In this section, we are going to be looking at the rate of completing a task. The method that we use to answer questions like this vary depending on the type of question that we are asked. We are going to have a look at a few different examples of different ways that these types of questions can be asked.

**Example 1**

It takes 6 men 8 days to build a wall. Estimate how long it would take 4 men to build the same wall?

The first step in answering a question like this is to work out how many men-days there are to build the wall, which means the number of days that it would take one man to build the wall. We work this out by multiplying the number of men (6) by the number of days that it takes them to build the wall (8).

This tells us that there are 48 men-days, which means that if one person was to build the wall on their own, it would take them 48 days.

We now need to find out how long it would take 4 men to build the wall. We work this out by dividing the number of men-days (48) by the number of men (4).

We now need to find out how long it would take 4 men to build the wall. We work this out by dividing the number of men-days (48) by the number of men (4).

Therefore, it takes 12 days for 4 men to build the wall.

When you have obtained an answer for these types of questions, it is a good idea to look at the question and your answer and check that it makes sense. This is because it is quite easy to make a mistake on these types of questions. We were told in this question that it takes 6 men 8 days to build a wall. We were asked to work out how long it would take 4 men to build the same wall. As there are less men working on the wall, we would expect the time taken to build the wall to be longer, and this was the case with our answer (it took 12 days rather than 8 days). This implies that we have obtained the correct answer.

When you have obtained an answer for these types of questions, it is a good idea to look at the question and your answer and check that it makes sense. This is because it is quite easy to make a mistake on these types of questions. We were told in this question that it takes 6 men 8 days to build a wall. We were asked to work out how long it would take 4 men to build the same wall. As there are less men working on the wall, we would expect the time taken to build the wall to be longer, and this was the case with our answer (it took 12 days rather than 8 days). This implies that we have obtained the correct answer.

**Example 2**

It takes 15 machines 10 hours to complete a task. Estimate how long it would take 20 machines to complete the same task?

The first step in answering this question is to work out the total number of machine-hours that it takes to the complete the task. We work this out by multiplying the number of machines (15) by the number of hours (10).

This tells us that there are 150 machine-hours to complete the task – this is the number of hours that it would take 1 machine to complete the task.

The question asks us to find how long it would take for 20 machines to complete the same task. We can work this out by dividing the number of machine hours (150) by the number of machines (20).

The question asks us to find how long it would take for 20 machines to complete the same task. We can work this out by dividing the number of machine hours (150) by the number of machines (20).

This tells us that it takes 7.5 hours for 20 machines to complete the task. We should now check whether this answer makes sense. The example in the question was 15 machines taking 10 hours. We now have more machines (20 rather than 15) and you'd expect that the time that it takes to complete the task would decrease, and this is exactly what happens; it now takes 7.5 hours rather than 10 hours. This implies that we have obtained the correct answer.

**Extension Question**

A common extension for a question like this is to ask about what assumptions you have made. The assumption that we have made is that all of the machines work at the same rate. If the machines did not work at the same rate, we would be unable to work out how long it would take for the 20 machines to complete the task.

This is also true for the first example with the workers building a wall. The assumption that we made when working out the number of days that it takes to build the wall is that the builders work at the same rate as each other.