Tricky Logic: Total time to complete the task Claire can finish the the task in $5$ hours, Ruth can finish it in $7$ hours. If Clair works for $1$ hour alone, then Ruth joins her to finish the task. What is the total time to complete the task? Answer: let $t$ = number of hours Claire & Ruth work together. $$(t+1)/5 + t/7 = 1 \tag{*}\label{*}$$ The total time is $(t + 1)$ hours. I don't understand the equation $\eqref{*}$. May anyone explain it to me? Thanks a lot!

Let $X$ be the amount of work to be done. Then the rates of works per hour of Claire and Ruth are: $$\frac{X}{5} \ \ \ \text{and} \ \ \ \frac{X}{7}.$$ For example, if the task is to solve $35$ problems, Claire will solve $7$ problems per hour and Ruth will solve $5$ problems per hour.

Now the problem states that Claire works for an hour. Then Ruth joins and they work together to complete the task. So, the total time is $t+1$ for Claire and $t$ for Ruth. We make up an equation: $$\frac{X}{5}\cdot (t+1)+\frac{X}{7}\cdot t=X \ \ \ \stackrel{\text{divide by}X}\Rightarrow \ \ \ \ \frac{t+1}{5}+\frac{t}{7}=1.$$ Is it clear now? Once you solve the equation and find $t$, will it be the final answer or $t+1$?