document.write( "Question 632103: The sum of the reciprocals of two consecutive integers is 9/20. Find the two integers. \n" ); document.write( "

Algebra.Com's Answer #398021 by KMST(5328) $\"\"$ $\"About$

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THE FIFTH GRADER WAY:
\n" ); document.write( "The reciprocals are fractions that have those two integers for denominators.
\n" ); document.write( "To add them, we can use the product of those two integers as the common denominator.
\n" ); document.write( "We know two numbers that are consecutive integers and whose product is 20.
\n" ); document.write( "They are 4 and 5.
\n" ); document.write( "Let's see if they work.
\n" ); document.write( " $\"1%2F4%2B1%2F5=1%2A5%2F4%2F5%2B1%2A4%2F5%2F4=5%2F20%2B4%2F20=9%2F20\"$
\n" ); document.write( "The numbers are $\"highlight%284%29\"$ and $\"highlight%285%29\"$ .
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\n" ); document.write( "THE ALGEBRA WAY:
\n" ); document.write( "Let the two consecutive numbers be $\"x\"$ and $\"x%2B1\"$ .
\n" ); document.write( "Their reciprocals are
\n" ); document.write( " $\"1%2Fx\"$ and $\"1%2F%28x%2B1%29\"$
\n" ); document.write( "The sum of those reciprocals is
\n" ); document.write( "

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

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