In previous lessons, we have considered some examples of Digit Word Problems.
In these lessons, we will consider examples of Digit Word Problems that involve the interchanging of digits.
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
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.
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)
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
Answer: The number is 83.
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