Finding the Unknown — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Finding the Unknown" — 7 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Finding the Unknown" — 7 important questions with detailed answers for CBSE board e…
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NCERT Solutions for Class 7 Mathematics — Finding the Unknown. Step-by-step answers to all 7 textbook questions from this chapter, written for CBSE board preparation and free to use.
Finding the Unknown — All 7 Questions Solved
Q1. What is a linear equation? Solve: 2x + 5 = 13.
Linear Equation:
Definition: A linear equation is an equation of the first degree where the variable(s) have a power of 1.
It is called "linear" because when plotted on a graph, it forms a straight line.
Standard form (one variable): ax + b = c
Where a, b, c are constants and x is the variable (a ≠ 0)
Examples:
• 2x + 5 = 13
• 3x - 7 = 2
• x + 10 = 25
• 5x = 20
---
Solving: 2x + 5 = 13
Step 1: Identify the equation
2x + 5 = 13
We need to find the value of x that makes this equation true.
Step 2: Subtract 5 from both sides
2x + 5 - 5 = 13 - 5
2x = 8
Note: Whatever we do to one side of the equation, we must do to the other side to maintain equality.
Step 3: Divide both sides by 2
2x ÷ 2 = 8 ÷ 2
x = 4
Step 4: Verify the solution
Substitute x = 4 back into the original equation:
2(4) + 5 = 13
8 + 5 = 13
13 = 13 ✓ (True)
Answer: x = 4
---
Method Summary:
To solve linear equations:
1. Simplify both sides (combine like terms)
2. Move variable terms to one side, constants to the other
3. Use inverse operations (opposite operations):
• Subtract if added; Add if subtracted
• Divide if multiplied; Multiply if divided
4. Simplify to isolate the variable
5. Verify the solution by substituting back
Q2. Solve: 3x - 7 = 2 and verify the solution.
Solving: 3x - 7 = 2
Step 1: Add 7 to both sides
3x - 7 + 7 = 2 + 7
3x = 9
Explanation: The inverse of subtraction is addition.
We add 7 to both sides to isolate the term with x.
Step 2: Divide both sides by 3
3x ÷ 3 = 9 ÷ 3
x = 3
Explanation: The inverse of multiplication is division.
We divide by 3 to isolate x.
---
Step 3: Verify the solution
Substitute x = 3 back into the original equation:
3(3) - 7 = 2
9 - 7 = 2
2 = 2 ✓ (True)
The solution is verified!
---
Answer: x = 3
---
Solution Method Flowchart:
Original: 3x - 7 = 2
↓ (Add 7 to both sides)
3x = 9
↓ (Divide both sides by 3)
x = 3
↓ (Substitute back to verify)
3(3) - 7 = 2 ✓
---
General Strategy:
• Identify what operation is being done to x
• Use the inverse operation to undo it
• Work towards isolating x on one side
• Always perform the same operation on both sides
Q3. Solve: x/4 = 5. What is the value of x?
Solving: x/4 = 5
Step 1: Identify the operation
The variable x is being divided by 4.
x/4 = 5
This means: x ÷ 4 = 5
Step 2: Use the inverse operation
The inverse of division is multiplication.
Multiply both sides by 4:
(x/4) × 4 = 5 × 4
x = 20
Explanation: When we multiply x/4 by 4, we get:
(x/4) × 4 = x × (4/4) = x × 1 = x
Step 3: Verify the solution
Substitute x = 20 back into the original equation:
20/4 = 5
5 = 5 ✓ (True)
The solution is correct!
---
Answer: x = 20
---
Alternative Method (Using Balancing):
Original equation: x/4 = 5
What this means: If x is divided into 4 equal parts, each part equals 5.
Therefore, the whole x = 5 × 4 = 20
---
Step-by-Step Solution:
Starting equation: x/4 = 5
Multiply both sides by 4: x/4 × 4 = 5 × 4
Simplify: x = 20
Verify: 20/4 = 5 ✓
---
Key Principle:
To isolate a variable that is divided by a number, multiply both sides by that number.
