Half the perimeter of a rectangular garden, whose length is more than its width, is . Find the dimensions of the garden.

Understand the Question Let the length of the rectangular garden be and its width be . The perimeter of a rectangle is , so half the perimeter is . The length is more than the width: . We can substitute the expression for length into the half-perimeter equation to find the dimensions of the garden. Step 1 · Form the Linear Equations Let the length of the garden be and the width be .Given that the length is more than the width: Given that half the perimeter is : Step 2 · Solve for the Width Substitute equation into equation : Step 3 · Solve for the Length Substitute into equation : Answer > Common Mistakes Perimeter vs. Half-Perimeter: Writing instead of . The question states that is half the perimeter, not the full perimeter. Variable Relation Reversal: Setting instead of . Since the length is greater than the width, must equal .

Pair of Linear Equations in Two Variables | Exercise 3.1

Question 5

Half the perimeter of a rectangular garden, whose length is $4 \text{ m}$ more than its width, is $36 \text{ m}$ . Find the dimensions of the garden.

Solution

Understand the Question

Let the length of the rectangular garden be $l$ and its width be $w$ .
The perimeter of a rectangle is $2(l + w)$ , so half the perimeter is $\dfrac{2(l + w)}{2} = l + w = 36\text{ m}$ .
The length is $4\text{ m}$ more than the width: $l = w + 4$ .
We can substitute the expression for length into the half-perimeter equation to find the dimensions of the garden.

Step 1 · Form the Linear Equations

Let the length of the garden be $l$ and the width be $w$ .Given that the length is $4\text{ m}$ more than the width: $l = w + 4 \quad \dots (1)$

Given that half the perimeter is $36\text{ m}$ : $\text{Half-perimeter} = \dfrac{2(l + w)}{2} = l + w$

$l + w = 36 \quad \dots (2)$

Step 2 · Solve for the Width

Substitute equation $(1)$ into equation $(2)$ :

\begin{aligned} (w + 4) + w &= 36 \\[0.6em] 2w + 4 &= 36 \\[0.6em] 2w &= 36 - 4 \\[0.6em] 2w &= 32 \\[0.6em] w &= \dfrac{32}{2} \\[0.6em] w &= 16\text{ m} \end{aligned}

Step 3 · Solve for the Length

Substitute $w = 16$ into equation $(1)$ :

\begin{aligned} l &= w + 4 \\[0.6em] &= 16 + 4 \\[0.6em] &= 20\text{ m} \end{aligned}

Answer

$\text{Length} = 20\text{ m}, \text{Width} = 16\text{ m}$

Common Mistakes

Perimeter vs. Half-Perimeter: Writing $2(l + w) = 36$ instead of $l + w = 36$ . The question states that $36\text{ m}$ is half the perimeter, not the full perimeter.
Variable Relation Reversal: Setting $w = l + 4$ instead of $l = w + 4$ . Since the length is greater than the width, $l$ must equal $w + 4$ .

Question 5

More questions in Exercise 3.1