Question

The sum of the reciprocals of two consecutive integers is 9/20. Determine the consecutive integers algebraically.

Answer 1

Answer : The two consecutive integers are, 4 and 5

Step-by-step explanation :

Let the two consecutive number be, x and (x+1)

The sum of the reciprocals of two consecutive integers is 9/20.

The expression will be:

$\frac{1}{x}+\frac{1}{x+1}=\frac{9}{20}$

Now solving the term 'x', we get:

$\frac{x+1+x}{(x)(x+1)}=\frac{9}{20}$

$\frac{2x+1}{x^2+x}=\frac{9}{20}$

$20(2x+1)=9(x^2+x)$

$40x+20=9x^2+9x$

$9x^2-31x-20=0$

We are solving the quadratic equation by middle term splitting.

$9x^2-36x-5x-20=0$

$9x(x-4)+5(x-4)=0$

$(9x+5)(x-4)=0$

$(9x+5)=0\text{ and }(x-4)=0$

$x=\frac{-5}{9}\text{ and }x=4$

We are neglecting the value of $x=\frac{-5}{9}$

Thus we are taking x = 4

When x = 4 then (x+1) = (4+1) = 5

Thus, the two consecutive integers are, 4 and 5

Answer 2

The answer is 2 1/5.