus," "together"
• Example: "The total cost of both items" → add the costs

Subtraction Keywords

• "Difference," "less than," "decreased by," "fewer," "remain," "how many more," "how much less"
• Example: "How many more students chose A than B?" → subtract

Multiplication Keywords

• "Times," "product," "each," "per," "of" (especially with fractions/percents), "double," "triple," "every"
• Example: "$15 per hour" → 15 × hours

Division Keywords

• "Per," "each," "ratio," "average," "split," "shared equally," "out of," "divided by"
• Example: "Split equally among 4 friends" → total ÷ 4

Equals Keywords

• "Is," "was," "will be," "gives," "yields," "results in," "costs"
• Example: "The total cost is $50" → expression = 50

Warning: "Less than" is tricky. "5 less than x" means x - 5, NOT 5 - x. The thing that comes after "less than" goes first in the equation.

Examples at Every Level

Pre-Algebra: Basic Word Problems

Problem: "A rectangular garden is twice as long as it is wide. The perimeter is 36 meters. Find the dimensions."

RUESC:

1. Read: Rectangle, length is twice the width, perimeter is 36m
2. Underline: "twice as long as wide" and "perimeter is 36 meters"
3. Equation: Let w = width, so length = 2w. Perimeter = 2(length) + 2(width) = 2(2w) + 2(w) = 36
4. Solve: 4w + 2w = 36 → 6w = 36 → w = 6m, length = 12m
5. Check: Perimeter = 2(12) + 2(6) = 24 + 12 = 36 ✓

Algebra: Systems of Equations

Problem: "A movie theater sold 250 tickets. Adult tickets cost $12 and child tickets cost $7. Total revenue was $2,400. How many of each were sold?"

RUESC:

1. Read: Two types of tickets, different prices, know total tickets and total revenue
2. Underline: 250 tickets, $12 adult, $7 child, $2,400 total
3. Equations: a + c = 250 and 12a + 7c = 2400
4. Solve: c = 250 - a → 12a + 7(250 - a) = 2400 → 12a + 1750 - 7a = 2400 → 5a = 650 → a = 130, c = 120
5. Check: 130 + 120 = 250 ✓; 12(130) + 7(120) = 1560 + 840 = 2400 ✓

Geometry: Applied Problems

Problem: "A ladder leans against a wall. The foot of the ladder is 5 feet from the wall, and the ladder reaches 12 feet up the wall. How long is the ladder?"

RUESC:

1. Read: Right triangle situation — wall, ground, ladder
2. Underline: 5 feet from wall, 12 feet up wall, length of ladder = ?
3. Equation: Pythagorean theorem: a² + b² = c² → 5² + 12² = c²
4. Solve: 25 + 144 = c² → 169 = c² → c = 13 feet
5. Check: 5² + 12² = 25 + 144 = 169 = 13² ✓ (Classic 5-12-13 right triangle)

Statistics: Probability Word Problems

Problem: "A bag contains 4 red balls, 3 blue balls, and 5 green balls. If you draw two balls without replacement, what's the probability both are red?"

RUESC:

1. Read: Multi-draw probability, without replacement (key phrase!)
2. Underline: 4 red, 3 blue, 5 green, two draws, without replacement, both red
3. Equation: P(both red) = P(first red) × P(second red | first was red) = (4/12) × (3/11)
4. Solve: (4/12)(3/11) = 12/132 = 1/11 ≈ 0.0909 or about 9.1%
5. Check: Makes intuitive sense — reds are 1/3 of the bag, so getting two in a row should be relatively unlikely ✓

Calculus: Optimization Problems

Problem: "A farmer has 200 meters of fencing and wants to enclose a rectangular area next to a river (no fence needed along the river). What dimensions maximize the area?"

RUESC:

1. Read: Rectangular area, one side is a river (so fencing on 3 sides only), maximize area
2. Underline: 200m of fencing, rectangular, river on one side, maximize area
3. Equation: Let x = width (two sides), y = length (one side). Constraint: 2x + y = 200 → y = 200 - 2x. Area: A = xy = x(200 - 2x) = 200x - 2x²
4. Solve: dA/dx = 200 - 4x = 0 → x = 50m → y = 200 - 100 = 100m → A = 5,000 m²
5. Check: Second derivative test: d²A/dx² = -4 < 0, so this is a maximum ✓. Fencing: 2(50) + 100 = 200 ✓

The 5 Most Common Mistakes (And How to Fix Them)

Mistake 1: Rushing to Calculate

Most errors happen because students start doing math before they fully understand the problem. The RUESC method forces you to read and set up before solving. The two minutes you spend understanding the problem saves ten minutes of wrong-direction calculations.

