Question

A two digit number is 4 times the Sum of its digits. If 18 is added to the number the digits are reversed.Find the number​

Answer 1

S O L U T I O N :

Let the ten's digit number be r

Let the one's digit number be m

$\boxed{\bf{The\:original\:number=10r+m}}}}}\\\boxed{\bf{The\:reversed\:number=10m+r}}}}}$

A/q

$\longrightarrow\rm{10r+m=4(r+m)}\\\\\longrightarrow\rm{10r+m=4r+4m}\\\\\longrightarrow\rm{10r-4r=4m-m}\\\\\longrightarrow\rm{\cancel{6}r=\cancel{3}m}\\\\\longrightarrow\rm{m=2r..............(1)}$

&

$\longrightarrow\rm{10r+m+18=10m+r}\\\\\longrightarrow\rm{10r-r+m-10m=-18}\\\\\longrightarrow\rm{9r-9m=-18}\\\\\longrightarrow\rm{9(r-m)=-18}\\\\\longrightarrow\rm{r-m=\cancel{-18/9}}\\\\\longrightarrow\rm{r-m=-2}\\\\\longrightarrow\rm{r-2r=-2\:\:\:[from(1)]}\\\\\longrightarrow\rm{\cancel{-}r=\cancel{-}2}\\\\\longrightarrow\bf{r=2}$

Putting the value of r in equation (1),we get;

$\longrightarrow\rm{m=2\times 2}\\\\\longrightarrow\bf{m=4}$

Thus;

$\underbrace{\sf{The\;original\:number\:(10r+m)=[10(2)+4]=[20+4]=\boxed{\bf{24}}}}}}$

Answer 2

GIVEN:

• A two digit number is 4 times the Sum of its digits
• If 18 is added to the number the digits are reversed

TO FIND:

• What is the original number ?

SOLUTION:

Let the digit at unit's place be 'y' and the digit at ten's place be 'x'

• Number = 10x + y

CASE:- ❶

▣ A two digit number is 4 times the Sum of its digits

According to question:-

$\sf{\rightharpoonup 10x + y = 4(x + y) }$

$\sf{\rightharpoonup 10x + y = 4x + 4y }$

$\sf{\rightharpoonup 10x - 4x = 4y - y }$

$\sf{\rightharpoonup 6x = 3y }$

Divide by '3' on both sides

$\large\bf{\rightharpoonup 2x = y...1) }$

CASE:- ❷

▣ If 18 is added to the number the digits are reversed

◉ Number obtained by reversing the digits = 10y + x

❮ Original number + 18 = Number obtained by reversing the digits ❯

According to question:-

$\sf{\rightharpoonup 10x + y + 18 = 10y + x}$

$\sf{\rightharpoonup 10x + y - (10y + x) = -18}$

$\sf{\rightharpoonup 10x + y -10y -x = -18}$

$\sf{\rightharpoonup 9x - 9y = -18}$

Divide by '9' on both sides

$\large\bf{\rightharpoonup x - y = -2...2) }$

Put the value of 'y' from equation 1) in equation 2)

$\sf{\rightharpoonup x - 2x = -2 }$

$\sf{\rightharpoonup \cancel{-} x = \cancel{-} 2 }$

$\large\bf{\rightharpoonup \star \: x = 2 \: \star}$

Put the value of 'x' in equation 2)

$\sf{\rightharpoonup 2 - y = -2 }$

$\sf{\rightharpoonup - y = -2-2}$

$\sf{\rightharpoonup \cancel{-} y = \cancel{-} 4 }$

$\large\bf{\rightharpoonup \star \: y = 4 \: \star}$

➜ Number = 10x + y

➜ Number = 10(2) + 4

➜ Number = 20 + 4

★ Number = 24 ★

❛ Hence, the number formed is 24 ❜

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