Tools for school

First we need to define the amount of job done in terms of time and rate $\large \frac{time}{rate}=Amount-of-job-done$ For example if you use use a tool that does a job in 3 hours for 6 hours you will have completed 2 jobs. $\large \frac{6hours}{3\frac{hours}{job}}=2job$ Since we want to know how long it will take the tools to do half the job $\large \frac{x}{rate}=\frac{1}{2}job$ the rate of the first tool is 3 hours per 1 job the second 5 hours per 1 job and the third tool 7 hours per one job. We want the sum of all three to do half the job so $\large \frac{x}{3}+\frac{x}{5}+\frac{x}{7}=\frac{1}{2}$ $\large \frac{35x+21x+15x}{105}=\frac{1}{2}$ multiply both sides by 105 and combine like terms $\large 71x=\frac{105}{2}$ now divide both sides by 71 $\large x=\frac{105 \times 1}{2 \times 71}=\frac{105}{142}hours$ now that we know how many hours it will take to complete half the job and we know that there are sixty minutes per every hour all we have left to do is get our units right. Hours to complete the job times minutes in an hour, 60. $\large \frac{105}{142} \times 60=\frac{6300}{142}≈44$ If you're answer is more than the tool with the fastest rate you know you're answer doesn't make sense. If you have 1 tool working on a job and you add more tools to work on the job and it takes you longer?

Full job = 1

Half job = 0.5

P1 = 1/3 , P2 = 1/5, P3 = 1/7

[0.5 / (1/3+1/5+1/7)] × 60 = 44.36619718

44 minutes.