# Digit Word Problems - Interchanging of digits

In these lessons, we will consider examples of Digit Word Problems that involve the interchanging of digits. This follows on from examples of Digit Word Problems.

### Convert Digits to Number

If the problem involves interchanging of the digits in the integer then you would need to convert from the digits to numbers and vice versa. To convert the digits to numbers, we need to multiply with the digit with the place value of the digit.

For example, the value of the number formed by the digit 4 in the ten’s place and the digit 3 in the one’s place is 4 × 10 + 3 × 1

This type of digit problems is shown in the following example

### Interchanging Of Digits Problems

Example:
The sum of the digits of a two-digit number is 11. If we interchange the digits then the new number formed is 45 less than the original. Find the original number.

Solution:
Step 1:
Assign variables
Let x = one’s digit and t = ten’s digit

Sentence: The sum of the digits of a two-digit number is 11.
x + t = 11

Isolate variable x
x = 11 – t             (equation 1)

Step 2:
Convert digits to number
Original number = t × 10 + x
Interchanged number = x × 10 + t
Sentence: If we interchange the digits then the new number formed is 45 less than the original.
Interchanged = Original – 45
x × 10 + t = t × 10 + x – 45
10x + t = 10t + x – 45
10x – x + t = 10t – 45 (–x to both sides)
10x – x = 10t – t – 45 (– t to both sides)
10x – x + 45 = 10t – t (+ 45 to both sides)
10t – t = 10x – x + 45 (Rewrite equation with t on the left hand side)

Combine like terms
10t – t = 10x – x + 45
9t = 9x + 45            (equation 2)

Substitute equation 1 into equation 2
9t = 9(11 – t) + 45
9t = 99 – 9t + 45

Isolate variable t
9t + 9t = 99 + 45
18t = 144 The ten’s digit is 8. The one’s digit is 11 – 8 = 3

### Videos

How to use Systems of Equations to solve Reversing Digits Word Problem?

Reversing digits word problem
Problem:
The sum of two digits of a 2-digit number is 11. Reversing the digits increase the number by 45. What is the number?

Example:
The sum of two digits of a 2-digit number is 13. Reversing the digits increase the number by 45. What is the number?

Example:
The sum of the digits of a two digit number is 9. When the digits are reversed the new number is 9 less than three times the original.

Examples:

1. The sum of the digits of a 2-digit number is 10. The tens digit is 4 times the ones digit. Find the number.

2. A 2-digit number is 10 times the sum of its digits. The tens digit is 2 greater than the units digit. Find the number.

3. The sum of the digits of a 2-digit number is 14. If 18 is added to the number, the result is the number with its digits reversed. Find the original number.

4. The sum of the digits of a 2 digit number is 9. If the digits are reversed, the new number is 9 less than 3 times the original number. Find the original number.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 