Chapter 9.9, Problem 8P

To determine

To find the original number.

Answer to Problem 8P

The original number is $678$ .

Explanation of Solution

Given:

The number is a $3$ digit number.

Calculation:

Given:

Let the number be $100x+10y+z$ .

Now, the sum of the digits of a $3$ digit number is $21$ .

  $x+y+z=21$

The sum of $100\text{'}s$ and $10\text{'}s$ digits is $3$ less than twice the unit’s digit.

  $x+y=2z-3$

If the digits are reversed, the number becomes - $100z+10y+x$

Thus, $100x+10y+z+198=100z+10y+x$

Solving this equation, we have:

  $\begin{array}{l}100x+10y+z+198=100z+10y+x\\ 99x-99z=-198\\ x-z=-2\end{array}$

Solving $x+y+z=21$ and $x+y=2z-3$ , we have:

  $x+y=21-z=2z-3$

  $\begin{array}{l}21-z=2z-3\\ -3z=-24\\ z=8\end{array}$

Now, solving $x-z=-2$ to find the value of $x$ , we have:

  $\begin{array}{l}x-8=-2\\ x=6\end{array}$

Now, solving $x+y+z=21$ to find the value of $y$ , we have

  $\begin{array}{l}6+y+8=21\\ y=7\end{array}$

Thus, $z=8$ , $y=7$ and $x=6$

The original number is $100x+10y+z=600+70+8=678$

Conclusion:

The original number is $678$ .