Question

The difference btw two no.s is 48, the difference btw the arithmetic mean and geometric mean of the no.s is 18. Find the no.

Answer 1

Let the numbers be x and y.
So according to question,
x - y = 48

or (√x - √y)(√x + √y) = 48......eqn .1

arithmetic mean of x and y = (x + y)/2

and geometric mean of x and y = √(xy)

so according to question,
(x + y)/2 - √(xy) = 18

so x + y - 2√xy = 36
(√x - √y)² = 36
√x - √y = ±6.....eqn. 2

Putting eqn. 2 in eqn. 1
we get,
√x + √y = ±8....eqn.3

Adding eqn.2 and eqn.3,
√x + √y + √x - √y =14
2√x = 14
√x =7

So x = 7² = 49....eqn.4

As x - y = 48
y = x - 48 = 49 - 48 = 1.

Therefore x = 49 and y = 1