Q4. Form an equation and solve: Five times a number decreased by 3 is 22. Find the number.
Problem: Five times a number decreased by 3 is 22. Find the number.
---
Step 1: Define the variable
Let x = the unknown number
Step 2: Form the equation
Break down the sentence:
• "Five times a number" = 5x
• "decreased by 3" = minus 3 = - 3
• "is 22" = = 22
Equation: 5x - 3 = 22
---
Step 3: Solve the equation
5x - 3 = 22
Add 3 to both sides:
5x - 3 + 3 = 22 + 3
5x = 25
Divide both sides by 5:
5x ÷ 5 = 25 ÷ 5
x = 5
---
Step 4: Verify the solution
Substitute x = 5 back into the equation:
5(5) - 3 = 22
25 - 3 = 22
22 = 22 ✓ (True)
Also verify with the original statement:
Five times 5 = 5 × 5 = 25
Decreased by 3 = 25 - 3 = 22 ✓
---
Answer: The number is 5.
---
Key Steps for Word Problems:
1. Read the problem carefully
2. Define a variable for the unknown
3. Translate words into mathematical expressions:
• "times" → multiplication (×)
• "decreased by" → subtraction (-)
• "increased by" → addition (+)
• "divided by" → division (÷)
4. Form the equation
5. Solve the equation
6. Verify the solution
7. State the answer in words
Q5. Solve: 2(x + 3) = 16 and verify.
Solving: 2(x + 3) = 16
Step 1: Expand the bracket (distribute the 2)
2(x + 3) = 2·x + 2·3
2x + 6 = 16
Explanation: Distributive property:
a(b + c) = ab + ac
Step 2: Subtract 6 from both sides
2x + 6 - 6 = 16 - 6
2x = 10
Step 3: Divide both sides by 2
2x ÷ 2 = 10 ÷ 2
x = 5
---
Step 4: Verify the solution
Substitute x = 5 back into the original equation:
2(5 + 3) = 16
2(8) = 16
16 = 16 ✓ (True)
The solution is verified!
---
Answer: x = 5
---
Alternative Method (Without Expanding):
Original: 2(x + 3) = 16
Step 1: Divide both sides by 2
2(x + 3)/2 = 16/2
x + 3 = 8
Step 2: Subtract 3 from both sides
x + 3 - 3 = 8 - 3
x = 5
This method is often faster when the whole equation is multiplied.
---
Solution Summary:
Method 1 (Expand first):
2(x + 3) = 16
2x + 6 = 16
2x = 10
x = 5 ✓
Method 2 (Divide first):
2(x + 3) = 16
x + 3 = 8
x = 5 ✓
Both methods give the same answer.
Q6. Solve the equation: (x - 4)/3 = 2 and check your answer.
Solving: (x - 4)/3 = 2
Step 1: Identify the operation
The expression (x - 4) is being divided by 3.
Step 2: Multiply both sides by 3
[(x - 4)/3] × 3 = 2 × 3
x - 4 = 6
Explanation: When we multiply (x - 4)/3 by 3, the division by 3 is canceled.
[(x - 4)/3] × 3 = (x - 4) × (3/3) = (x - 4) × 1 = x - 4
Step 3: Add 4 to both sides
x - 4 + 4 = 6 + 4
x = 10
Step 4: Check (Verify) the solution
Substitute x = 10 back into the original equation:
(10 - 4)/3 = 2
6/3 = 2
2 = 2 ✓ (True)
The solution is verified!
