Question

The circumference of two circles are in the ratio 4:5. If the area of the larger circle is 625 square units, find the area of the smaller circle.

• A. 400 sq. units
• B. 300 sq. units
• C. 500 sq. units
• D. 450 sq. units

Answer 1

Answer:

A. 400 sq. units

To find the area of the smaller circle, let's break down the problem step-by-step:

Given Data:

• Ratio of the circumferences ($C_1 : C_2$) = $4 : 5$

• Area of the larger circle ($A_2$) = $625$ square units

Step-by-Step Calculation:

1. Find the ratio of the radii: The circumference of a circle is given by $C = 2\pi r$. Since the ratio of the circumferences is $4:5$:

   
   

   $\frac{2\pi r_1}{2\pi r_2} = \frac{4}{5} \implies \frac{r_1}{r_2} = \frac{4}{5}$

   (The ratio of the radii of the two circles is also $4:5$)

2. Find the ratio of their areas: The area of a circle is given by $A = \pi r^2$. The ratio of the areas is the square of the ratio of their radii:

   
   

   $\frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25}$

3. Calculate the area of the smaller circle ($A_1$): Substitute the given area of the larger circle ($A_2 = 625$) into the ratio:

   
   

   $\frac{A_1}{625} = \frac{16}{25}$

   $A_1 = 625 \times \frac{16}{25}$

   $A_1 = 25 \times 16$

   $A_1 = 400$

Answer:

The area of the smaller circle is $400$ square units.