Question

a two digit number is 5 times the sum of the digits and is also equal to 5 more than twice the product of the digits . find the number

Answer 1

Let xy be the required two digit number.

Let x be the digit which is in unit's digit.

Let y be the number which is in ten's digit.

Therefore the decimal expansion is 10x+y.   ----- (1)


Given that two digit number is 5 times the sum of its digits.

10x + y = 5(x + y)

10x + y = 5x + 5y

5x - 4y = 0   

y = 5x/4   ------- (2)


Given that it is equal to five more than twice the product of its digits.

10x + y = 5 + 2 * xy

10x + y = 5 + 2xy   ---------- (3)

Substitute (2) in (3), we get

10x + 5x/4 = 5 + 2x(5x/4)

45x = 20 + 10x^2

10x^2 - 45x + 20 = 0

2x^2 - 9x + 4 = 0

2x^2 - 8x - 1x + 4 = 0

2x(x - 4) -1( x - 4) = 0

(x - 4)(2x - 1) = 0

x = 4 (or) x = 1/2.

Therefore x = 4.

Substitute x = 4 in (2), we get

y = 5 * 4/4

y = 20/4

y = 5.


Substitute x and y in (1), we get

The required number = 10(x) + y

                                     = 10 * 4 + 5

                                     = 45.


Therefore the required number is 45.


Hope this helps!