Question

The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the gm of the same numbers exceeds by 12 the smaller of the numbers.Find the numbers.(kindly provide the solution as well)

Answer 1

(a + b)/2 = b - 24
____
/(a*b)= 12 + a

b - a = 48 or b = a + 48

a*b = a^2 + 24*a + 144

a*(a + 48) = a^2 + 24*a + 144

a^2 + 48*a = a^2 + 24*a + 144

24*a = 144

a = 144/24 = 6

B = 6 + 48 = 54

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Answer 2

Answer:

Required two numbers are 6 and 54.

Step-by-step explanation:

Let x and y be two numbers such that a<b.

Given the arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers.

So,$\frac{x + y}{2} = y - 24$

Again the gm of the same numbers exceeds by 12 the smaller of the numbers.

${(x \times y)}^{2} = 12 + x$

$y - x = 48 \\ y = x + 48$

and $x \times y = {x}^{2} + 24x + 144$

moreover,$x \times (x + 48) = {x}^{2} + 24x + 144$

By middle term method,

$24 \times x = 144$

$x = \frac{144}{24} = 6$

$y = 6 + 48 = 54$

Required two numbers are 6 and 54.