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

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Question 632103: The sum of the reciprocals of two consecutive integers is 9/20. Find the two integers.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
THE FIFTH GRADER WAY:
The reciprocals are fractions that have those two integers for denominators.
To add them, we can use the product of those two integers as the common denominator.
We know two numbers that are consecutive integers and whose product is 20.
They are 4 and 5.
Let's see if they work.
1%2F4%2B1%2F5=1%2A5%2F4%2F5%2B1%2A4%2F5%2F4=5%2F20%2B4%2F20=9%2F20
The numbers are highlight%284%29 and highlight%285%29.

THE ALGEBRA WAY:
Let the two consecutive numbers be x and x%2B1.
Their reciprocals are
1%2Fx and 1%2F%28x%2B1%29
The sum of those reciprocals is

We are told that the sum is 9%2F20, so
%282x%2B1%29%2F%28x%5E2%2Bx%29=9%2F20 is our equation.
Multiplying both sides times 20%28x%5E2%2Bx%29, or "equating the cross products", we get
20%282x%2B1%29=9%28x%5E2%2Bx%29
We transform that into the standard form of a quadratic equation.
20%282x%2B1%29=9%28x%5E2%2Bx%29 --> 40x%2B20=9x%5E2%2B9x --> 40x%2B20-40x-20=9x%5E2%2B9x-40x-20 --> 9x%5E2-31x-20=0
Since we expect at least one integer solution for x , factoring should work to solve the equation.
If we are not good at factoring, we solve using the quadratic formula.
If we are good at factoring, we factor to find that
9x%5E2-31x-20=%289x%2B5%29%28x-4%29
We write the equation as %289x%2B5%29%28x-4%29=0
and realize that the solutions to that equation are
x=-5%2F9 and highlight%28x=4%29
We discard the solution that is not an integer, and find that the two integers are x=highlight%284%29
and x%2B1=4%2B1=highlight%285%29 .