Question

A two digit number is 8 times the sum of its
digits. The number obtained by interchanging
the digits is 45 less than the original number.
Find the two digit number.​

Answer 1

Answer:

✳ The two digit number is 72.

SOLUTION

Let us assume that the digit in the ten's place is x.

Let us assume that the digit in the one's place is y.

The two digit number = 10x+y

Case - 1

It is said that the two digit number is 8 times the sum of its digits .

⇒ 10x + y = 8(x+y)

⇒ 10x + y = 8x + 8y

⇒ 10x - 8x = 8y - y

⇒ 2x = 7y

$\implies \sf x = \dfrac{7}{2} y \longrightarrow (1)$

⇒ x = 3.5 y

Case - 2

The number obtained by interchanging the digits is 45 less than the original number.

⇒ 10y + x = (10x + y) - 45

⇒ 10y + x = 10x + y - 45

⇒ 10y - y = 10x - x - 45

⇒ 9y = 9x - 45

⇒ 9y - 9x = - 45

Dividing throughout by 9,

⇒ y - x = -5

⇒ y - 3.5 y = 5

$\implies \sf y = \dfrac{5}{2.5}$

⇒ y = 2

Now let us substitute y = 2 in (1) ,

$\implies \sf x = \dfrac{7}{2} * 2$

⇒ x = 7.

∴ The required two digit number , 10x+y = 7 × 10 + 2 = 72