document.write( "Question 1199522: if $\"32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29\"$ , where a,b and c are positive integers, and b < c, evaluate the smallest possible value of abc \n" ); document.write( "

Algebra.Com's Answer #833454 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( " $\"a%2B1%2F%28%28c%2B1%29%2Fb%29\"$ = $\"a%2Bb%2F%28c%2B1%29\"$ = $\"%28a%28c%2B1%29%2Bb%29%2F%28c%2B1%29\"$ = $\"%28ac%2Ba%2Bb%29%2F%28c%2B1%29\"$

\n" ); document.write( " $\"%28ac%2Ba%2Bb%29%2F%28c%2B1%29=32%2F9\"$

\n" ); document.write( "Certainly if we are wanting the product abc to be minimum, we want c to be as small as possible; so try c=8.

\n" ); document.write( " $\"%288a%2Ba%2Bb%29%2F9=32%2F9\"$
\n" ); document.write( " $\"9a%2Bb=32\"$

\n" ); document.write( "Again we want the product abc to be minimum; that means the product ab should be minimum. The minimum value of the product ab, subject to the constraint 9a+b=32, is when a=3 and b=5.

\n" ); document.write( "ANSWER: abc = 3*5*8 = 120

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