The ratio of radii of two circles is 3:2. What is the ratio of their circumferences?

The ratio of the radii.

\$3 : 2 \$

Let the radii of the two circles be 3r and 2r respectively.

Also let \$C_1\$ and \$C_2\$ be the circumference of the two circles of radii 3r and 2r respectively.

Circumference of a circle.

\$C_1 = 2 \pi r \$.

Substitute \$r = 3r\$.

\$ C_1 = 2 \times \pi \times 3r \$

Multiply.

\$ C_2 = 6 \pi r \$

Circumference of a circle.

\$C_2 = 2 \pi r \$

Substitute \$r = 3r\$.

\$ C_1 = 2 \times \pi \times 2r \$

Multiply.

\$ C_2 = 4 \pi r \$

Ratio of circumferences.

\$ C_1 : C_2 \$

Substitute the values of \$C_1 and C_2\$.

\$ 6 \pi r : 4 \pi r \$

Cancel out common factor.

\$ 6 \cancel{\pi} r : 4 \cancel{\pi} r \$

Cancel out common factor.

\$ \cancel{6}^3 r : \cancel{4}^2 r \$

Cancel out common factor.

\$ 3 \cancel{r} : 2 \cancel{r} \$

Answer

Ratio of circumferences of the two circles of radii 3r and 2r is 3 : 2