Ola and Petro together sort a warehouse delivery in 6 days, Petro and Roma together in 10 days, and Ola and Roma together in 15/2 days. Each person always works at the same constant rate. How many days do all 3 take working together?

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Pairs at the warehouse

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Ola and Petro together sort a warehouse delivery in $6$ days, Petro and Roma together in $10$ days, and Ola and Roma together in $\frac{15}{2}$ days. Each person always works at the same constant rate. How many days do all $3$ take working together?

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Hint

Add the $3$ pair rates: each person is counted exactly $2$ times.

Solution

The pair rates are $\frac{1}{6}$ , $\frac{1}{10}$ and $\frac{1}{\frac{15}{2}} = \frac{2}{15}$ of the job per day.
Their sum is $\frac{1}{6} + \frac{1}{10} + \frac{2}{15} = \frac{2}{5}$ , and this counts every person's rate $2$ times.
So the $3$ together work at $\frac{2}{5} \div 2 = \frac{1}{5}$ of the job per day.
The job then takes $1 \div (\frac{1}{5}) = 5$ days, so the answer is $5$ .

Answer: $5$

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