Interesting. Meanwhile I just had the experience of trying to tutor a not-very-bright kid who eventually got it and was even able to produce an appropriate equation, but it took some doing. I tried it on a familiar old logic problem that other folks were using as an illustration of how it can solve it handily, and then it even produced a generalized equation. It suddenly got stupid when I tried it, even using exactly the same numbers! I did get finally get it to that point, but it needed lots of help. It’s like for some reason it’s suddenly become stupider than the experience that others reported.
Herewith the painful session:
There are 5 machines that can produce 5 devices in 5 minutes. How long will it take 100 machines to produce 100 devices?
We can start by using the rate of production for one machine: one machine produces one device in one minute.
So, five machines can produce five devices in one minute.
If we want to know how long it will take 100 machines to produce 100 devices, we can use the following proportion:
5 machines / 5 minutes = 100 machines / x minutes
where x is the number of minutes it will take for 100 machines to produce 100 devices.
We can solve for x by cross-multiplying:
5 machines * x minutes = 5 minutes * 100 machines
5x = 500
x = 100
Therefore, it will take 100 machines 100 minutes to produce 100 devices.
That is not correct. Let me give you a hint. What is the production rate per machine per minute?
You are correct, I apologize for the mistake in my previous answer.
The production rate per machine per minute is 1 device / 5 minutes = 1/5 device per minute.
Therefore, the total production rate for 5 machines is:
5 machines x 1/5 device per minute per machine = 1 device per minute
This means that 5 machines can produce 1 device in 1 minute.
To find out how long it will take 100 machines to produce 100 devices, we can use the following proportion:
5 machines / 1 minute = 100 machines / x minutes
where x is the number of minutes it will take for 100 machines to produce 100 devices.
We can solve for x by cross-multiplying:
5 * x = 100 * 1
x = 20
Therefore, it will take 100 machines 20 minutes to produce 100 devices.
Also not correct. Since realistically each machine can only achieve partial completion in one minute, it’s more realistic to say that each machine produces one device every five minutes. Hence five machines working together produce five devices, as stated. Now, how long will it take for 100 machines each producing one device every five minutes to produce 100 devices total?
You are correct, thank you for pointing that out.
If each machine produces one device every 5 minutes, then the production rate for one machine is 1/5 device per minute.
Therefore, the total production rate for 100 machines is:
100 machines x 1/5 device per minute per machine = 20 devices per minute
To produce 100 devices, we can use the following proportion:
20 devices / 1 minute = 100 devices / x minutes
where x is the number of minutes it will take for 100 machines to produce 100 devices.
We can solve for x by cross-multiplying:
20 * x = 100 * 1
x = 5
Therefore, it will take 100 machines 5 minutes to produce 100 devices.
That is correct! Can you now write out a general formula for the time T it would take for M machines to produce D devices at a production rate of P devices per minute?
Yes, I can write a general formula for the time T it would take for M machines to produce D devices at a production rate of P devices per minute.
The formula is:
T = D / (M * P)
where:
T is the time it will take for M machines to produce D devices
M is the number of machines
D is the number of devices to be produced
P is the production rate per machine per minute (in devices per minute)We can derive this formula using the following proportion:
M machines / P devices per minute = D devices / T minutes
Cross-multiplying gives:
M * P * T = D
Solving for T, we get:
T = D / (M * P)
Therefore, if we know the number of machines, the production rate per machine, and the number of devices to be produced, we can use this formula to calculate the time it will take to produce them.
Yes, very good, your formula is correct!