Question 1 – All Parts
Part (i)
📋 Given Information
- Total number of students = 10
- Number of girls is 4 more than the number of boys
- We need to find the number of boys and girls
🎯 To Find
The number of boys and the number of girls who took part in the quiz.
📐 Forming the Equations
Let the number of boys be \( x \) and the number of girls be \( y \).
From the given conditions:
Explanation: Equation (1) represents the total number of students, and equation (2) represents that girls are 4 more than boys.
📝 Step-by-Step Solution
From equation (1): \( x + y = 10 \)
From equation (2): \( y – x = 4 \) or \( x – y = -4 \)
For \( x + y = 10 \):
When \( x = 0 \): \( y = 10 \) → Point: \( (0, 10) \)
When \( y = 0 \): \( x = 10 \) → Point: \( (10, 0) \)
When \( x = 5 \): \( y = 5 \) → Point: \( (5, 5) \)
For \( y = x + 4 \):
When \( x = 0 \): \( y = 4 \) → Point: \( (0, 4) \)
When \( x = 3 \): \( y = 7 \) → Point: \( (3, 7) \)
When \( x = 6 \): \( y = 10 \) → Point: \( (6, 10) \)
When we plot both lines on a graph paper with x-axis representing boys and y-axis representing girls, the two lines intersect at the point \( (3, 7) \).
This means: \( x = 3 \) and \( y = 7 \)
Check in equation (1): \( 3 + 7 = 10 \) ✓
Check in equation (2): \( 7 = 3 + 4 = 7 \) ✓
Both equations are satisfied!
💡 Alternative Method: Substitution Method
Algebraic Solution
From equation (2): \( y = x + 4 \)
Substitute this in equation (1):
\[ x + (x + 4) = 10 \] \[ 2x + 4 = 10 \] \[ 2x = 6 \] \[ x = 3 \]Now substitute \( x = 3 \) in equation (2):
\[ y = 3 + 4 = 7 \]Therefore, \( x = 3 \) and \( y = 7 \)
Number of boys = 3
Number of girls = 7
Part (ii)
📋 Given Information
- Cost of 5 pencils and 7 pens = ₹50
- Cost of 7 pencils and 5 pens = ₹46
- We need to find the cost of one pencil and one pen
🎯 To Find
The cost of one pencil and the cost of one pen.
📐 Forming the Equations
Let the cost of one pencil be ₹\( x \) and the cost of one pen be ₹\( y \).
From the given conditions:
Explanation: These equations represent the total cost for different combinations of pencils and pens.
📝 Step-by-Step Solution
When \( x = 0 \): \( 7y = 50 \) → \( y = \frac{50}{7} \approx 7.14 \) → Point: \( (0, 7.14) \)
When \( y = 0 \): \( 5x = 50 \) → \( x = 10 \) → Point: \( (10, 0) \)
When \( x = 3 \): \( 5(3) + 7y = 50 \) → \( 15 + 7y = 50 \) → \( 7y = 35 \) → \( y = 5 \) → Point: \( (3, 5) \)
When \( x = 0 \): \( 5y = 46 \) → \( y = \frac{46}{5} = 9.2 \) → Point: \( (0, 9.2) \)
When \( y = 0 \): \( 7x = 46 \) → \( x = \frac{46}{7} \approx 6.57 \) → Point: \( (6.57, 0) \)
When \( x = 3 \): \( 7(3) + 5y = 46 \) → \( 21 + 5y = 46 \) → \( 5y = 25 \) → \( y = 5 \) → Point: \( (3, 5) \)
When we plot both lines on a graph paper with x-axis representing cost of pencil and y-axis representing cost of pen, we observe that both lines intersect at the point \( (3, 5) \).
This means: \( x = 3 \) and \( y = 5 \)
Check in equation (1): \( 5(3) + 7(5) = 15 + 35 = 50 \) ✓
Check in equation (2): \( 7(3) + 5(5) = 21 + 25 = 46 \) ✓
Both equations are satisfied!
💡 Alternative Method: Elimination Method
Algebraic Solution by Elimination
Given equations:
\[ 5x + 7y = 50 \quad \text{…(1)} \] \[ 7x + 5y = 46 \quad \text{…(2)} \]Step 1: Multiply equation (1) by 7 and equation (2) by 5:
\[ 35x + 49y = 350 \quad \text{…(3)} \] \[ 35x + 25y = 230 \quad \text{…(4)} \]Step 2: Subtract equation (4) from equation (3):
\[ (35x + 49y) – (35x + 25y) = 350 – 230 \] \[ 24y = 120 \] \[ y = 5 \]Step 3: Substitute \( y = 5 \) in equation (1):
\[ 5x + 7(5) = 50 \] \[ 5x + 35 = 50 \] \[ 5x = 15 \] \[ x = 3 \]Therefore, \( x = 3 \) and \( y = 5 \)
Cost of one pencil = ₹3
Cost of one pen = ₹5
⚠️ Common Mistakes to Avoid
❌ Mistake 1: Confusing which variable represents which quantity (boys/girls or pencils/pens).
✅ Correct: Always clearly define your variables at the start. For example, “Let x = number of boys” or “Let x = cost of one pencil”.
❌ Mistake 2: Plotting points incorrectly on the graph or using wrong scale.
✅ Correct: Choose an appropriate scale (like 1 cm = 1 unit) and mark points carefully. Use a ruler for drawing straight lines.
❌ Mistake 3: Not verifying the solution in both equations.
✅ Correct: Always substitute your answer back into both original equations to confirm it satisfies both conditions.
❌ Mistake 4: Writing equations incorrectly from word problems.
✅ Correct: Read the problem carefully. “Girls are 4 more than boys” means y = x + 4, not x = y + 4.
💡 Key Points to Remember
- Graphical Method: The solution of a pair of linear equations is the point where the two lines intersect.
- Consistent Equations: If lines intersect at exactly one point, the pair has a unique solution and is called consistent.
- Variable Definition: Always clearly define what each variable represents before forming equations.
- Verification: The solution must satisfy both equations simultaneously.
- Real-world Context: In word problems, the solution should make sense in the given context (e.g., number of students must be whole numbers).
- Graph Accuracy: Use graph paper, appropriate scale, and a ruler for accurate plotting.
- Multiple Methods: You can solve using graphical method, substitution method, or elimination method – all should give the same answer.
📝 Practice Similar Problems
Master these concepts by solving:
