Question

Question 4: 11 p The sum of the digits of a two-digit number is 14. When the digits are reversed the new number is 36 less than the original number. Find the original number. Check your answer.

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Answer 1

95

Explanation:

Let x represent the ones digit. The tens digit will be higher, because reversing the digits results in a lower number. Its value is (14 -x). The given relation is ...

(10(14-x) +x) -(10x +(14 -x)) = 36 . . . . reversing the digits gives 36 less

140 -9x -9x -14 = 36 . . . . eliminate parentheses

90 = 18x . . . . . . . . add 18x-36

5 = x . . . . . . ones digit

14-5 = 9 . . . tens digit

The original number is 95.

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Check

95 -59 = 36

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Additional comment

The difference in value when the digits of a 2-digit number are reversed is always 9 times the difference in the digits. This means the difference in the digits is 36/9 = 4. Since the sum of digits is 14, the two digits are (14+4)/2 = 9 and (14-4)/2 = 5. 95 is the number of interest.

The solutions to a "sum and difference" problem are half the sum and half the difference of the given sum and difference. That is how we know the larger digit is (1/2)(14 + 4), for example.

We worked this "the hard way" using the above equation. It can actually be worked in your head if you're familiar with these generic solutions.