Question 5 – Rectangular Garden Perimeter Problem
Understanding the Problem
Given Information:
- The garden is rectangular in shape
- Length is 4 m more than width
- Half the perimeter = 36 m
To Find:
- Length of the garden
- Width of the garden
Visual Representation of the Garden
┌─────────────────────────┐
│ │
│ │ Width = x m
│ RECTANGULAR │
│ GARDEN │
│ │
└─────────────────────────┘
Length = y m
Given: y = x + 4
Half Perimeter = 36 m
Step 1: Define Variables
Width of the garden = \( x \) meters
Length of the garden = \( y \) meters
Step 2: Form the Equations
We know that perimeter of rectangle = \( 2(length + width) = 2(x + y) \)
Half the perimeter = \( \frac{2(x + y)}{2} = x + y \)
Given: Half the perimeter = 36 m
\[ x + y = 36 \quad \text{…(2)} \]System of Linear Equations:
\[ y = x + 4 \quad \text{…(1)} \] \[ x + y = 36 \quad \text{…(2)} \]Step 3: Check Consistency
Equation (1): \( x – y + 4 = 0 \) → \( a_1 = 1, b_1 = -1, c_1 = 4 \)
Equation (2): \( x + y – 36 = 0 \) → \( a_2 = 1, b_2 = 1, c_2 = -36 \)
Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) (1 ≠ -1)
The lines intersect at one point. The system is consistent with a unique solution.
Step 4: Graphical Solution
For Equation (1): \( y = x + 4 \)
| x | 0 | 10 | 16 |
|---|---|---|---|
| y = x + 4 | 4 | 14 | 20 |
| Points | (0, 4) | (10, 14) | (16, 20) |
For Equation (2): \( x + y = 36 \) or \( y = 36 – x \)
| x | 0 | 36 | 16 |
|---|---|---|---|
| y = 36 – x | 36 | 0 | 20 |
| Points | (0, 36) | (36, 0) | (16, 20) |
Finding the Intersection Point
Both lines pass through the point (16, 20).
This is the point of intersection, which gives us our solution.
- Line 1 (\( y = x + 4 \)) passes through (0, 4), (10, 14), and (16, 20)
- Line 2 (\( x + y = 36 \)) passes through (0, 36), (36, 0), and (16, 20)
- Both lines intersect at the point (16, 20)
Step 5: Verify the Solution
This means: Width = 16 m, Length = 20 m
LHS = \( y = 20 \)
RHS = \( x + 4 = 16 + 4 = 20 \)
LHS = RHS ✓
LHS = \( x + y = 16 + 20 = 36 \)
RHS = \( 36 \)
LHS = RHS ✓
1. Is length 4 m more than width? → 20 = 16 + 4 ✓
2. Is half the perimeter 36 m? → 16 + 20 = 36 ✓
3. Full perimeter = 2(16 + 20) = 2(36) = 72 m
4. Half of 72 m = 36 m ✓
Width of the garden = 16 meters
Length of the garden = 20 meters
Alternative Method: Algebraic Solution
💡 Solving by Substitution Method
Given Equations:
\[ y = x + 4 \quad \text{…(1)} \] \[ x + y = 36 \quad \text{…(2)} \]Step 1: Substitute equation (1) into equation (2):
\[ x + (x + 4) = 36 \] \[ 2x + 4 = 36 \] \[ 2x = 32 \] \[ x = 16 \]Step 2: Substitute \( x = 16 \) in equation (1):
\[ y = 16 + 4 = 20 \]Solution: Width = 16 m, Length = 20 m
⚠️ Common Mistakes to Avoid
✅ Correct: Half the perimeter of a rectangle = \( x + y \), not \( 2(x + y) \).
✅ Correct: If length (y) is more, then \( y = x + 4 \). Read the statement carefully to identify which variable is larger.
✅ Correct: Always verify that length > width (since length is 4 m more) and both dimensions are positive.
✅ Correct: Use a suitable scale like 1 cm = 2 units or 1 cm = 4 units for this problem, as values go up to 36.
💡 Key Points to Remember
- Perimeter Formula: Perimeter of rectangle = \( 2(length + width) \)
- Half Perimeter: Half the perimeter = \( length + width \)
- Variable Definition: Clearly define which variable represents length and which represents width
- “More than” Statement: “A is 4 more than B” means \( A = B + 4 \)
- Real-world Context: In rectangles, length is typically the longer dimension
- Verification: Always check your solution in both equations and against the original problem statement
- Graphical Accuracy: Choose appropriate scale and plot at least 3 points per line
- Intersection Point: The coordinates of the intersection point give the values of both variables
- Units: Don’t forget to include units (meters) in your final answer
📝 Similar Word Problems
Practice These Related Problems:
- Problem 1: The perimeter of a rectangle is 50 cm. If the length is 5 cm more than the width, find the dimensions.
- Problem 2: Half the perimeter of a rectangular field is 60 m. If the length exceeds the width by 10 m, find the dimensions.
- Problem 3: The sum of length and breadth of a rectangle is 25 m. If the length is 7 m more than the breadth, find both dimensions.
- Question 1(i): Students in Mathematics quiz problem
🎯 Tips for Solving Word Problems
Step-by-Step Approach:
- Read Carefully: Understand what is given and what needs to be found
- Define Variables: Assign variables to unknown quantities
- Translate to Equations: Convert word statements into mathematical equations
- Check Consistency: Verify that the system has a solution
- Solve: Use graphical or algebraic method
- Verify: Check the solution in original problem context
- State Answer: Write the final answer with proper units
📚 Related Questions
Farhan Mansuri
M.Sc. Mathematics, B.Ed.
15+ Years Teaching Experience
Passionate about making mathematics easy and enjoyable for students.