# If Steven can mix 20 drinks in 5 minutes, Sue can mix 20 drinks in 10 minutes, and Jack can mix 20 drinks in 15 minutes, how much time will it take all 3 of them working together to mix the 20 drinks?

Mar 19, 2017

It will take $2.73$ minutes for all the $3$ of them working together to mix the $20$ drinks.

#### Explanation:

As Steven can mix $20$ drinks in $5$ minutes,

he can mix $\frac{20}{5} = 4$ drinks in $1$ minute.

As Sue can mix $20$ drinks in $10$ minutes,

she can mix $\frac{20}{10} = 2$ drinks in $1$ minute.

and as Jack can mix $20$ drinks in $15$ minutes,

he can mix $\frac{20}{15} = \frac{4}{3}$ drinks in $1$ minute.

Hence, together all $3$ can mix $4 + 2 + \frac{4}{3} = \frac{22}{3}$ drinks in $1$ minute

i.e. $1$ drink in $\frac{3}{22}$ minutes

and $20$ drinks in $20 \times \frac{3}{22} = \frac{60}{22} = 2.73$ minutes.

Mar 19, 2017

$2. \overline{72}$ minutes or $3$ minutes for a whole number of drinks.

#### Explanation:

The least common multiple of $5$, $10$ and $15$ is $30$. So let's look at how many drinks the team could mix in $30$ minutes:

Steven: $20 \cdot \frac{30}{5} = 120$

Sue: $20 \cdot \frac{30}{10} = 60$

Jack: $20 \cdot \frac{30}{15} = 40$

So a total of $120 + 60 + 40 = 220$ drinks.

So to mix $20$ drinks would take the team:

$30 \cdot \frac{20}{220} = \frac{30}{11} = 2. \overline{72}$ minutes.

If each was working on their own, then at the end of $2. \overline{72}$ minutes they would each be part way through making a drink.

Let us see how far they would get:

Steven: $\frac{120}{11} = 10 \frac{10}{11}$

Sue: $\frac{60}{11} = 5 \frac{5}{11}$

Jack: $\frac{40}{11} = 3 \frac{7}{11}$

So that's $18$ whole drinks. So we would require two of the team to finish to get the whole $20$ drinks, or one of the team to finish one and make a complete extra one.

Let us see how much more time they need to complete the partial drink:

Steven: $\frac{1}{11} \cdot \frac{5}{20} = \frac{1}{44}$ minutes

Sue: $\frac{6}{11} \cdot \frac{10}{20} = \frac{3}{11}$ minutes

Jack: $\frac{4}{11} \cdot \frac{15}{20} = \frac{3}{11}$ minutes

So we would have to wait $\frac{3}{11}$ minutes for two more of the team to finish, by which time all three would have finished.

$\frac{30}{11} + \frac{3}{11} = \frac{33}{11} = 3$

So $3$ minutes total, by which time Steven could have finished making his $12$th drink, giving a total of $12 + 6 + 4 = 22$ drinks.

If it is possible to break down the stages of preparation of a drink in order that more than one person can work on it, then it may be possible to achieve the $20$ whole drinks in $2. \overline{72}$ minutes.

For example:

• Jack completes $\frac{1}{11}$th of a drink before handing it to Steven to complete. (Steven may finish making $10$ whole drinks before he gets around to it)
• Jack completes $\frac{6}{11}$ths of a second drink before handing it to Sue to complete. (Sue may finish making $5$ whole drinks before she gets around to it)
• Jack completes $3$ whole drinks in the remaining time.