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Question 9
A two – digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

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Solution

Let the two – digit number = 10x + y
Case I: multiplying the sum of the digits by 8 and then subtracting 5 = two – digit number

8 × (x + y) – 5 = 10x + y
8x + 8y – 5 = 10x + y
2x – 7y = - 5...(i)
Case II: Multiplying the difference of the digits by 16 and then adding 3 = two – digit number
16 × (x –y) + 3 = 10x + y
16x – 16y + 3 = 10x + y
6x – 17y = - 3...(ii)

Now, multiplying Eq. (i) by 3 and then subtracting from Eq. (ii), we get
6x – 17y = - 3
6x – 21y = - 15
4y = 12
y = 3

Now put the value of y in Eq. (i) we get
2x – 7 × 3 = - 5
2x = 21 – 5 = 16 .
So,
x = 8
Hence, the required two – digit number
= 10x + y
= 10 × 8 + 3 = 80 + 3 = 83


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