The sum of the length of two rectangles P and Q is 60 cm, while their breadths are 16 cm and 12 cm respectively. If the ratio of the area of rectangles P to that of Q is 8 : 3, then find the difference between perimeters of these two rectangles?

The correct answer is 48 cm.

Let's break down the problem step-by-step to find the difference between the perimeters of the two rectangles P and Q.

Step 1: Define the given variables.
Let $L_P$ be the length of rectangle P and $L_Q$ be the length of rectangle Q.
We are given:
Sum of the lengths: $L_P+L_Q=60\text{cm}$
Breadth of rectangle P ($B_P$) = $16\text{cm}$
Breadth of rectangle Q ($B_Q$) = $12\text{cm}$

Step 2: Use the ratio of their areas to find the relation between their lengths.
The area of a rectangle is given by the formula:
$\text{Area}=\text{Length}\×\text{Breadth}$
So, the area of rectangle P ($A_P$) = $L_P\×16$
And the area of rectangle Q ($A_Q$) = $L_Q\×12$

We are given that the ratio of their areas is 8 : 3:
$\frac{A_P}{A_Q}=\frac{8}{3}$

Substitute the formulas for $A_P$ and $A_Q$ into the ratio:
$\frac{L_P\×16}{L_Q\×12}=\frac{8}{3}$

Simplify the fraction on the left side by dividing 16 and 12 by 4:
$\frac{L_P\×4}{L_Q\×3}=\frac{8}{3}$

Multiply both sides by 3:
$L_P\×4=L_Q\×8$

Divide both sides by 4:
$L_P=2L_Q$

Step 3: Calculate the individual lengths.
Substitute $L_P=2L_Q$ into the equation for the sum of the lengths:
$2L_Q+L_Q=60$
$3L_Q=60$
$L_Q=20\text{cm}$

Now, calculate $L_P$:
$L_P=2\×20=40\text{cm}$

Step 4: Find the perimeters of both rectangles.
The perimeter of a rectangle is given by:
$\text{Perimeter}=2\×(\text{Length}+\text{Breadth})$

Perimeter of rectangle P (${\text{Perimeter}}_P$):
${\text{Perimeter}}_P=2\×(40+16)=2\×56=112\text{cm}$

Perimeter of rectangle Q (${\text{Perimeter}}_Q$):
${\text{Perimeter}}_Q=2\×(20+12)=2\×32=64\text{cm}$

Step 5: Find the difference between their perimeters.
$\text{Difference}={\text{Perimeter}}_P-{\text{Perimeter}}_Q$
$\text{Difference}=112-64=48\text{cm}$

Thus, the difference between the perimeters of the two rectangles is 48 cm.