Question

Q1.Two numbers a,b are such that their gm is less than 40% less than am the ratio between the numbers.

Answer 1

Step-by-step explanation:

Given numbers are a, b

$</strong></p><p><strong>[tex]\begin{gathered} \sf \: Arithmetic \: \: mean = (a + b) / 2 \\ \\ \sf \: Geometric \: \: mean = \sqrt{ab} \end{gathered}$

According to the Question,

G. M is 40% less than AM.

We can say,

G. M + 40% of AM = AM

Further,

G. M = 60% of AM.

Now,

$\begin{gathered} \sqrt{ab} = 60 \% \: \times \frac{a + b}{2} \\ \\ \sqrt{ab} = \frac{6}{10} \times \frac{a + b}{2} \\ \\ \sqrt{ab} = \frac{3}{10} \times {a + b} \\ \\ \end{gathered}$

Squaring on both sides,

$\begin{gathered}ab \: = ( { \frac{3}{10} })^{2} ( {a + b})^{2} \\ \\ 100ab = 9( {a}^{2} + {b}^{2} + 2ab) \\ \\ 9a ^{2} + 9 {b}^{2} - 82ab = 0\end{gathered}$

Divide by ab,

$9( \frac{a}{b} ) + 9( \frac{b}{a} ) = 82$

Let a/b = x ( This is the required ratio)

$</p><p>\begin{gathered}9x + \frac{9}{x} = 82 \\ \\ 9 {x}^{2} + 9 = 82x \\ \\ 9 {x}^{2} - 82x + 9 = 0 \\ \\( 9x - 1)(x - 9) = 0\end{gathered}$

Now, x = 1/9 or x = 9

So, The ratio of numbers can be a : b = 1 : 9 if a < b And 9 : 1 if a > b.