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Question
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 equation (i) by 3 and then subtracting from equation (ii), we get
(6x – 17y) – (6x – 21y) = – 3 – (–15)
⇒ 4y = 12
⇒ y = 3
Now, put the value of y in equation (i), we get
2x – 7 × 3 = –5
⇒ 2x = 21 – 5 = 16
⇒ x = 8
Hence, the required two-digit number
= 10x + y
= 10 × 8 + 3
= 80 + 3
= 83
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