The sum of three numbers in A.P. is 30 and the ratio of the first number to the third number is 3 : 7. Find the numbers.

Let the three numbers that are in A.P. be a - d, a, a + d.

Given, sum of three numbers = 30
⇒ a - d + a + a + d = 30
⇒ 3a = 30
⇒ a = 10.

Given, the ratio of the first number to the third number = 3 : 7

$\Rightarrow \dfrac{a - d}{a + d} = \dfrac{3}{7} \\[0.5em] \Rightarrow 7(a - d) = 3(a + d) \\[0.5em] \Rightarrow 7a - 7d = 3a + 3d \\[0.5em] \Rightarrow 7a - 3a = 3d + 7d \\[0.5em] \Rightarrow 4a = 10d \\[0.5em] \Rightarrow d = \dfrac{4a}{10} \\[0.5em]$

Putting the value of a = 10 we get,

$\Rightarrow d = \dfrac{4 \times 10}{10} \\[0.5em] \Rightarrow d = 4.$

∴ a - d = 10 - 4 = 6 and a + d = 10 + 4 = 14.

Hence, the numbers are 6, 10, 14.