The sum of diameters of two circles is 14 cm and the difference of their circumferences is 8 cm. Find the circumferences of the two circles.

Let radius of larger circle be R cm and smaller circle r cm.

Diameter = 2R and 2r

Given, sum of diameters of two circles is 14 cm

⇒ 2r + 2R = 14

⇒ r + R = 7 ………..(1)

Given, difference in circumferences is 8 cm

⇒ 2πR - 2πr = 8

⇒ 2π(R - r) = 8

⇒ π(R - r) = 4

⇒ $\dfrac{22}{7}(R - r) = 4$

⇒ $R - r = \dfrac{28}{22}$ ………(2)

Adding equation 1 and 2,

$\Rightarrow r + R + R - r = 7 + \dfrac{28}{22} \\[1em] \Rightarrow 2R = \dfrac{154 + 28}{22} \\[1em] \Rightarrow R = \dfrac{182}{44} = \dfrac{91}{22}.$

Substituting value of R in Eq 2 we get,

$\Rightarrow \dfrac{91}{22} - r = \dfrac{28}{22} \\[1em] \Rightarrow r = \dfrac{91}{22} - \dfrac{28}{22} \\[1em] \Rightarrow r = \dfrac{91 - 28}{22} \\[1em] \Rightarrow r = \dfrac{63}{22}.$

Circumference of larger circle = 2πR

= $2 \times \dfrac{22}{7} \times \dfrac{91}{22}$

= 2 x 13

= 26 cm.

Circumference of smaller circle = 2πr

= $2 \times \dfrac{22}{7} \times \dfrac{63}{22}$

= 2 x 9

= 18 cm.

Hence, circumference of larger circle = 26 cm and smaller circle = 18 cm.