Form the pair of linear equations in the following problems and find their solution (if they exist) by the elimination method: (i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction? (ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu? (iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

Solutions: (i) Fraction = 3/5; (ii) Nuri = 50 years, Sonu = 20 years; (iii) Number = 18

Exercise 3.3 – Question 3: Elimination Method

📚 About This Question

Focus: Form pairs of linear equations from word problems and solve using elimination method.

Elimination Method: Add or subtract equations to eliminate one variable, then solve for the other.

📐 Elimination Method – Steps

Step 1: Write both equations in standard form
Step 2: Make coefficients of one variable equal (by multiplying)
Step 3: Add or subtract equations to eliminate that variable
Step 4: Solve for the remaining variable
Step 5: Substitute back to find the other variable
Step 6: Verify the solution

Part (i) – Fraction Problem

Problem: If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes \( \frac{1}{2} \) if we only add 1 to the denominator. What is the fraction?

Step 1: Define Variables

Let:

Numerator of the fraction = \( x \)

Denominator of the fraction = \( y \)

Original fraction = \( \frac{x}{y} \)

Step 2: Form Equations

Condition 1: “Add 1 to numerator and subtract 1 from denominator, fraction becomes 1” \[ \frac{x + 1}{y – 1} = 1 \]

Cross multiply:

\[ x + 1 = y – 1 \] \[ x – y = -2 \quad \text{…(1)} \]

Condition 2: “Add 1 to denominator, fraction becomes \( \frac{1}{2} \)” \[ \frac{x}{y + 1} = \frac{1}{2} \]

Cross multiply:

\[ 2x = y + 1 \] \[ 2x – y = 1 \quad \text{…(2)} \]

Step 3: Solve by Elimination Method

We have:

Equation (1): \( x – y = -2 \)

Equation (2): \( 2x – y = 1 \)

Subtract equation (1) from equation (2):

\[ (2x – y) – (x – y) = 1 – (-2) \] \[ 2x – y – x + y = 1 + 2 \] \[ x = 3 \]

Substitute \( x = 3 \) in equation (1): \[ 3 – y = -2 \] \[ -y = -5 \] \[ y = 5 \]

Step 4: Visual Representation

🔢 Fraction Transformation

Original Fraction: \( \frac{3}{5} \)

⬇️

Add 1 to numerator, subtract 1 from denominator:

\( \frac{3+1}{5-1} = \frac{4}{4} = 1 \) ✓

⬇️

Add 1 to denominator only:

\( \frac{3}{5+1} = \frac{3}{6} = \frac{1}{2} \) ✓

Step 5: Verification

Original fraction: \( \frac{3}{5} \)

Check condition 1:

\( \frac{3 + 1}{5 – 1} = \frac{4}{4} = 1 \) ✓

Check condition 2:

\( \frac{3}{5 + 1} = \frac{3}{6} = \frac{1}{2} \) ✓

✅ Answer: The fraction is \( \frac{3}{5} \)

Part (ii) – Age Problem (Nuri and Sonu)

Problem: Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

Step 1: Define Variables

Let:

Present age of Nuri = \( x \) years

Present age of Sonu = \( y \) years

Step 2: Form Equations

Condition 1: “Five years ago, Nuri was thrice as old as Sonu”

Five years ago: Nuri = \( x – 5 \), Sonu = \( y – 5 \)

\[ x – 5 = 3(y – 5) \] \[ x – 5 = 3y – 15 \] \[ x – 3y = -10 \quad \text{…(1)} \]

Condition 2: “Ten years later, Nuri will be twice as old as Sonu”

After 10 years: Nuri = \( x + 10 \), Sonu = \( y + 10 \)

\[ x + 10 = 2(y + 10) \] \[ x + 10 = 2y + 20 \] \[ x – 2y = 10 \quad \text{…(2)} \]

Step 3: Solve by Elimination Method

We have:

Equation (1): \( x – 3y = -10 \)

Equation (2): \( x – 2y = 10 \)

Subtract equation (2) from equation (1):

\[ (x – 3y) – (x – 2y) = -10 – 10 \] \[ x – 3y – x + 2y = -20 \] \[ -y = -20 \] \[ y = 20 \]

Substitute \( y = 20 \) in equation (2): \[ x – 2(20) = 10 \] \[ x – 40 = 10 \] \[ x = 50 \]

Step 4: Visual Timeline

📊 Age Timeline

Timeline showing ages at different points in time

Step 5: Verification

Check condition 1 (5 years ago):

Nuri: \( 50 – 5 = 45 \) years

Sonu: \( 20 – 5 = 15 \) years

\( 45 = 3 \times 15 \) ✓

Check condition 2 (10 years later):

Nuri: \( 50 + 10 = 60 \) years

Sonu: \( 20 + 10 = 30 \) years

\( 60 = 2 \times 30 \) ✓

✅ Answer: Present age of Nuri = 50 years, Present age of Sonu = 20 years

Part (iii) – Two-Digit Number Problem

Problem: The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

Step 1: Define Variables

Let:

Tens digit = \( x \)

Units digit = \( y \)

Original number = \( 10x + y \)

Reversed number = \( 10y + x \)

Step 2: Form Equations

Condition 1: “Sum of digits is 9” \[ x + y = 9 \quad \text{…(1)} \]

