The difference between two positive numbers is 40 and the ratio of these integers is 1 : 3. Find the integers.

Let one integer be x.
Then, Other integer = x – 40
As per the conditions, we have
$\frac{\mathrm{x} - 40}{\mathrm{x}} = \frac{1}{3}$
⇒ 3(x – 40) = x
⇒ 3x – 120 = x
⇒ 3x – x = 120
⇒ 2x = 120
⇒ x = 60
Hence the integers are 60, and 60 – 40 = 20.