Mistake 2: Assigning Variables Randomly

Always define your variables clearly. Write "Let x = the number of adult tickets" — don't just throw x's and y's around hoping they'll sort themselves out. Clear variable definitions make the rest of the problem easier AND help you catch mistakes.

Mistake 3: Ignoring Units

Units are your friend. If you're solving for speed and your answer is in dollars, something went wrong. Keep units attached to every number, and make sure they cancel properly.

Example: 60 miles/hour × 3 hours = 180 miles. The "hours" cancel, leaving you with miles. If your units don't work out, your setup is wrong.

Mistake 4: Not Drawing a Picture

For any problem involving physical situations (distances, shapes, movement, mixtures), sketch it out. Even a terrible drawing helps you visualize what's happening. Label everything with the numbers from the problem.

This is especially important for:

• Distance/rate/time problems
• Geometry problems
• Physics word problems
• Any problem with multiple objects or people

Mistake 5: Skipping the Check Step

"I got an answer, so I'm done." No! Plug your answer back into the original problem. Not into your equation — into the original English description. Does it make sense?

If you calculated that a person is -3 years old, or that a car travels at 5,000 mph, or that a company's profit is negative when the problem says they made money — something went wrong. Common sense is your best error detector.

Category-Specific Strategies

Distance/Rate/Time Problems

Use the formula: Distance = Rate × Time

Set up a table:

	Distance	Rate	Time
Object 1
Object 2

Fill in what you know, use variables for what you don't, and write an equation based on the relationship (they meet, they're a certain distance apart, etc.).

Mixture Problems

Set up: Amount × Concentration = Total

	Amount	Concentration	Total
Solution 1
Solution 2
Mixture

The mixture row comes from adding the previous rows (usually: amounts add up, totals add up, but concentrations do NOT add).

Age Problems

Set "now" as your baseline. If someone is x years old now:

• They were (x - 5) years old five years ago
• They'll be (x + 3) years old in three years

Write equations based on the age relationships described.

Work/Rate Problems

If person A can do a job in 5 hours, their rate is 1/5 of the job per hour.

Combined rate: 1/A + 1/B = 1/T (where T is time to finish together)

When You're Completely Stuck

Sometimes you read a word problem three times and you still have no idea how to start. Here's what to do:

1. Identify what you're solving for. What is the question asking? Write it as a sentence: "I need to find ___."
2. List everything you know. Write down every number and fact from the problem.
3. Ask: what formula connects these? Look at the quantities you have and the quantity you need. What mathematical relationship connects them?
4. Try a simpler version. Can you solve the problem with easier numbers? If "a pool fills in 3.7 hours and drains in 5.2 hours," first try it with 2 hours and 4 hours to understand the setup.
5. Work backwards. If the answer were 10, would the problem make sense? Try a guess and see what happens — this sometimes reveals the structure.

And if you're still stuck, tools like Gradily can walk you through the setup step by step. The key is understanding how to translate the words into math — once you see the equation, you can usually solve it. Gradily is especially good at explaining that translation step, which is the part most textbooks breeze past.

Practice Is Everything

Here's the uncomfortable truth: you can read this article ten times and still struggle with word problems. The only way to get better is practice. But not mindless practice — deliberate practice:

1. Do problems by category. Spend a session on just distance problems, then just mixture problems. Pattern recognition builds faster when you focus on one type.
2. After solving, categorize the problem. Was it a system of equations? Rate problem? Optimization? Knowing the type helps you recognize similar problems on exams.
3. Study your mistakes. When you get a problem wrong, don't just look at the answer — figure out where your setup went wrong. Was it the translation step? The algebra? The interpretation?

Word problems are the bridge between math class and the real world. They're harder than pure computation, but they're also more useful. Every time your phone calculates a tip, every time you figure out how long a road trip will take, every time you compare prices per unit at the grocery store — you're solving word problems.

You can do this. Read carefully, set up methodically, solve patiently, and always check your work.