---
Answer: x = 10
---
Step-by-Step Solution Table:
| Step | Operation | Equation | Reason |
|------|-----------|----------|--------|
| 1 | Original | (x - 4)/3 = 2 | Starting equation |
| 2 | Multiply by 3 | x - 4 = 6 | Remove division by 3 |
| 3 | Add 4 | x = 10 | Isolate x |
| 4 | Verify | (10 - 4)/3 = 2 | Check: 6/3 = 2 ✓ |
---
Key Principle:
To solve equations with fractions:
1. Multiply both sides by the denominator to eliminate the fraction
2. Continue solving as before
3. Always verify by substituting back
Q7. A number when multiplied by 6 and then 5 is added gives 47. Find the number.
Problem: A number when multiplied by 6 and then 5 is added gives 47. Find the number.
---
Step 1: Define the variable
Let x = the unknown number
Step 2: Form the equation
Break down the sentence:
• "A number" = x
• "when multiplied by 6" = 6x
• "and then 5 is added" = + 5
• "gives 47" = = 47
Equation: 6x + 5 = 47
---
Step 3: Solve the equation
6x + 5 = 47
Subtract 5 from both sides:
6x + 5 - 5 = 47 - 5
6x = 42
Divide both sides by 6:
6x ÷ 6 = 42 ÷ 6
x = 7
---
Step 4: Verify the solution
Substitute x = 7 back into the equation:
6(7) + 5 = 47
42 + 5 = 47
47 = 47 ✓ (True)
Also check with the original statement:
Multiply 7 by 6: 7 × 6 = 42
Add 5: 42 + 5 = 47 ✓
---
Answer: The number is 7.
---
Solution Process:
Given number = x
↓
Multiply by 6 → 6x
↓
Add 5 → 6x + 5
↓
Result is 47 → 6x + 5 = 47
↓ (Solve)
Subtract 5 → 6x = 42
↓
Divide by 6 → x = 7
↓ (Verify)
Check: 7 × 6 + 5 = 42 + 5 = 47 ✓
---
General Strategy for Word Problems:
1. Identify the unknown → assign variable x
2. Translate each operation in order:
- "multiplied by 6" → multiply first
- "then 5 is added" → add after
3. Form equation with the final result
4. Solve using inverse operations in reverse order
5. Always verify the answer
Finding the Unknown — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. If x + 8 = 15, what is the value of x?
- A. 6
- B. 7
- C. 8
- D. 9
Answer: B. 7 — To find x, subtract 8 from both sides of the equation: x = 15 - 8 = 7.
MCQ 2. Solve for y: y - 12 = 5.
- A. 7
- B. 10
- C. 15
- D. 17
Answer: D. 17 — To find y, add 12 to both sides of the equation: y = 5 + 12 = 17.
MCQ 3. Find the value of z if 6z = 42.
- A. 6
- B. 7
- C. 8
- D. 9
Answer: B. 7 — To find z, divide both sides by 6: z = 42 / 6 = 7.
MCQ 4. If a number divided by 4 gives 9, what is the number?
- A. 24
- B. 28
- C. 32
- D. 36
Answer: D. 36 — Let the number be x. The equation is x/4 = 9. Multiply both sides by 4: x = 9 × 4 = 36.
MCQ 5. What is the number if you add 7 to it and get 23?
- A. 14
- B. 15
- C. 16
- D. 17
Answer: C. 16 — Let the number be x. The equation is x + 7 = 23. Subtract 7 from both sides: x = 23 - 7 = 16.
MCQ 6. Solve the equation: 2x + 5 = 19.
- A. 6
- B. 7
- C. 8
- D. 9
Answer: B. 7 — First, subtract 5 from both sides: 2x = 19 - 5 = 14. Then, divide by 2: x = 14 / 2 = 7.
MCQ 7. If 3y - 8 = 13, what is y?
- A. 5
- B. 6
- C. 7
- D. 8
Answer: C. 7 — First, add 8 to both sides: 3y = 13 + 8 = 21. Then, divide by 3: y = 21 / 3 = 7.
MCQ 8. Find the value of p in p/3 - 2 = 4.
