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CBSE · Class 7 · Mathematics

Chapter 15: Finding the Unknown

5 concepts · 40 practice questions with answers · Free to read

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Forming equations

From daily life

A taxi charges ₹30 plus ₹12 per km. If your bill is ₹150, finding the distance is solving a puzzle written as an equation.

Simple definition

An equation says two expressions are equal. It has an unknown (a letter such as x) whose value we must find. A value that makes both sides equal is the solution.

x + 37x + 3 – 3 = 7 – 3 → x = 4

In the figure: An equation is like a balanced scale.

Live figure

Forming equations

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Rule 1: Words to symbols — more than → +, less than → –, times or twice → ×, shared or half → ÷, is → =.

  • Easy: "A number plus 7 is 19" → x + 7 = 19.
  • Medium: "5 more than twice a number is 17" → 2x + 5 = 17.
  • Exam-type: "Half of a number is 12" → x ÷ 2 = 12.

Rule 2: Substituting a value — Replace the letter by the number and compare both sides.

  • Easy: x = 4 in x + 9 = 13: 4 + 9 = 13 ✔.
  • Medium: x = 3 in 2x + 5 = 12: 11 ≠ 12 ✘.
  • Exam-type: x = 5 in 4x – 3 = 2x + 7: 17 = 17 ✔.

Rule 3: Situations as equations — Pick the unknown, then write what stays equal.

  • Easy: Square with perimeter 36: 4s = 36.
  • Medium: Priya is 4 years older than Sam, ages total 26: s + (s + 4) = 26.
  • Exam-type: Three consecutive numbers total 48: n + (n + 1) + (n + 2) = 48.
Remember

An equation has "=" ; an expression has no "=".

Common mistake

Writing "5 less than x" as 5 – x. It is x – 5.

Shortcut

Test your equation with a number you can guess. Memory clue: "Less than: flip the order."

Spot the difference

TermHas "="?
Expression: 2x + 5No
Equation: 2x + 5 = 17Yes

Practice questions

Easy
  1. 1Value of 4x – 3 when x = 5?
  2. 2Is x = 3 a solution of 2x + 7 = 13?
  3. 3Which equation: "twice a number plus 5 is 19"?
Medium
  1. 4Value of 3x + 8 when x = –1?
  2. 5Which value of x makes x + 14 = 20?
  3. 6"A number divided by 4 is 9": equation?
Challenging
  1. 7Three consecutive numbers total 36. The smallest?
  2. 8Which x makes 5x – 2 = 3x + 8?
Show answers (try them yourself first)
  1. 117
  2. 2Yes
  3. 32x + 5 = 19
  4. 45
  5. 56
  6. 6x ÷ 4 = 9
  7. 711
  8. 85

One-step equations

From daily life

A balanced scale stays balanced if you add or remove the same weight from both pans. That is how equations are solved.

Simple definition

To solve, do the same thing to both sides until the unknown stands alone. Undo + by –, – by +, × by ÷ and ÷ by ×.

x + 37x + 3 – 3 = 7 – 3 → x = 4

In the figure: Subtract 3 from both pans: x = 4.

Live figure

One-step equations

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Rule 1: Add and subtract — Undo the addition or subtraction on both sides.

  • Easy: x + 7 = 15 → x = 15 – 7 = 8.
  • Medium: x – 13 = 24 → x = 24 + 13 = 37.
  • Exam-type: x + 9 = 2 → x = 2 – 9 = –7.

Rule 2: Multiply and divide — Undo × by ÷ and ÷ by ×.

  • Easy: 4x = 28 → x = 28 ÷ 4 = 7.
  • Medium: x ÷ 5 = 6 → x = 6 × 5 = 30.
  • Exam-type: –5x = 35 → x = 35 ÷ (–5) = –7.

Rule 3: Check by substituting — Put the answer back in the original equation.

  • Easy: x = 8: 8 + 7 = 15 ✔.
  • Medium: x = 30: 30 ÷ 5 = 6 ✔.
  • Exam-type: x = –7: –5 × (–7) = 35 ✔.
Remember

Whatever you do to one side, do to the other.

Common mistake

Moving a number without changing its sign. Use the opposite operation.

