Chapter 5.2, Problem 35E

To determine

To find: The original number if the sum of the digit of a two digit number is 11 and when the digits are reversed, the number increases by 27.

Answer to Problem 35E

The required original number is 47.

Explanation of Solution

Given information:

The sum of the digits of a two digit number is 11.

When the digits are reversed, the number increases by 27.

Calculation:

Let x be the units digit and y the tens digit.

Therefore, the units digit will have a factor of 1 and the tens digit will have a factor of 10.

The sum of 2-digit number is 11:

  $y+x=11$

(Equation 1)

Reversing the digits, we have the original number increased by 27:

  $10x+y=10y+x+27$

(Equation 2)

Solve for one variable in equation 1:

  $y=11-x$

Solve for x:

  $10x+\left(11-x\right)=10\left(11-x\right)+x+27$

(Substitute in equation 2)

  $10x+11-x=110-10x+x+27$

(Simplify)

  $10x+10x-x+11=110-10x+10x+x+27$

(Adding 10x on both sides)

  $20x-x-x+11=110+x-x+27$

(Subtract x on both sides)

  $18x+11=137$

(Simplify)

  $18x+11-11=137-11$

(Subtract 11 from both sides)

  $18x=126$

(Simplify)

  $\frac{18x}{18}=\frac{126}{18}$

(Divide by 18 on both sides)

  $x=7$

(Simplify)

Solve for y:

  $\begin{array}{l}y=11-x\\ y=11-7\end{array}$ (Putting the value of x)

  $y=4$

Therefore, the original number is 47.