SOLUTION: 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

Algebra ->  Expressions-with-variables -> SOLUTION: 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      Log On


   

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.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
N= the original number
N%2885%2F100%29= 85% of the original number
N%2BNk%2F100=N%28%28100%2Bk%29%2F100%29= the original number increased by %22k+%25%22 .
Increasing a number by %22k+%25%22 is multiplying it by %28%28100%2Bk%29%2F100%29 .
Similarly, decreasing a number S by %22w+%25%22 is multiplying it by %28%28100-w%29%2F100%29 :
S-Sw%2F100=S%281-w%2F100%29=S%28%28100-w%29%2F100%29 .
So,
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 .
Our equation is
N%28%28100%2Bk%29%2F100%29%28%28100-w%29%2F100%29=N%2885%2F100%29
%28%28100%2Bk%29%2F100%29%28%28100-w%29%2F100%29=%2885%2F100%29
%28%28100%2Bk%29%2F100%29%28100-w%29=85
%28100%2Bk%29%28100-w%29=85%2A100
%28100%2Bk%29%28100-w%29=8500
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
There should be %282%2B1%29%283%2B1%29%281%2B1%29=3%2A4%2A2=24 factors, which come in 24%2F2=12 pairs:
1%2A8500=8500 ,
2%2A4250=8500 ,
4%2A2125=8500 ,
5%2A1700=8500 ,
10%2A850=8500 ,
17%2A500=8500 ,
20%2A425=8500 ,
25%2A340=8500 ,
34%2A250=8500 ,
50%2A170=8500 ,
68%2A125=8500 , and
85%2A100=8500 .
Since 0%3Ck%3C173 , 100%3C100%2Bk%3C273 .
The only factors that could be 100%2Bk are 125 , 170 , and 250 .
100%2Bk=125--->k=125-100--->highlight%28k=25%29 , or
100%2Bk=170--->k=170-100--->highlight%28k=70%29 , or
100%2Bk=250--->k=250-100--->highlight%28k=150%29 .