document.write( "Question 1103077: A positive integer is 11 less than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is 7/6, then find the two integers. \n" ); document.write( "

Algebra.Com's Answer #717775 by KMST(5328) $\"\"$ $\"About$

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THE FIFTH-GRADER APPROACH:
\n" ); document.write( "Let us try positive integers,
\n" ); document.write( "starting with $\"1\"$ and $\"1%2B11=12\"$ ,
\n" ); document.write( "and see if we bump into the solution.
\n" ); document.write( "The reciprocal are $\"1%2F1=1\"$ and $\"1%2F12\"$ ,
\n" ); document.write( "so the sum is
\n" ); document.write( " $\"1%2B2%281%2F12%29=1%2B2%2F12=1%2B1%2F6=6%2F6%2B1%2F6=7%2F6\"$ ,
\n" ); document.write( "and that works, so if there is just one solution,
\n" ); document.write( "as the question suggests,
\n" ); document.write( "the numbers are $\"highlight%281%29\"$ and $\"highlight%2812%29\"$ .
\n" ); document.write( "
\n" ); document.write( "THE ALGEBRA STUDENT APPROACH:
\n" ); document.write( " $\"x\"$ = the smaller positive integer
\n" ); document.write( " $\"x%2B11\"$ = the larger positive integer
\n" ); document.write( " $\"1%2Fx\"$ = the reciprocal of the smaller integer
\n" ); document.write( " $\"2%281%2F%28x%2B11%29%29=2%2F%28x%2B11%29\"$ = twice the reciprocal of the larger integer
\n" ); document.write( "The sum is
\n" ); document.write( " $\"1%2Fx%2B2%2F%28x%2B11%29=7%2F6\"$
\n" ); document.write( "From here, we could say that
\n" ); document.write( "we multiply both sides of the equal sign times $\"6x%28x%2B11%29\"$
\n" ); document.write( "to eliminate denominators,
\n" ); document.write( "and get
\n" ); document.write( " $\"6x%28x%2B11%29%281%2Fx%2B2%2F%28x%2B11%29%29=%287%2F6%29%2A%286x%28x%2B11%29%29\"$
\n" ); document.write( " $\"6x%28x%2B11%29%2Fx%2B2%2A6x%28x%2B11%29%2F%28x%2B11%29=7x%28x%2B11%29\"$
\n" ); document.write( " $\"6%28x%2B11%29%2B12x=7%28x%5E2%2B11x%29\"$
\n" ); document.write( " $\"6x%2B66%2B12x=7x%5E2%2B77x\"$
\n" ); document.write( " $\"18x%2B66=7x%5E2%2B77x\"$
\n" ); document.write( " $\"0=7x%5E2%2B77x-18x-66\"$
\n" ); document.write( " $\"7x%5E2%2B59x-66=0\"$
\n" ); document.write( "At that point, knowing that x is a positive integer,
\n" ); document.write( "you know that factoring is an option.
\n" ); document.write( "Looking at the numbers, you see that $\"x=1\"$ is a solution,
\n" ); document.write( "because $\"7%2B59-66=66-66=0\"$ .
\n" ); document.write( "The factoring gives you
\n" ); document.write( " $\"%28x-1%29%287x%2B66%29=0\"$ ,
\n" ); document.write( "and the positive inter solution is $\"x=1\"$ , with $\"x%2B11=1%2B11=12\"$ .
\n" ); document.write( "The other solution is $\"x=-66%2F7\"$ , which is not a positive integer.
\n" ); document.write( "
\n" ); document.write( "You could instead think of finding a common denominator,
\n" ); document.write( "and calculating step by step:
\n" ); document.write( " $\"%28x%2B11%29%2F%28x%28x%2B11%29%29%2B2x%2F%28%28%2B11%29%29=7%2F6\"$
\n" ); document.write( " $\"%28x%2B11%29%2F%28x%5E2%2B11x%29%2B2x%2F%28x%5E2%2B11x%29=7%2F6\"$
\n" ); document.write( " $\"%28x%2B11%2B2x%29%2F%28x%5E2%2B11x%29=7%2F6\"$
\n" ); document.write( " $\"%283x%2B11%29%2F%28x%5E2%2B11x%29=7%2F6\"$
\n" ); document.write( " $\"6%283x%2B11%29%2F%28x%5E2%2B11x%29=7\"$
\n" ); document.write( " $\"6%283x%2B11%29=7%28x%5E2%2B11x%29\"$
\n" ); document.write( " $\"18x%2B66=7x%5E2%2B77x\"$
\n" ); document.write( "Same result, just not as elegant. \n" ); document.write( "

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