document.write( "Question 131457: The difference of 2 numbers is 4. The difference between their reciprocals is 4/21. Find the 2 numbers \n" ); document.write( "

Algebra.Com's Answer #95955 by josmiceli(19441) $\"\"$ $\"About$

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Call the larger number $\"a\"$
\n" ); document.write( "Call the smaller number $\"b\"$
\n" ); document.write( " $\"a+-+b+=+4\"$
\n" ); document.write( " $\"a+=+b+%2B+4\"$
\n" ); document.write( " $\"%281%2Fb%29+-+%281%2Fa%29+=+4%2F21\"$
\n" ); document.write( "Notice that the reciprocal of a smaller
\n" ); document.write( "number will be larger than the reciprocal
\n" ); document.write( "of a larger number
\n" ); document.write( "Substituting:
\n" ); document.write( " $\"%281%2Fb%29+-+%281%2F%28b%2B4%29%29+=+4%2F21\"$
\n" ); document.write( "multiply both sides by $\"b%2A%28b%2B4%29%2A21\"$
\n" ); document.write( " $\"21%2A%28b%2B4%29+-+21%2Ab+=+4%2Ab%2A%28b%2B4%29\"$
\n" ); document.write( " $\"21b+%2B+84+-+21b+=+4%2Ab%5E2+%2B+16b\"$
\n" ); document.write( " $\"4%2Ab%5E2+%2B+16b+-+84+=+0\"$
\n" ); document.write( " $\"b%5E2+%2B+4b+-+21+=+0\"$
\n" ); document.write( "solve by completing the square
\n" ); document.write( " $\"b%5E2+%2B+4b+%2B+%284%2F2%29%5E2+=+21+%2B+%284%2F2%29%5E2\"$
\n" ); document.write( " $\"b%5E2+%2B+4b+%2B+4+=+25\"$
\n" ); document.write( " $\"%28b+%2B+2%29%5E2+=+25\"$
\n" ); document.write( " $\"b+%2B+2+=+5\"$ (only the + square root makes sense here)
\n" ); document.write( " $\"b+=+3\"$
\n" ); document.write( " $\"a+-+b+=+4\"$
\n" ); document.write( " $\"a+-+3+=+4\"$
\n" ); document.write( " $\"a+=+7\"$
\n" ); document.write( "The numbers are 7 and 3
\n" ); document.write( "check:
\n" ); document.write( " $\"%281%2Fb%29+-+%281%2Fa%29+=+4%2F21\"$
\n" ); document.write( " $\"%281%2F3%29+-+%281%2F7%29+=+4%2F21\"$
\n" ); document.write( " $\"%287%2F21%29+-+%283%2F21%29+=+4%2F21\"$
\n" ); document.write( " $\"7+-+3+=+4\"$
\n" ); document.write( " $\"4+=+4\"$
\n" ); document.write( "OK \n" ); document.write( "

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