Each machine has the same constant rate at a factory. 4 machines work together and take 30 hours to fill out an order, if 5 machines worked together how many fewer hours does it take?? The answer is 6 hours as (4 times 30) divided by 5 equals 24. 30 minus 24 equals 6.
For me, I did 30 divided by 4 equals 7.5, which is the rate that each machine can work. 7.5 times 5 equals 37.5 and 37.5 minus 30 equals 7.5. I did 30 divided by 4 because I wanted to find the rate of each machine. I don't get why they multiplied 30 and 4 together and then divide everything by 5.
I understand why you would divide 30 by 4 and
think that was the rate (hours per machine); but machines working together don't combine that way! You would do that division, for example, if you were
making machines (
one at a time) and it took 30 hours to make 4. Making more takes
longer. But here, more machines working together take
less time, because they are working
simultaneously, not
sequentially.
One way to express this is that the amount of effort required to fill one order can be measured in
machine-hours: one machine working for one hour. Four machines working for one hour do 4 times as much work as one (4 machine-hours); and 4 machines working for 30 hours do 30 times that much work (120 machine-hours to fill one order). So we
multiply the number of machines by the time that each works, to get the amount of work done. Then if 5 machines work together, how long will it take to fill an order? They need to do 120 machine-hours of work; 120/5 = 24 machines.
The rate that is actually involved here is "orders per machine-hour", not "machines per hour".
A similar issue arises in electronics, where resistances add in
series (one after another), but their reciprocals add when they are in
parallel (sharing the load) You always have to pay attention to how the numbers combine, and not just do what you would do in a different kind of problem.