Question

If the arithmetic mean of two number a and b is 5 times their geometric mean, find a+b/a-b

Answer 1

$given \: \frac{a + b}{2} = 5 \sqrt{ab}$
Squaring and rearranging,
${a}^{2} - 98ab + {b}^{2} = 0$
$dividing \: throughout \: by \: {b}^{2}$
${( \frac{a}{b} })^{2} - 98( \frac{a}{b} ) + 1 = 0$
Applying standard formula for quadratic equation,
$\frac{a}{b} = 49 + 20 \sqrt{6} \: \: or \: 49 - 20 \sqrt{6}$
$applying \: the \: relation \: if \: \frac{a}{b} = \frac{x}{y}$
$then \: \frac{a + b}{a - b} = \frac{x + y}{x - y}$
$thus \: \frac{a + b}{a - b} = \frac{(49 + 20 \sqrt{6 } ) + 1}{(49 + 20 \sqrt{6} ) - 1} = \frac{25 + 10 \sqrt{6 }}{24+ 10 \sqrt{6}}$
Taking another value for a/b,
$\frac{a + b}{a - b} = \frac{(49 - 20 \sqrt{6 } ) + 1}{(49 - 20 \sqrt{6} ) - 1} = \frac{25 - 10 \sqrt{6 }}{24 - 10 \sqrt{6}}$

Answer 2

From the information we are given:

2 ( a + b ) = 5 √ a b

Multiply both sides by  

2

to get:

a + b = 10 √ a b

Square both sides to get:

( a + b )²  = a²  + 2 a b + b²  = 100 a b

Subtract  4 a b

from both ends to get:

( a b ) ² = a ²− 2 a b + b ² = 96 a b

So:

( a + b /a − b ) ² = 100 a b /96 a b = 25 /24 = 5 ²* 6 /12 ² = ( 5 √ 6/ 12 ) ²

So since  

a > b

we can take the positive square root to find:

a + b/ a − b = 5/√ 6/ 12