Question

I need good, clearly explaining for this qustion.NOTE: Harry can fill an order in s minutes.

Answer 1

We have to find how long does it take for each person to fill an order.

We will use the concept of "capacity", that is the amount of order they can make in a minute. Then, when two persons work together, their capacities can be added.

We have Harry, Hermione and Ron.

Lets call C1: capacity of Harry, C2: capacity of Hermione, and C3: capacity of Ron.

Then, we can write:

`\begin{gathered} C_1=(1)/(s) \\ C_2=(1)/(s-2) \\ C_3=(1)/(s+1) \end{gathered}`

Note that the capacity is the inverse of the time it takes for them to fill one order.

We know that when Harry and Hermione work together, they take 1 minute and 20 seconds to fill an order. We can express this in minutes as:

`T=1\min +20s\cdot(1\min)/(60s)=1+(20)/(60)=1+(1)/(3)=(4)/(3)\min`

Then, we can write that the sum of the capacities of Harry and Hermione is equal to the inverse of the time it took them to fill an order:

`C_1+C_2=(1)/(T)`

If we replace C1, C2 and T, we get:

`\begin{gathered} C_1+C_2=(1)/(T) \\ (1)/(s)+(1)/(s-2)=(1)/((4)/(3)) \\ (1)/(s)+(1)/(s-2)=(3)/(4) \end{gathered}`

We can solve for s as:

`\begin{gathered} (1)/(s)+(1)/(s-2)=(3)/(4) \\ (s+s-2)/(s(s-2))=(3)/(4) \\ (2s-2)/(s^2-2s)=(3)/(4) \\ 4\cdot(2s-2)=3\cdot(s^2-2s) \\ 8s-8=3s^2-6s \\ 0=3s^2-6s-8s+8 \\ 0=3s^2-14s+8 \end{gathered}`

We have a quadratic equation, so we can find the solutions for s as:

`\begin{gathered} s=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ s=\frac{-(-14)\pm\sqrt[]{(-14)^2-4\cdot3\cdot8}}{2\cdot3} \\ s=\frac{14\pm\sqrt[]{196-96}}{6} \\ s=\frac{14\pm\sqrt[]{100}}{6} \\ s=(14\pm10)/(6) \\ s_1=(14-10)/(6)=(4)/(6)=(2)/(3) \\ s_2=(14+10)/(6)=(24)/(6)=4 \end{gathered}`

We have 2 solutions for s.

We can check that s = 2/3 is not a valid solution because it would make Hermione fill the order in negative time: s has to be greater than 2 for Hermione to have a valid time to fill the order:

`s-2>0\Rightarrow s>2`

Then, the value of s is 4.

Then, if we replace s we can calculate the time it takes for each member to fill an order:

`\begin{gathered} \text{Harry}\longrightarrow t_1=s=4\min \\ \text{Hermione}\longrightarrow t_2=s-2=4-2=2\min \\ \text{Ron}\longrightarrow t_3=s+1=4+1=5\min \end{gathered}`

Now, if the three work together, the capacities add, so we can express the time T it would take as:

`C_1+C_2+C_3=(1)/(T)`

We can replace the capacities and calculate T as:

`\begin{gathered} (1)/(T)=(1)/(s)+(1)/(s-2)+(1)/(s+1)=(1)/(4)+(1)/(2)+(1)/(5) \\ (1)/(T)=(1)/(4)\cdot(5)/(5)+(1)/(2)\cdot(10)/(10)+(1)/(5)\cdot(4)/(4) \\ (1)/(T)=(5)/(20)+(10)/(20)+(4)/(20) \\ (1)/(T)=(5+10+4)/(20) \\ (1)/(T)=(19)/(20) \\ T=(20)/(19) \\ T\approx1.0526 \end{gathered}`

It would take them approximately 1.0526 minutes, that can be expressed as 1 minute and 3 seconds. Note that this time is less than the time it takes Hermione and Harry when they work together, as we are adding a new person and increasing the capacity.

Answer:

a) Harry: 4 min, Hermione: 2 min, Ron: 5 min

b) It would take 1 minute and 3 seconds approximately.