Question

The sum of a two digit number is 9. If 27 is added to the number, the digits are reversed. Find
then number Find three consecutive even numbers whose sum is 234​

Answer 1

S O L U T I O N  :

$\bf{\large{\underline{\bf{Given\::}}}}}$

The sum of a two digit number is 9. If 27 is added to the number, the digits are reversed.

$\bf{\large{\underline{\bf{To\:find\::}}}}}$

The number.

$\bf{\large{\underline{\bf{Explanation\::}}}}}$

Let the tens digit place be r

Let the ones digit place be m

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

A/q

$\longrightarrow\sf{r+m=9}\\\\\longrightarrow\sf{r=9-m....................(1)}$

&

$\longrightarrow\sf{10r+m+27=10m+r}\\\\\longrightarrow\sf{10r-r+m-10m=-27}\\\\\longrightarrow\sf{9r-9m=-27}\\\\\longrightarrow\sf{9(r-m)=-27}\\\\\longrightarrow\sf{r-m=\cancel{\dfrac{-27}{9} }}\\\\\longrightarrow\sf{r-m=-3}\\\\\longrightarrow\sf{9-m-m=-3\:\:\:[from(1)]}\\\\\longrightarrow\sf{9-2m=-3}\\\\\longrightarrow\sf{-2m=-3-9}\\\\\longrightarrow\sf{-2m=-12}\\\\\longrightarrow\sf{m=\cancel{\dfrac{-12}{-2} }}\\\\\longrightarrow\bf{m=6}$

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

$\longrightarrow\sf{r=9-6}\\\\\longrightarrow\bf{r=3}$

Thus;

$\underbrace{\sf{The\:number\:(10r+m)=[10(3)+6]=[30+6]=\boxed{\bf{36}}}}}$

&

Let the three consecutive even number be x, x + 1 and x + 2 respectively

A/q

$\longrightarrow\rm{x+x+1+x+2=234}\\\\\longrightarrow\rm{3x+3=234}\\\\\longrightarrow\rm{3x=234-3}\\\\\longrightarrow\rm{3x=231}\\\\\longrightarrow\rm{x=\cancel{231/3}}\\\\\longrightarrow\bf{x=77}$

Thus;

$\bullet\:\sf{1st\:number=x=\boxed{77}}}\\\\\bullet\:\sf{2nd\:number=(x+1)=(77+1)=\boxed{78}}}\\\\\bullet\:\sf{3rd\:number=(x+2)=(77+2)=\boxed{79}}}$