# The sum of a two digit number and the number formed by reversing the order of digit is 66. If the two digits differ by 2, find out the number. How many such numbers are there?

Answer

Verified

363.9k+ views

Hint: Consider the number as $xy$ . Try to understand the place values of the number. Apply the conditions carefully and try to make an equation.

Complete step-by-step answer:

Let us assume that the number is $xy$ .

In one place we have $y$ . In tens place we have $x$ . That means:

$xy=\left( 10\times x \right)+\left( 1\times y \right)=10x+y$

Now if we reverse the number, the number becomes $yx$ .

Now in ones place we have $x$ and in tens place we have $y$ . That means:

$yx=\left( 10\times y \right)+\left( 1\times x \right)=10y+x$

It is given in the question that if we add these two numbers we will get 66. So this is our first condition.

$\left( 10x+y \right)+\left( 10y+x \right)=66$

Now, add the terms with same variables:

$\begin{align}

& \Rightarrow \left( 10x+x \right)+\left( 10y+y \right)=66 \\

& \Rightarrow 11x+11y=66 \\

\end{align}$

Take 11 common from the left hand side,

$\Rightarrow 11(x+y)=66$

Divide both sides by 11,

$\begin{align}

& \Rightarrow \dfrac{11\left( x+y \right)}{11}=\dfrac{66}{11} \\

& \Rightarrow x+y=6........(1) \\

\end{align}$

Now the second condition is the two digits differ by 2.

So either $x$ is greater than $y$ or $y$ is greater than $x$ . The difference between them is 2.

Let us first assume that $x$ is greater than $y$ . So we will get:

$\begin{align}

& x-y=2 \\

& \Rightarrow x=2+y........(2) \\

\end{align}$

Put the value of $x$ in equation (1)

$\begin{align}

& (y+2)+y=6 \\

& \Rightarrow (y+y)+2=6 \\

& \Rightarrow 2y+2=6 \\

& \Rightarrow 2y=6-2 \\

& \Rightarrow 2y=4 \\

& \Rightarrow y=\dfrac{4}{2} \\

& \Rightarrow y=2 \\

\end{align}$

Now put the value of $y$ in (2)

$x=2+2=4$

Hence, $x=4,y=2$

Therefore the number is 42.

If $y$ is greater than $x$ ,

$\begin{align}

& y-x=2 \\

& \Rightarrow y=2+x...........(3) \\

\end{align}$

Now put the value of $y$ in equation (1). We will get,

$\begin{align}

& x+(x+2)=6 \\

& \Rightarrow 2x+2=6 \\

& \Rightarrow 2x=6-2 \\

& \Rightarrow 2x=4 \\

& \Rightarrow x=\dfrac{4}{2} \\

& \Rightarrow x=2 \\

\end{align}$

Put the value of $x$ in equation (3) to get the value of $y$ .

$\begin{align}

& y=2+2 \\

& \Rightarrow y=4 \\

\end{align}$

Hence, $x=2,y=4$

Therefore the number is 24.

We can get two such numbers. One is 24 and another one is 42.

Note: We can assume the number either as $xy$ or as $yx$ . If we take the number as $yx$ , then $x$ will be our ones position and $y$ will be our tens position. We have to apply the conditions accordingly. In both the cases we will get the same answer.

Complete step-by-step answer:

Let us assume that the number is $xy$ .

In one place we have $y$ . In tens place we have $x$ . That means:

$xy=\left( 10\times x \right)+\left( 1\times y \right)=10x+y$

Now if we reverse the number, the number becomes $yx$ .

Now in ones place we have $x$ and in tens place we have $y$ . That means:

$yx=\left( 10\times y \right)+\left( 1\times x \right)=10y+x$

It is given in the question that if we add these two numbers we will get 66. So this is our first condition.

$\left( 10x+y \right)+\left( 10y+x \right)=66$

Now, add the terms with same variables:

$\begin{align}

& \Rightarrow \left( 10x+x \right)+\left( 10y+y \right)=66 \\

& \Rightarrow 11x+11y=66 \\

\end{align}$

Take 11 common from the left hand side,

$\Rightarrow 11(x+y)=66$

Divide both sides by 11,

$\begin{align}

& \Rightarrow \dfrac{11\left( x+y \right)}{11}=\dfrac{66}{11} \\

& \Rightarrow x+y=6........(1) \\

\end{align}$

Now the second condition is the two digits differ by 2.

So either $x$ is greater than $y$ or $y$ is greater than $x$ . The difference between them is 2.

Let us first assume that $x$ is greater than $y$ . So we will get:

$\begin{align}

& x-y=2 \\

& \Rightarrow x=2+y........(2) \\

\end{align}$

Put the value of $x$ in equation (1)

$\begin{align}

& (y+2)+y=6 \\

& \Rightarrow (y+y)+2=6 \\

& \Rightarrow 2y+2=6 \\

& \Rightarrow 2y=6-2 \\

& \Rightarrow 2y=4 \\

& \Rightarrow y=\dfrac{4}{2} \\

& \Rightarrow y=2 \\

\end{align}$

Now put the value of $y$ in (2)

$x=2+2=4$

Hence, $x=4,y=2$

Therefore the number is 42.

If $y$ is greater than $x$ ,

$\begin{align}

& y-x=2 \\

& \Rightarrow y=2+x...........(3) \\

\end{align}$

Now put the value of $y$ in equation (1). We will get,

$\begin{align}

& x+(x+2)=6 \\

& \Rightarrow 2x+2=6 \\

& \Rightarrow 2x=6-2 \\

& \Rightarrow 2x=4 \\

& \Rightarrow x=\dfrac{4}{2} \\

& \Rightarrow x=2 \\

\end{align}$

Put the value of $x$ in equation (3) to get the value of $y$ .

$\begin{align}

& y=2+2 \\

& \Rightarrow y=4 \\

\end{align}$

Hence, $x=2,y=4$

Therefore the number is 24.

We can get two such numbers. One is 24 and another one is 42.

Note: We can assume the number either as $xy$ or as $yx$ . If we take the number as $yx$ , then $x$ will be our ones position and $y$ will be our tens position. We have to apply the conditions accordingly. In both the cases we will get the same answer.

Last updated date: 30th Sep 2023

â€¢

Total views: 363.9k

â€¢

Views today: 3.63k

Recently Updated Pages

What do you mean by public facilities

Please Write an Essay on Disaster Management

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is meant by shramdaan AVoluntary contribution class 11 social science CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

An alternating current can be produced by A a transformer class 12 physics CBSE

What is the value of 01+23+45+67++1617+1819+20 class 11 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers