The sum of 2 positive numbers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the greater number minus the lesser number?

Solution:

Let the two positive numbers be x and y; x > y

ATQ,

x + y = 151 ---------- (1)

y = 19 + √x​ ---------- (2)

Putting the value y = 19 + √x​ in equ (1)

x + 19 + √x = 151

⇒ x + √x = 151 - 19

⇒ x + √x = 132

⇒ √x = 132 - x

⇒ $(\sqrt{x}{)}^2=(132-x{)}^2$

⇒ $x={132}^2-2.132.x+x^2$

${\mathrm{<br>}}^{}$ ⇒ $x^2$ - 265x + 17424 = 0

∴ x = 121, 144

The value of x can be 121 and 144.

The value of x = 144 cannot be possible as it doesn’t satisfies the two condition.

Therefore the value of x is 121​.

From equ. (1) we get,

y = 151 - 121

y = 30

∴ The value of greatest number minus the lesser would be x-y, i.e, 121 - 30 = 91