Question

two digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 by multiplying the difference of digits by 16 and then adding 3 then find the number

Answer 1

Solution :-

Let the two digit number be 10x + y where x is the tens digit and y is the ones digit.

Now, according to the question.
10x + y = 8(x + y) - 5
10x + y = 8x + 8y - 5
10x - 8x + y - 8y = - 5
2x - 7y = - 5  .................(1)

And,
10x + y = 16(x - y) + 3
10x + y = 16x - 16y + 3
10x - 16x + y + 16y = 3
- 6x + 17y = 3  ................(2)

Now, multiplying the equation (1) by 17 and (2) by 7, we get
34x - 119y = - 85 ...............(3)
- 42x + 119y = 21 ..............(4)

Now, adding (3) and (4), we get

   34x - 119y = - 85
- 42x + 119y =   21
_________________
 - 8x             = - 64
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⇒ 8x = 64
x = 64/8
x = 8 
So, tens digit is 8.  

Substituting the value of x = 8 in (1), we get
2x - 7y = - 5
2*8 - 7y = - 5
16 - 7y = - 5
- 7y = - 5 - 16
- 7y = - 21
7y = 21
y = 21/7
y = 3

Ones digit is 3.
So, the required number is 83.

Answer.

Answer 2

Let us assume, x and y are the two digits of the two-digit number

Therefore, the two-digit number is 10x + y

Given,

10x + y = 8(x + y) – 5

10x + y = 8x + 8y – 5

7y – 2x = 5 ----------------1

Also given,

10x + y = 16(x – y) + 3

10x + y = 16x – 16y + 3

17y – 6x = 3 --------------------2

Multiply equation 1 by 3

21y – 6x = 15 -------------3

Subtract equation 2 from equation 3

4y = 12

y = 3

substitute y = 3 in equation 1

7*3 – 2x = 5

21 – 2x = 5

2x = 16

x = 8

Therefore, the two-digit number = 10x + y = 10*8 + 3 = 83

Ans – The required two-digit number = 83.