ANSWER
Length of the playing alley = 9m
Width of the playing alley = 3m
STEP-BY-STEP EXPLANATION:
Given information
<em>The perimeter of a rectangular playing alley = 24 m</em>
<em>The length of the alley is three times the width</em>
Let l represents the length of the alley
Let w represents the width of the alley
Step 1: Write the formula for calculating the perimeter of a rectangle
![\text{Perimeter of a rectangle = 2(l + w)}](https://tex.z-dn.net/?f=%5Ctext%7BPerimeter%20of%20a%20rectangle%20%3D%202%28l%20%2B%20w%29%7D)
Where l is the length and w is the width of the rectangle
Recall, length = 3 times the width of the alley
Mathematically,
![\begin{gathered} l\text{ = 3 }\times\text{ w} \\ l\text{ = 3w} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20l%5Ctext%7B%20%3D%203%20%7D%5Ctimes%5Ctext%7B%20w%7D%20%5C%5C%20l%5Ctext%7B%20%3D%203w%7D%20%5Cend%7Bgathered%7D)
Step 2: Substitute the value of l = 3w into the above formula
![\begin{gathered} P\text{ = 2(l + w)} \\ p\text{ = 24m} \\ l\text{ = 3w} \\ 24\text{ = 2(3w + w)} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%5Ctext%7B%20%3D%202%28l%20%2B%20w%29%7D%20%5C%5C%20p%5Ctext%7B%20%3D%2024m%7D%20%5C%5C%20l%5Ctext%7B%20%3D%203w%7D%20%5C%5C%2024%5Ctext%7B%20%3D%202%283w%20%2B%20w%29%7D%20%5Cend%7Bgathered%7D)
Step 3: Solve for w
![\begin{gathered} 24\text{ = 2(4w)} \\ 24\text{ = 8w} \\ \text{Divide both sides by 8} \\ \frac{24}{8}\text{ = }\frac{8w}{8} \\ w\text{ = 3 m} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%2024%5Ctext%7B%20%3D%202%284w%29%7D%20%5C%5C%2024%5Ctext%7B%20%3D%208w%7D%20%5C%5C%20%5Ctext%7BDivide%20both%20sides%20by%208%7D%20%5C%5C%20%5Cfrac%7B24%7D%7B8%7D%5Ctext%7B%20%3D%20%7D%5Cfrac%7B8w%7D%7B8%7D%20%5C%5C%20w%5Ctext%7B%20%3D%203%20m%7D%20%5Cend%7Bgathered%7D)
From the calculations above, you will see that the width of the playing alley is 3m
Step 4: Solve for l
![\begin{gathered} \text{Recall, l = 3w} \\ w\text{ = 3} \\ l\text{ = 3 }\times3 \\ l\text{ = 9m} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Ctext%7BRecall%2C%20l%20%3D%203w%7D%20%5C%5C%20w%5Ctext%7B%20%3D%203%7D%20%5C%5C%20l%5Ctext%7B%20%3D%203%20%7D%5Ctimes3%20%5C%5C%20l%5Ctext%7B%20%3D%209m%7D%20%5Cend%7Bgathered%7D)
Hence, the length of the playing alley is 9m