It would take one machine 4 hours to complete a production order and another machine 2 hour to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.

This problem is a classic example of a work-rate problem, where we need to determine the total time it takes for multiple entities (in this case, two machines) to complete a single task (a production order) when working together. The key is to first understand the individual work rate of each machine and then combine these rates to find their collective efficiency. The total work to be completed is considered as '1 unit' representing the entire production order.

Machine Work Rates Explained

The work rate of a machine indicates how much of the total production order it can complete in one hour. If a machine takes a certain number of hours to complete an entire job, its rate is the reciprocal of that time.

- Machine 1 Rate: We are given that one machine takes 4 hours to complete the production order. This means that in 1 hour, Machine 1 completes $\frac{1}{4}$ of the production order. So, the work rate of Machine 1 $=\frac{1}{4}$ order per hour.
- Machine 2 Rate: We are also given that the second machine takes 2 hours to complete the same production order. This means that in 1 hour, Machine 2 completes $\frac{1}{2}$ of the production order. So, the work rate of Machine 2 $=\frac{1}{2}$ order per hour.

Combined Production Rate Calculation

When both machines work simultaneously on the same production order, their individual work rates add up. This sum gives us the combined rate at which they complete the order together in one hour.

Combined Work Rate $=$ Work Rate of Machine 1 $+$ Work Rate of Machine 2

Combined Work Rate $=\frac{1}{4}+\frac{1}{2}$

To add these fractions, we need a common denominator, which is 4:

Combined Work Rate $=\frac{1}{4}+\frac{1\times 2}{2\times 2}$

Combined Work Rate $=\frac{1}{4}+\frac{2}{4}$

Combined Work Rate $=\frac{1+2}{4}$

Combined Work Rate $=\frac{3}{4}$ order per hour.

This result means that together, the two machines can complete $\frac{3}{4}$ of the total production order in one hour.

Time to Complete Production Order

To find the total time taken for both machines to complete the entire production order, we use the formula:

Time Taken $=\frac{\text{Total Work}}{\text{Combined Work Rate}}$

Since the 'Total Work' is one complete production order (1 unit of work):

Time Taken $=\frac{1}{\frac{3}{4}}$

To divide by a fraction, we multiply by its reciprocal:

Time Taken $=1\times \frac{4}{3}$

Time Taken $=\frac{4}{3}$ hours.

Therefore, if both machines work simultaneously at their respective constant rates, they will complete the production order in $\frac{4}{3}$ hours.

Production Order Calculation Summary

Here is a quick summary of the work rates and the final calculated time:The calculated time of $\frac{4}{3}$ hours matches the provided correct answer option.