Question

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.

Answer 1

Let the two – digit number = 10x + y
Case I: multiplying the sum of the digits by 8 and then subtracting 5 = two – digit number
$\Rightarrow$ 8 $\times$ (x + y) – 5 = 10x + y
$\Rightarrow$ 8x + 8y – 5 = 10x + y
$\Rightarrow$ 2x – 7y = - 5...(i)
Case II: Multiplying the difference of the digits by 16 and then adding 3 = two – digit number
$\Rightarrow$16 $\times$ (x –y) + 3 = 10x + y
$\Rightarrow$16x – 16y + 3 = 10x + y
$\Rightarrow$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
$\Rightarrow$ y = 3
Now put the value of y in Eq. (i) we get
2x – 7 × 3 = - 5
$\Rightarrow$2x = 21 – 5 = 16 .
So, x = 8
Hence, the required two – digit number
= 10x + y
= 10 × 8 + 3 = 80 + 3 = 83