Writing and Solving Equations from Word Problems

Grade: 7-8 | Topic: Algebra

What You Will Learn

After this lesson you will be able to read a word problem, identify the unknown, translate the English sentences into a mathematical equation, solve that equation, and verify that the answer makes sense. These translation skills are the bridge between classroom algebra and real-world problem solving.

Theory

The 5-step word problem strategy

Every word problem can be tackled with the same approach:

1. Read the problem carefully — at least twice.
2. Define the variable: decide what $x$ (or another letter) represents and write it down.
3. Translate the words into an equation using math operations.
4. Solve the equation.
5. Check — substitute your answer back into the original words (not just the equation) and ask: "Does this answer make sense?"

Translating words to math

Knowing which words map to which operations is the key skill:

English phrase	Mathematical operation
sum, total, more than, increased by, plus	$+$
difference, less than, decreased by, minus	$-$
product, times, of, twice, triple	$\times$
quotient, divided by, per, split equally	$\div$
is, equals, gives, results in, was	$=$

Watch out for "less than" — it reverses the order:

• "5 less than a number" means $x - 5$ (not $5 - x$).
• "A number decreased by 5" also means $x - 5$.

Choosing what $x$ represents

Always let $x$ stand for the quantity the problem asks you to find. If it asks "how many apples," then let $x$ = the number of apples. If other unknowns depend on $x$, express them in terms of $x$ too.

For example: "Maria is 4 years older than Tom." If Tom's age is $x$, then Maria's age is $x + 4$.

Worked Examples

Example 1: A number problem (easy)

Problem: Three times a number, decreased by 7, equals 20. Find the number.

Step 1: Let $x$ = the number.

Step 2: Translate: "three times a number" = $3x$; "decreased by 7" = $- 7$; "equals 20" = $= 20$.

$3x - 7 = 20$

Step 3: Add 7 to both sides.

$3x = 27$

Step 4: Divide both sides by 3.

$x = 9$

Answer: The number is 9.

Check: $3(9) - 7 = 27 - 7 = 20$ ✓

Example 2: Age problem (medium)

Problem: Sam is 5 years older than his sister Lily. The sum of their ages is 31. How old is each person?

Step 1: Let $x$ = Lily's age. Then Sam's age = $x + 5$.

Step 2: Translate "the sum of their ages is 31":

$x + (x + 5) = 31$

Step 3: Combine like terms.

$2x + 5 = 31$

Step 4: Subtract 5 from both sides.

$2x = 26$

Step 5: Divide both sides by 2.

$x = 13$

Answer: Lily is 13 years old and Sam is 18 years old.

Check: $13 + 18 = 31$ ✓, and $18 - 13 = 5$ ✓

Example 3: Perimeter problem (medium)

Problem: The length of a rectangle is 3 cm more than twice its width. The perimeter is 54 cm. Find the dimensions.

Step 1: Let $w$ = width (in cm). Then length = $2w + 3$.

Step 2: Use the perimeter formula $P = 2l + 2w$:

$2(2w + 3) + 2w = 54$

Step 3: Distribute.

$4w + 6 + 2w = 54$

Step 4: Combine like terms.

$6w + 6 = 54$

Step 5: Subtract 6 and divide by 6.

$6w = 48 \implies w = 8$

Step 6: Find the length: $l = 2(8) + 3 = 19$.

Answer: Width = 8 cm, Length = 19 cm.

Check: Perimeter = $2(19) + 2(8) = 38 + 16 = 54$ ✓

Example 4: Money problem (medium)

Problem: A concert ticket costs $12 and there is a $5 service fee per order. If you spent $65 total, how many tickets did you buy?

Step 1: Let $t$ = number of tickets.

Step 2: Total cost = (price per ticket $\times$ number of tickets) + service fee:

$12t + 5 = 65$

Step 3: Subtract 5 from both sides.

$12t = 60$

Step 4: Divide by 12.

$t = 5$

Answer: You bought 5 tickets.

Check: $12(5) + 5 = 60 + 5 = 65$ ✓

Example 5: Consecutive integers (challenging)

Problem: The sum of three consecutive integers is 72. Find the integers.

Step 1: Let $x$ = the first integer. Then the three consecutive integers are $x$, $x + 1$, and $x + 2$.

Step 2: Write the equation:

$x + (x + 1) + (x + 2) = 72$

Step 3: Combine like terms.

$3x + 3 = 72$

Step 4: Subtract 3.

$3x = 69$

Step 5: Divide by 3.

$x = 23$

Answer: The three integers are 23, 24, and 25.

Check: $23 + 24 + 25 = 72$ ✓

Common Mistakes

Mistake 1: Reversing "less than"

❌ "7 less than a number" $\rightarrow 7 - x$

✅ "7 less than a number" $\rightarrow x - 7$

Why this matters: "Less than" subtracts from the main quantity, so the number being reduced comes first. "7 less than $x$" means start at $x$ and take away 7.

Mistake 2: Forgetting to define what $x$ represents

❌ Jumping straight to writing an equation without clearly stating what the variable means.

✅ Always write "Let $x$ = ..." before forming the equation.

Why this matters: Without a clear definition, you might assign $x$ to the wrong quantity and get an answer that does not match what the problem asks for.

Mistake 3: Not checking if the answer makes sense

❌ Finding $x = -3$ for "How many apples did she buy?" and writing that as the final answer.

✅ Recognizing that a negative count is impossible, then re-reading the problem to find the setup error.

Why this matters: Math can produce technically correct solutions that are impossible in real life. Always interpret your answer in the context of the problem.

Practice Problems

Try these on your own before checking the answers:

1. Five times a number, plus 8, equals 43. Find the number.
2. A triangle's second side is twice the first side, and the third side is 4 cm longer than the first. The perimeter is 40 cm. Find each side length.
3. Emma has $3 more than twice what Jake has. Together they have $33. How much does each person have?
4. The sum of two consecutive odd integers is 56. Find the integers.
5. A phone plan charges $25 per month plus $0.10 per text message. If the bill is $37, how many texts were sent?

Click to see answers

1. $5x + 8 = 43 \implies 5x = 35 \implies x = 7$. The number is 7.
2. Let $x$ = first side. $x + 2x + (x + 4) = 40 \implies 4x + 4 = 40 \implies 4x = 36 \implies x = 9$. Sides: 9 cm, 18 cm, 13 cm.
3. Let $x$ = Jake's amount. $x + (2x + 3) = 33 \implies 3x + 3 = 33 \implies 3x = 30 \implies x = 10$. Jake: $10, Emma: $23.
4. Let $x$ = first odd integer. $x + (x + 2) = 56 \implies 2x + 2 = 56 \implies 2x = 54 \implies x = 27$. Integers: 27 and 29.
5. $25 + 0.10t = 37 \implies 0.10t = 12 \implies t = 120$. 120 texts were sent.

Summary

• Use the 5-step strategy: read, define, translate, solve, check.
• Learn the key words that map to math operations (sum = add, difference = subtract, product = multiply).
• "Less than" reverses the order: "$a$ less than $b$" means $b - a$.
• Always define your variable before writing the equation.
• Check that your answer makes sense in the real-world context of the problem.

Related Topics

• Linear Equations — How to Solve Step by Step
• Solving Two-Step Equations Step by Step
• Ratio Word Problems with Step-by-Step Solutions

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