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A two-digit number can be 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 $3$ . What is the number?

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answer is C.

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Detailed Solution

Given, the two-digit number can be obtained by multiplying the sum of the digits by

8

and then subtracting

5

.
Same number can be obtained by multiplying the difference of the digits by

16

and then adding

3

.
Let us assume that the tenth’s place is x, and one’s place is y.
So, the two-digit number will be

10 x + y

.
From the first condition, we get,

8 x + y − 5 = 10 x + y ⇒ 8 x + 8 y − 5 = 10 x + y ⇒ 2 x − 7 y = − 5 .... (1)

And, according to the second condition, we get,

16 x − y + 3 = 10 x + y ⇒ 16 x − 16 y + 3 = 10 x + y ⇒ 6 x − 17 y = − 3 .... 2

On multiplying the equation (1) by

3

and subtracting equation (2) from it, we get,

6 x − 21 y − 6 x + 17 y = − 15 + 3 ⇒ − 4 y = − 12 ⇒ y = 3

On substituting the value

y = 3

in equation (1), we get,

⇒ 2 x − 73 = − 5 ⇒ 2 x = − 5 + 21 ⇒ 2 x = 16 ⇒ x = 8

Thus, the two-digit number is,

10 x + y ⇒ 108 + 3 ⇒ 83

Hence, the correct option is 3.

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