MathsICSE Class 8 · Linear Equations in One Variable

Word Problems on Linear Equations: Numbers, Ages, Money and Speed (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

Choose a variable for the unknown, translate each sentence into algebra, form one equation, solve it, and check against the words of the problem. Use $x, x + 2$ for consecutive even or odd numbers, and total value $=$ number of items $\times$ value of each item.

Follow the five steps in order, then go back to the words of the problem to check that your answer fits, as 14 + 16 = 30 does.

Illustration drawn for QuestionBuddy.

Understand Word Problems on Linear Equations

The curriculum asks for linear equations in contextual problems with multiplication and division, avoiding complex coefficients.

Steps.
1. Read twice; identify the unknown.
2. Let a letter stand for it (with units).
3. Write other quantities in terms of that letter.
4. Form an equation from the single equality stated in the problem.
5. Solve and check against the original wording.

Typical patterns.

Numbers: consecutive integers $x, x + 1, x + 2$ ; consecutive even or odd numbers $x, x + 2, x + 4$ ; "five times a number decreased by 8" is $5 x - 8$ .
Ages: present ages $x$ and $y$ ; after $n$ years $x + n$ ; $n$ years ago $x - n$ . Use the stated relation at the stated time.
Money: value $=$ count $\times$ denomination. If 40 notes are of Rs 50 and Rs 100 and $a$ are Rs 50, then $40 - a$ are Rs 100.
Fractions: fraction $\frac{x}{x + 5}$ when the denominator exceeds the numerator by $5$ .
Geometry: perimeter $2 (l + b)$ with $l = 2 b$ .
Distance, speed, time: $d = v t$ ; for a round trip with equal distance $d$ , time $= \frac{d}{v _{1}} + \frac{d}{v _{2}}$ .

Check. Verify each condition in words: do the numbers have the stated sum, difference or ratio? Is the answer sensible (positive age, whole number of coins)?

Word Problems on Linear Equations: formulas to remember

Consecutive numbers: $x, x + 1, x + 2 (even/odd: x, x + 2, x + 4)$; Choose by the wording.
Total value: $total = count \times value of each$; For coins and notes.
Distance: $d = v \times t$; Keep units consistent.

How to solve Word Problems on Linear Equations questions

Define the unknown, translate each phrase, then solve and check against the words.

1Underline what is asked and let a letter be that.
2Express the other quantities in terms of the letter.
3Form the equation from the single equality in the problem.
4Solve the equation.
5Check each condition in the problem and write the answer in words with units.

Word Problems on Linear Equations in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Two-way trip

Suppose a bus goes from town A to town B at 60 km/h and returns along the same road at 40 km/h, taking 5 hours in all.

Find the distance between the towns.

Try it first, then show the solutionSolution

$\frac{d}{60} + \frac{d}{40} = 5$ .
Multiply by $120$ : $2 d + 3 d = 600$ , so $d = 120$ .

Answer: 120 km.

Memory hook: Same distance, two speeds: add the times.

Tricks to remember Word Problems on Linear Equations

Short memory aids. Say each one aloud once, then use it on a question the same day.

Even and odd steps
Consecutive evens or odds differ by 2, not 1.
Use it when: Number problems.
Check with words
Substitute back into the story, not just the equation.
Use it when: Catching translation errors.

Word Problems on Linear Equations: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Consecutive even numbers · 1 mark · MCQBoard level

The sum of two consecutive even numbers is 74. The numbers are:
(a) 35, 39 (b) 36, 38 (c) 34, 40 (d) 37, 37

Try it first, then show the solutionSolution

$x + (x + 2) = 74 \Rightarrow 2 x = 72 \Rightarrow x = 36$ ; numbers $36, 38$ .

Answer: (b)

Example 2: Number problem · 3 marks · Short answerBoard level

Five times a number decreased by 8 equals three times the number increased by 12. Find the number.

Try it first, then show the solutionSolution

$5 x - 8 = 3 x + 12$ .
$2 x = 20 \Rightarrow x = 10$ .
Check: $50 - 8 = 42$ and $30 + 12 = 42$ .

Answer: 10

Example 3: Ages · 3 marks · Short answerHarder board level

A father is 4 times as old as his son. In 6 years he will be 3 times as old as his son. Find their present ages.

Try it first, then show the solutionSolution

Son $x$ , father $4 x$ . After 6 years: $4 x + 6 = 3 (x + 6)$ .
$4 x + 6 = 3 x + 18 \Rightarrow x = 12$ .
Son $12$ , father $48$ . Check: $54 = 3 \times 18$ .

