+1222 A mathematician works for t hours per day and solves p problems per hour, where t and p are positive integers. One day, the mathematician drinks some coffee and discovers that he can now solve 3p + 12 problems per hour. In fact, he only works for t - 7 hours that day, but he still solves twice as many problems as he would in a normal day. How many problems does he solve the day he drinks coffee?
+1910 We can write an equation to solve this problem. We have
\((3p+12)(t-7)=2tp\)
Now, factoring the left side, we get
\(3pt-21p+12t-84=2tp\\ 12t+pt-21p=84\\ pt=21p-12t+84\\ pt=7(3p+12)-12t\\ pt+12t=7(3p+12)\\ t(p+12)=7(3p+12)\\ t/7=\frac{3p+12}{p+12}\)
We could test some values here, since no time was given. t has to be less than 24 and a multiple of 7.
When t is 7, p is 0.
When t is 14, p is 12.
When t is 21, p is undefined.
Thanks! :)
