Question

The sum of two digit number is 10 and sum of reciprocal of each digit is 10÷21

Answer 1

Let the one number be x
and the other number be y
x+y =10
x= 10-y ....(1)
According to the question
$\frac{1}{x} + \frac{1}{y} = \frac{10}{21} \\ \frac{y + x}{xy} = \frac{10}{21} \\ 21y + 21x = 10xy \\ 21x + 21y = 10xy \\ 21(x + y) = 10xy \\ 21(10) = 10(10 - y)( y) \: \: \: [from \: equation \: (1) ] \\ 210 = 100y - 10 {y}^{2} \\ 21 = 10y - {y}^{2} \\ {y}^{2} - 10y + 21 = 0 \\ {y}^{2} - 7y - 3y + 21 = 0 \\ y(y - 7) - 3(y -7 ) = 0 \\ (y - 3)(y - 7) = 0 \\ y = 3 \: or \: 7 \\we \: know \: that \: \: x + y = 10 \\when \: y = 3 , \: \: x = 7 \\ when \: y = 7 , \: \: x = 3 \\ thus , \: the \: two \: numbers \: are \: \: 3 \: and \: 7$