Answer: Son 12 years; father 48 years

Example 4: Notes of two kinds · 4 marks · Short answerHarder board level

A cashier has 40 notes of Rs 50 and Rs 100 worth Rs 2,750 in all. How many notes of each kind are there?

Try it first, then show the solutionSolution

Let there be $a$ notes of Rs 50; then $40 - a$ of Rs 100.
$50 a + 100 (40 - a) = 2750 \Rightarrow 4000 - 50 a = 2750$ .
$50 a = 1250 \Rightarrow a = 25$ ; notes of Rs 100: $15$ .
Check: $1250 + 1500 = 2750$ .

Answer: 25 notes of Rs 50 and 15 notes of Rs 100

Example 5: A fraction problem · 4 marks · Long answerHOTS / competency

The denominator of a fraction is 5 more than its numerator. If 3 is added to both numerator and denominator, the fraction becomes $\frac{3}{4}$ . Find the fraction.

Try it first, then show the solutionSolution

Fraction $= \frac{x}{x + 5}$ . New fraction $\frac{x + 3}{x + 8} = \frac{3}{4}$ .
$4 (x + 3) = 3 (x + 8) \Rightarrow 4 x + 12 = 3 x + 24 \Rightarrow x = 12$ .
Fraction $= \frac{12}{17}$ . Check: $\frac{15}{20} = \frac{3}{4}$ .

Answer: $\frac{12}{17}$

Where students lose marks in Word Problems on Linear Equations

The slips examiners see most often, and what to write instead.

Mistake: Using consecutive integers when evens are asked
Consecutive even numbers differ by 2.
Wrong working
$x + (x + 1) = 74$ .
Right working
$x + (x + 2) = 74$ .
Do this: Read the word "even" or "odd" before choosing the step.
Usually costs: Whole question.
Mistake: Not interpreting "years ago" correctly
Past ages are smaller.
Wrong working
Four years ago: $(x + 4) + (x + 9) = 33$ .
Right working
$(x - 4) + (x + 1) = 33$ .
Do this: Subtract for the past, add for the future.
Usually costs: Answer mark.

Exam tips

State "Let ... be ..." with units.
Write the equation in words first, then in symbols.
Finish with a sentence answer and a check.

Practice questions on Word Problems on Linear Equations

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The length of a rectangle is twice its breadth and its perimeter is 36 cm. The breadth is:
(a) 6 cm (b) 8 cm (c) 12 cm (d) 18 cm
Hint
Perimeter = 2(2b + b).
Answer
(a)
Q2 · 2 marks · Short answerBoard level
Twice a number increased by 15 is 41. Find the number.
Hint
Form 2x + 15 = 41.
Answer
$2 x = 26 \Rightarrow x = 13$ .
Q3 · 3 marks · Short answerHarder board level
Ravi is 5 years older than Sita. Four years ago the sum of their ages was 33. Find their present ages.
Hint
Sita x, Ravi x + 5.
Answer
Four years ago: $(x - 4) + (x + 1) = 33 \Rightarrow 2 x - 3 = 33 \Rightarrow x = 18$ . Sita $18$ , Ravi $23$ .
Q4 · 3 marks · Short answerHarder board level
The sum of three consecutive multiples of 7 is 147. Find them.
Hint
Let them be 7(n-1), 7n, 7(n+1).
Answer
$21 n = 147 \Rightarrow n = 7$ . Numbers: $42, 49, 56$ .
Q5 · 4 marks · Case-basedHOTS / competency
A car goes from A to B at 60 km/h and returns at 40 km/h. The whole journey takes 5 hours. Find the distance AB and the average speed for the round trip.
Hint
Time = distance/speed on each leg.
Answer
$\frac{d}{60} + \frac{d}{40} = 5 \Rightarrow 5 d = 600 \Rightarrow d = 120$ km. Total distance $240$ km in $5$ h: average speed $48$ km/h.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Three consecutive odd numbers starting at x?Show answer
x, x + 2, x + 4.
2.Value of n notes of Rs 50?Show answer
Rs 50n.
3.Round trip time formula for equal distance d?Show answer
d/v1 + d/v2.
4.How do you represent age 5 years ago?Show answer
x - 5.

Word Problems on Linear Equations: frequently asked questions

What if my answer is negative or a fraction of a person?

Re-check the equation; ages and counts must be positive whole numbers.

Should I always define a letter?

Yes; it earns a method mark.