Question

The ratio of the factorial of a number x to
the square of the factorial of another number, which when
increased by 50% gives the required number, is 1.25. Find
the number x.
(a) 6
(b) 5.
(c) 9
(d) None of these .
I want the steps not the answer , i have the answer as 6 but I want the steps​

Answer 1

✿$\underline{\textbf{\textsf{ \purple{Solution}:- }}}$

$\sf{\\}$

Let the first number be $\sf{x}$. This is the required number.

Let the second number be $\sf{y}$

$\sf{\\}$

The first step is to form a linear equation.

There is a relation given between x and y.

"Another number, increased by 50% gives required number"

On increasing y by 50% we get x

$\sf{y + \dfrac{50}{100}y = x}$

$\longrightarrow \sf{y + \dfrac{y}{2} = x}$

$\longrightarrow \sf{ \dfrac{3}{2}y = x }$

$\longrightarrow \sf{y = \Big( \dfrac{2}{3} x \Big)}$

$\sf{\\}$

Now, it is given that, ratio of factorial of x to square of factorial of y is 1.25

$\sf{ \dfrac{ (x!) }{ { (y!) }^{2} } = 1.25 = \dfrac{5}{4} }$

$\longrightarrow \sf{x! = \dfrac{5}{4} {(y!)}^{2}}$

$\longrightarrow \sf{ (y!)^2 = \dfrac{4}{5} x!}$

$\longrightarrow \sf{ y! = \sqrt{ \dfrac{4}{5} x! } }$

We have already found out that, $\sf{y = \Big( \dfrac{2}{3} x \Big)}$

So,

$\longrightarrow \sf{ \Big(\dfrac{2}{3}x\Big)! = 2 \sqrt{ \dfrac{x!}{5} }}$

Now, we have to put values from the options in the equation and see which value satisfies,

$\underline{\texttt{ \red{Putting x = 6}:- }}$

$\underline{\textbf{ \pink{L.H.S}:- }}$

$\sf{\Big(\dfrac{2}{3}x\Big)!}$

$\longrightarrow \sf{ \Big(\dfrac{2}{3}\:\:6\Big)!}$

$\longrightarrow \sf{4!}$

$\longrightarrow \sf{24}$

$\underline{\textbf{ \pink{R.H.S}:- }}$

$\sf{2\sqrt{ \dfrac{x!}{5} }}$

$\longrightarrow \sf{2\sqrt{ \dfrac{6!}{5} }}$

$\longrightarrow \sf{2 \sqrt{\dfrac{720}{5}}}$

$\longrightarrow \sf{2 \sqrt{144}}$

$\longrightarrow \sf{2 * 12}$

$\longrightarrow \sf{24}$

$\sf{\\}$

Hence, $\sf{L.H.S = R.H.S}$

So the answer is,

$\longrightarrow \underline{\underline{\sf{ \green{x = 6} }}}$