document.write( "Question 1017725: An original number is increased by k%, and then the result is decreased by w%. The final result is 85% of the original number. If k is a positive integer less than 173 and if w is a positive integer, find the sum of all possible distinct values of k. \n" ); document.write( "

Algebra.Com's Answer #633973 by KMST(5328) $\"\"$ $\"About$

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$\"N\"$ = the original number
\n" ); document.write( " $\"N%2885%2F100%29\"$ = 85% of the original number
\n" ); document.write( " $\"N%2BNk%2F100=N%28%28100%2Bk%29%2F100%29\"$ = the original number increased by $\"%22k+%25%22\"$ .
\n" ); document.write( "Increasing a number by $\"%22k+%25%22\"$ is multiplying it by $\"%28%28100%2Bk%29%2F100%29\"$ .
\n" ); document.write( "Similarly, decreasing a number $\"S\"$ by $\"%22w+%25%22\"$ is multiplying it by $\"%28%28100-w%29%2F100%29\"$ :
\n" ); document.write( " $\"S-Sw%2F100=S%281-w%2F100%29=S%28%28100-w%29%2F100%29\"$ .
\n" ); document.write( "So,
\n" ); document.write( " $\"N%28%28100%2Bk%29%2F100%29%28%28100-w%29%2F100%29\"$ = the original number first increased by $\"%22k+%25%22\"$ , and then decreased by $\"%22w+%25%22\"$ .
\n" ); document.write( "Our equation is
\n" ); document.write( " $\"N%28%28100%2Bk%29%2F100%29%28%28100-w%29%2F100%29=N%2885%2F100%29\"$
\n" ); document.write( " $\"%28%28100%2Bk%29%2F100%29%28%28100-w%29%2F100%29=%2885%2F100%29\"$
\n" ); document.write( " $\"%28%28100%2Bk%29%2F100%29%28100-w%29=85\"$
\n" ); document.write( " $\"%28100%2Bk%29%28100-w%29=85%2A100\"$
\n" ); document.write( " $\"%28100%2Bk%29%28100-w%29=8500\"$
\n" ); document.write( "Now we need pairs of factors of $\"8500=85%2A10%2A10=17%2A5%2A2%2A5%2A2%2A5=2%5E2%2A5%5E3%2A17=2%5E2%2A5%5E3%2A17%5E1\"$
\n" ); document.write( "There should be $\"%282%2B1%29%283%2B1%29%281%2B1%29=3%2A4%2A2=24\"$ factors, which come in $\"24%2F2=12\"$ pairs:
\n" ); document.write( " $\"1%2A8500=8500\"$ ,
\n" ); document.write( " $\"2%2A4250=8500\"$ ,
\n" ); document.write( " $\"4%2A2125=8500\"$ ,
\n" ); document.write( " $\"5%2A1700=8500\"$ ,
\n" ); document.write( " $\"10%2A850=8500\"$ ,
\n" ); document.write( " $\"17%2A500=8500\"$ ,
\n" ); document.write( " $\"20%2A425=8500\"$ ,
\n" ); document.write( " $\"25%2A340=8500\"$ ,
\n" ); document.write( " $\"34%2A250=8500\"$ ,
\n" ); document.write( " $\"50%2A170=8500\"$ ,
\n" ); document.write( " $\"68%2A125=8500\"$ , and
\n" ); document.write( " $\"85%2A100=8500\"$ .
\n" ); document.write( "Since $\"0%3Ck%3C173\"$ , $\"100%3C100%2Bk%3C273\"$ .
\n" ); document.write( "The only factors that could be $\"100%2Bk\"$ are $\"125\"$ , $\"170\"$ , and $\"250\"$ .
\n" ); document.write( " $\"100%2Bk=125\"$ ---> $\"k=125-100\"$ ---> $\"highlight%28k=25%29\"$ , or
\n" ); document.write( " $\"100%2Bk=170\"$ ---> $\"k=170-100\"$ ---> $\"highlight%28k=70%29\"$ , or
\n" ); document.write( " $\"100%2Bk=250\"$ ---> $\"k=250-100\"$ ---> $\"highlight%28k=150%29\"$ . \n" ); document.write( "

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