Shortcut

Ask: what was done to x? Then undo it. Memory clue: "Do and undo on both sides."

Spot the difference

Equation hasUndo with
x + a– a
x – a+ a
a × x÷ a
x ÷ a× a

Practice questions

Easy
  1. 1Solve x + 12 = 30.
  2. 2Solve x – 9 = 14.
  3. 3Solve 6x = 54.
Medium
  1. 4Solve x ÷ 7 = 4.
  2. 5Solve 25 = x + 8.
  3. 6Solve –4x = 28.
Challenging
  1. 7Solve x + 15 = 6.
  2. 8Solve x ÷ (–3) = 5.
Show answers (try them yourself first)
  1. 118
  2. 223
  3. 39
  4. 428
  5. 517
  6. 6–7
  7. 7–9
  8. 8–15

Two-step equations

From daily life

Think of a number, double it and add 5 to get 17. To find the number, undo the steps backwards: subtract 5, then halve.

Simple definition

Two operations were applied to x. Undo them in reverse order: first the add/subtract, then the multiply/divide, always on both sides.

x→ ×2 →2x→ +5 →176← ÷2 ←12← –5 ←172x + 5 = 17 → x = 6

In the figure: Reverse the steps: –5 first, then ÷2.

Live figure

Two-step equations

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Rule 1: Form ax + b = c — Subtract b (or add), then divide by a.

  • Easy: 2x + 3 = 11 → 2x = 8 → x = 4.
  • Medium: 3x – 5 = 16 → 3x = 21 → x = 7.
  • Exam-type: 7x + 9 = 2 → 7x = –7 → x = –1.

Rule 2: Form x ÷ a + b = c — Remove b first, then multiply by a.

  • Easy: x ÷ 2 + 3 = 8 → x ÷ 2 = 5 → x = 10.
  • Medium: x ÷ 3 – 4 = 2 → x ÷ 3 = 6 → x = 18.
  • Exam-type: (x + 3) ÷ 4 = 5 → x + 3 = 20 → x = 17.

Rule 3: Verify — Put x back into the left side.

  • Easy: x = 4: 2 × 4 + 3 = 11 ✔.
  • Medium: x = 7: 3 × 7 – 5 = 16 ✔.
  • Exam-type: x = –1: 7 × (–1) + 9 = 2 ✔.
Remember

Undo the last operation first.

Common mistake

Dividing before removing the +b: 2x + 3 = 11 is not x + 3 = 5.

Shortcut

Add/subtract first, multiply/divide second. Memory clue: "Peel the outside first."

Spot the difference

Order of undoingStep
1st+ or – (c ± b)
2nd× or ÷

Practice questions

Easy
  1. 1Solve 3x + 2 = 20.
  2. 2Solve 5x – 4 = 21.
  3. 3Solve x ÷ 4 + 3 = 7.
Medium
  1. 4Solve 9x + 7 = –2.
  2. 5Solve 4x – 11 = 13.
  3. 6Solve 2x + 9 = 1.
Challenging
  1. 7Solve (x – 5) ÷ 3 = 4.
  2. 8Solve 6x ÷ 5 = 12.
Show answers (try them yourself first)
  1. 16
  2. 25
  3. 316
  4. 4–1
  5. 56
  6. 6–4
  7. 717
  8. 810

Brackets and variables on both sides

From daily life

Two taxis charge differently: ₹5 + ₹3 per km and ₹1 + ₹4 per km. They cost the same at some distance, an equation with x on both sides.

Simple definition

Remove brackets using the distributive property, collect the x terms on one side and the numbers on the other, then solve. Moving a term across “=” changes its sign.

3(x + 2) = 213x + 6 = 213x = 15x = 5

In the figure: Expand 3(x + 2) = 3x + 6, then solve.

Live figure

Brackets and variables on both sides

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Rule 1: Expand brackets — Multiply the outside number with each term inside.

  • Easy: 3(x + 2) = 21 → 3x + 6 = 21 → x = 5.
  • Medium: 2(x – 3) = 10 → x – 3 = 5 → x = 8.
  • Exam-type: 2(3x – 1) = 16 → 6x – 2 = 16 → x = 3.

Rule 2: Unknown on both sides — Subtract the smaller x-term from both sides.