Condition 2: “Nine times the number = twice the reversed number” \[ 9(10x + y) = 2(10y + x) \] \[ 90x + 9y = 20y + 2x \] \[ 90x – 2x + 9y – 20y = 0 \] \[ 88x – 11y = 0 \]

Divide by 11:

\[ 8x – y = 0 \] \[ 8x = y \quad \text{…(2)} \]

Step 3: Solve by Elimination Method

From equation (2): \[ y = 8x \quad \text{…(3)} \] Substitute in equation (1): \[ x + 8x = 9 \] \[ 9x = 9 \] \[ x = 1 \]

Substitute \( x = 1 \) in equation (3): \[ y = 8(1) = 8 \]

Therefore:

Original number = \( 10x + y = 10(1) + 8 = 18 \)

Step 4: Visual Representation

🔢 Number Transformation

Original Number: 18

(Tens digit: 1, Units digit: 8)

⬇️

Sum of digits: 1 + 8 = 9 ✓

⬇️

Reversed Number: 81

(Tens digit: 8, Units digit: 1)

⬇️

Check: 9 × 18 = 162 and 2 × 81 = 162 ✓

Step 5: Verification

Original number: 18

Reversed number: 81

Check condition 1:

Sum of digits: \( 1 + 8 = 9 \) ✓

Check condition 2:

Nine times original: \( 9 \times 18 = 162 \)

Twice the reversed: \( 2 \times 81 = 162 \)

\( 162 = 162 \) ✓

✅ Answer: The number is 18

📊 Summary of All Solutions

Part	Problem Type	Solution
(i)	Fraction problem	\( \frac{3}{5} \)
(ii)	Age problem	Nuri: 50 years, Sonu: 20 years
(iii)	Two-digit number	18

⚠️ Common Mistakes to Avoid

❌ Mistake 1: In fraction problems, forgetting to cross-multiply correctly.

✅ Correct: \( \frac{a}{b} = \frac{c}{d} \) means \( ad = bc \)

❌ Mistake 2: In age problems, confusing past, present, and future ages.

✅ Correct: Draw a timeline to visualize clearly.

❌ Mistake 3: In digit problems, writing the number incorrectly.

✅ Correct: Two-digit number with tens digit x and units digit y is \( 10x + y \), NOT \( x + y \)

❌ Mistake 4: Sign errors when subtracting equations in elimination method.

✅ Correct: Be careful: \( (a – b) – (c – d) = a – b – c + d \)

💡 Key Points to Remember

Elimination Method:
- Make coefficients equal by multiplication
- Add equations if signs are opposite
- Subtract equations if signs are same
Fraction Problems:
- Let numerator = x, denominator = y
- Fraction = \( \frac{x}{y} \)
- Always cross-multiply to form equations
Age Problems:
- Define present ages clearly
- Past age = Present age – years
- Future age = Present age + years
- Use timeline for visualization
Digit Problems:
- Two-digit number: \( 10x + y \)
- Reversed number: \( 10y + x \)
- Three-digit number: \( 100x + 10y + z \)
Always verify: Check solution in both original conditions

Farhan Mansuri

M.Sc. Mathematics, B.Ed.

15+ Years Teaching Experience

Passionate about making mathematics easy and enjoyable for students.

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Class 10 Maths Chapter 3 Exercise 3.3 Question 3

Exercise 3.3 – Question 3: Elimination Method

📚 About This Question

📐 Elimination Method – Steps

📑 Question Parts

Part (i) – Fraction Problem

Step 1: Define Variables

Step 2: Form Equations

Step 3: Solve by Elimination Method

Step 4: Visual Representation

🔢 Fraction Transformation

Step 5: Verification

Part (ii) – Age Problem (Nuri and Sonu)

Step 1: Define Variables

Step 2: Form Equations

Step 3: Solve by Elimination Method

Step 4: Visual Timeline

📊 Age Timeline

Step 5: Verification

Part (iii) – Two-Digit Number Problem

Step 1: Define Variables

Step 2: Form Equations

Step 3: Solve by Elimination Method

Step 4: Visual Representation

🔢 Number Transformation

Step 5: Verification

📊 Summary of All Solutions

⚠️ Common Mistakes to Avoid

💡 Key Points to Remember

Farhan Mansuri

Leave a Comment Cancel Reply

Exercise 3.3 – Question 3: Elimination Method

📚 About This Question

📐 Elimination Method – Steps

📑 Question Parts

Part (i) – Fraction Problem

Step 1: Define Variables

Step 2: Form Equations

Step 3: Solve by Elimination Method

Step 4: Visual Representation

🔢 Fraction Transformation

Step 5: Verification

Part (ii) – Age Problem (Nuri and Sonu)

Step 1: Define Variables

Step 2: Form Equations

Step 3: Solve by Elimination Method

Step 4: Visual Timeline

📊 Age Timeline

Step 5: Verification

Part (iii) – Two-Digit Number Problem

Step 1: Define Variables

Step 2: Form Equations

Step 3: Solve by Elimination Method

Step 4: Visual Representation

🔢 Number Transformation

Step 5: Verification

📊 Summary of All Solutions

⚠️ Common Mistakes to Avoid

💡 Key Points to Remember

📚 Related Questions

Farhan Mansuri

Leave a Comment Cancel Reply