Question

number consisting of two digits is four times the sum of its digits and if 27 be added to it the
s are reversed. the number is
​

Answer 1

$\sf\purple{Answer:-}$

${\underline{\boxed{\sf\green{Number = 36 }}}}$

${\underline{\underline{\bold\blue{Step-by-step-explanation:-}}}}$

Let the one's digit be x and the ten's digit be y.

$\therefore\sf\blue{Number \: formed = x + 10y}$

${\underline{\underline{\bold{According\: to \: question :-}}}}$

1st case:-

$\hookrightarrow\sf x + 10y = 4(x + y) \\\\\hookrightarrow\sf x + 10y = 4x + 4y \\\\\hookrightarrow\sf x - 4x = 4y - 10y \\\\\hookrightarrow\sf -3x = - 6y \\\\\hookrightarrow\sf x =\dfrac{\cancel{-6}y}{\cancel{-3}}\\\\\hookrightarrow{\boxed{\sf\green{x = 2y ..........(i)}}}$

$\rule{200}{1}$

2nd case:-

$\hookrightarrow\sf x + 10y + 27 = 10x + y \\\\\hookrightarrow\sf x - 10x + 10y - y = -27 \\\\\hookrightarrow\sf -9x + 9y = -27 \\\\\hookrightarrow\sf -(9x - 9y ) = -27 \\\\\hookrightarrow\sf 9x - 9y = 27 \\\\\hookrightarrow\sf 9(x - y) = 27 \\\\\hookrightarrow\sf x - y =\cancel{\dfrac{27}{9}} \\\\\hookrightarrow\sf x - y = 3 \\\\\hookrightarrow{\boxed{\sf\green{x = y + 3..........(ii)}}}$

Now from both (i) & (ii) we get :-

$\hookrightarrow\sf y + 3 = 2y \\\\\hookrightarrow\sf y - 2y = -3 \\\\\hookrightarrow\sf -y = -3 \\\\\therefore{\boxed{\sf\green{y = 3}}}$

Putting value of y in (i) :-

$\hookrightarrow\sf x = 2 \times 3 \\\\\therefore{\boxed{\sf\blue{x = 6}}}$

$\rule{200}{1}$

$\sf Number \: formed = x + 10y \\\\\hookrightarrow\sf Number \: formed = 6 + 10 \times 3 \\\\\hookrightarrow\sf Number \: formed = 6 + 30 \\\\\hookrightarrow{\boxed{\sf\green{Number \: formed = 36}}}$

${\underline{\boxed{\therefore{\sf\blue{Number \: formed = 36}}}}}$

Answer 2

$\LARGE\textrm{\underline{\underline \green{Answer:}}}$

$\star$ $\large\sf \purple{ Number \: is \: 36}$

$\Large\textrm{\underline{\underline \green{Step \: by \: step\: explanation:}}}$

GivEn :

$\large\sf{Number \: is- \:}$

•$\large\sf{ \: \:Four \: times \: the \: sum \: of \: digits}$

•$\large\sf{\: \:If \: 27 \: is \: added \: = \: Reversed \: number \: }$

To FinD :

•$\large\sf{\: \: The \: Number}$

SoluTion :

$\star$ $\normalsize\sf \pink{Let \: the \: one's \: place \: digit \: be \: x}$

$\star$ $\normalsize\sf \pink{ Let \: the \: ten's \: place \: digit \: be \: y}$

$\star$ ${\boxed{\sf \orange{Number \: = \: x \: + \: 10y}}}$

$\scriptsize\sf{\: \: \: \: \: (Formula \: used \: = 10 × one's \: place \: digit \; number \: + \: ten's \: place \: digit \: number)}$

$\large{\textrm{\underline{\underline{According \: to \: Question :-}}}}$

✰$\normalsize\sf \red{Case \: 1 :\:}$

➠ $\large\sf\ x \: + \: 10y \: = \: 4(x \: + \: y )$

$\scriptsize\sf{ \: \: (Number \: is \: 4 \: times\: the \: sum \: of \: digits)}$

➠$\large\sf\ x \: + \: 10y \: = \: 4 x \: + \: 4y$

➠$\large\sf\ 10y \: - \: 4y \: = \: 4x \: - \: x$

➠$\large\sf\ 6y \: = \: 3x$

➠$\large\sf\ x \: = \: \frac{6y}{3}$

$\scriptsize\sf{\: \: \: \: \: (Cancelling \: the \: values)}$

➠${\boxed{\sf{x \: = \: 2y }}}$

$\scriptsize\sf{\: \: \: \: \:(Let \: it \: be \: equation\: 1)}$

$\rule{300}{2}$

✰$\normalsize\sf \red{Case \: 2: \:}$

➠$\large\sf\ x \: + \: 10y \: + \: 27 \: = \: 10x\: + \: y$

$\scriptsize\sf{\: \: (If \: 27 \: is \: added \: to \: number \: then \: it \: is \: equal \: to \: its \: reversed)}$

➠$\large\sf\ x \: + \: 10y \: - \: 10x \: - \: y \: = \: -27$

➠$\large\sf\ -9x \: + 9y \: = \: -27$

➠$\large\sf\ -(9x \: - \: 9y) \: \: -(27)$

$\scriptsize\sf{ \: \: \: \:(Take \: negative \: sign \: common \: and \: then \: cancel \: both \: sides)}$

➠$\large\sf\ 9(x \: - \: y ) \: = \: 27$

➠$\large\sf\ x \: - \: y \: = \: \frac{27}{9}$

➠$\large\sf\ x \: - \: y \: = \: 3$

➠$\large\sf \ x \: = \: y \: + \: 3$

➠${\boxed{\sf{ x \: = \: y \: + \: 3}}}$

$\scriptsize\sf{\: \: \: \: \: (let \: it \: be \: second \: equation \:)}$

$\rule{300}{2}$

✰$\normalsize\sf \red{ Solving \: from \: equation \: 1 \: and \: 2, \: to \: find \: y :}$

➠$\large\sf\ 2y \: = \: y \: + \: 3$

➠$\large\sf\ 2y \: - y \: = 3$

➠$\large\sf\ y \: = \: 3$

➠${\boxed{\sf \blue{ y \: = \: 3}}}$

✰$\normalsize\sf \red{Putting \: the \: value of \: y \: in \: equation \: 2 }$

➠$\large\sf\ x \: = \: y \: + \: 3$

➠$\large\sf\ x \: = \: 3 \: + \: 3 \: = \: 6$

➠${\boxed{\sf \blue{x \: = \: 6}}}$

✰$\normalsize\sf \red{Now, \: find \: the \: number: \:}$

➠$\large\sf\ Number \: = \: x \: + \: 10y$

➠$\large\sf\ Number \: = 6 \: + \: 10 \times\ 3$

➠$\large\sf\ Number \: = \: 6 \: + \: 30 \: = \: 36$

➠${\boxed{\sf \purple{Number \: = \: 36}}}$