  • Easy: 4x = x + 15 → 3x = 15 → x = 5.
  • Medium: 5x – 3 = 2x + 9 → 3x = 12 → x = 4.
  • Exam-type: 9x – 5 = 4x + 20 → 5x = 25 → x = 5.

Rule 3: Brackets on both sides — Expand both, then collect like terms.

  • Easy: 5(x + 1) = 3(x + 5) → 5x + 5 = 3x + 15 → x = 5.
  • Medium: 4(x – 2) = 2(x + 4) → 4x – 8 = 2x + 8 → x = 8.
  • Exam-type: 3(2x + 1) = 5x + 10 → x = 7.
Remember

Multiply every term inside the bracket.

Common mistake

Writing 3(x + 2) = 3x + 2. The 2 must be multiplied too: 3x + 6.

Shortcut

Draw arrows from the outside number to each inside term. Memory clue: "Bracket: share with all."

Spot the difference

StepAction
1Expand brackets
2Collect x terms
3Solve

Practice questions

Easy
  1. 1Solve 2(x + 4) = 18.
  2. 2Solve 3(x – 2) = 12.
  3. 3Solve 6x = 2x + 20.
Medium
  1. 4Solve 8x – 5 = 3x + 15.
  2. 5Solve 4(2x – 1) = 28.
  3. 6Solve 3(x + 2) = 2x + 11.
Challenging
  1. 7Solve 5(x – 1) = 3(x + 3).
  2. 8Solve (3x – 2) ÷ 4 = x.
Show answers (try them yourself first)
  1. 15
  2. 26
  3. 35
  4. 44
  5. 54
  6. 65
  7. 77
  8. 8–2

Word problems and checking answers

From daily life

A taxi bill, a shared meal, ages in a family: each real story can become an equation, and the answer must also make sense in the story.

Simple definition

Read, choose the unknown, write the equation, solve, then check in the story. Typical mistakes are wrong signs, forgetting to multiply the whole bracket, and answers that do not fit the story.

ssss4s = 36Square of perimeter 36 → s = 9

In the figure: A square with perimeter 36 gives 4s = 36, so s = 9.

Rule 1: Numbers and amounts — Translate each sentence into one side of the equation.

  • Easy: A number plus 15 is 40: x + 15 = 40 → 25.
  • Medium: Twice a number is 38: 2x = 38 → 19.
  • Exam-type: Taxi ₹30 + ₹12 per km = ₹150: 12k + 30 = 150 → k = 10.

Rule 2: Ages, perimeters, consecutive numbers — Express every unknown with one letter.

  • Easy: Rectangle length = breadth + 3, perimeter 34: 4b + 6 = 34 → b = 7.
  • Medium: Three consecutive numbers total 48: 3n + 3 = 48 → n = 15.
  • Exam-type: Father is 3 times son's age, total 56: 4s = 56 → s = 14.

Rule 3: Check the story — Put the answer into the story, not just the equation.

  • Easy: Taxi 10 km: 30 + 120 = 150 ✔.
  • Medium: Ages 14 and 42 total 56 ✔.
  • Exam-type: Mistake: 2x + 3 = 11 → 2x = 14 is wrong; it is 2x = 8, x = 4.
Remember

Always check the answer in the story.

Common mistake

Adding 3 to both sides when 3 was added on one side. Subtract it.

Shortcut

Write "Let x be …" first so the letter has a meaning. Memory clue: "Read, Let, Write, Solve, Check."

Spot the difference

PhraseSymbol
more than+
less than–
times, twice×

Practice questions

Easy
  1. 1Three times a number is 51. The number?
  2. 2Ages total 40, differ by 6. Younger?
  3. 3Three consecutive numbers total 72. Largest?
Medium
  1. 48 pens cost ₹96. Price of one?
  2. 5Fee ₹20 plus ₹15 per kg, total ₹110. How many kg?
  3. 6Square of perimeter 52. Side?
Challenging
  1. 75x = 20 was solved as x = 20 – 5. Correct x?
  2. 8Twice a number minus 7 equals thrice the number minus 15. Number?
Show answers (try them yourself first)
  1. 117
  2. 217
  3. 325
  4. 412
  5. 56
  6. 613
  7. 74
  8. 88