If the same number be added to the numbers 5, 11, 15 and 31, the resulting numbers are in proportion. Find the number.

Let x be added to each number. The numbers will then be 5 + x, 11 + x, 15 + x and 31 + x.

According to question,

$\dfrac{5 + x}{11 + x} = \dfrac{15 + x}{31 + x}$

By cross multiplying, we get :

⇒ (5 + x)(31 + x) = (15 + x)(11 + x)

⇒ 155 + 5x + 31x + x² = 165 + 11x + 15x + x²

⇒ 155 + 36x + x² = 165 + 26x + x²

⇒ 36x - 26x + x² - x² = 165 - 155

⇒ 10x = 10

⇒ x = $\dfrac{10}{10}$

⇒ x = 1

Hence, 1 is the number that should be added to the given numbers.