- A. 12
- B. 15
- C. 18
- D. 21
Answer: C. 18 — First, add 2 to both sides: p/3 = 4 + 2 = 6. Then, multiply by 3: p = 6 × 3 = 18.
MCQ 9. A number decreased by 9 gives 11. What is the number?
- A. 2
- B. 18
- C. 20
- D. 22
Answer: C. 20 — Let the number be n. The equation is n - 9 = 11. Add 9 to both sides: n = 11 + 9 = 20.
MCQ 10. If x/5 = 7, then x equals:
- A. 12
- B. 25
- C. 30
- D. 35
Answer: D. 35 — To find x, multiply both sides of the equation by 5: x = 7 × 5 = 35.
MCQ 11. The sum of two consecutive integers is 45. What is the smaller integer?
- A. 21
- B. 22
- C. 23
- D. 24
Answer: B. 22 — Let the integers be x and x+1. So, x + (x+1) = 45 implies 2x + 1 = 45 implies 2x = 44 implies x = 22. The smaller integer is 22.
MCQ 12. Ravi's father is 3 times as old as Ravi. If the sum of their ages is 48, how old is Ravi?
- A. 10
- B. 12
- C. 14
- D. 16
Answer: B. 12 — Let Ravi's age be x. His father's age is 3x. Their sum is x + 3x = 48 implies 4x = 48 implies x = 12. Ravi is 12 years old.
MCQ 13. A number multiplied by 4 and then decreased by 6 gives 26. Find the number.
- A. 6
- B. 7
- C. 8
- D. 9
Answer: C. 8 — Let the number be x. The equation is 4x - 6 = 26. Add 6 to both sides: 4x = 32. Divide by 4: x = 8.
MCQ 14. Solve for x: 5x - 3 = 2x + 12.
- A. 3
- B. 4
- C. 5
- D. 6
Answer: C. 5 — Subtract 2x from both sides: 3x - 3 = 12. Add 3 to both sides: 3x = 15. Divide by 3: x = 5.
MCQ 15. The perimeter of a square is 60 cm. What is the length of one side?
- A. 10 cm
- B. 12 cm
- C. 15 cm
- D. 20 cm
Answer: C. 15 cm — The perimeter of a square is 4 times the side length. So, 4s = 60 implies s = 60 / 4 = 15 cm.
MCQ 16. If x/2 + 1/4 = 3/4, find x.
- A. 0.5
- B. 1
- C. 1.5
- D. 2
Answer: B. 1 — Subtract 1/4 from both sides: x/2 = 3/4 - 1/4 = 2/4 = 1/2. Multiply by 2: x = 1.
MCQ 17. What is the number if 0.5x + 3 = 5.5?
- A. 4
- B. 5
- C. 6
- D. 7
Answer: B. 5 — Subtract 3 from both sides: 0.5x = 5.5 - 3 = 2.5. Divide by 0.5: x = 2.5 / 0.5 = 5.
MCQ 18. If you think of a number, multiply it by 3, and add 8, you get 29. What is the number?
- A. 5
- B. 7
- C. 9
- D. 11
Answer: B. 7 — Let the number be n. The equation is 3n + 8 = 29. Subtract 8: 3n = 21. Divide by 3: n = 7.
MCQ 19. Solve for x: (x+5)/2 = 8.
- A. 9
- B. 11
- C. 13
- D. 15
Answer: B. 11 — Multiply both sides by 2: x + 5 = 16. Subtract 5 from both sides: x = 16 - 5 = 11.
MCQ 20. Two complementary angles differ by 10 degrees. Find the larger angle. (Complementary angles sum to 90 degrees).
- A. 40 degrees
- B. 45 degrees
- C. 50 degrees
- D. 55 degrees
Answer: C. 50 degrees — Let the angles be x and x+10. Their sum is x + (x+10) = 90 implies 2x + 10 = 90 implies 2x = 80 implies x = 40. The larger angle is x+10 = 40+10 = 50 degrees.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.