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3.1 L) Rate Questions – Part 2
3.1 L) Rate Questions – Part 2
Rate questions can be made harder by machines breaking down or by more machines being added. We are going to have a few examples where this is the case.
The content in this section builds on the content that was discussed in the previous section. Before working through this section, make sure that you have worked through the previous section (click here to be taken back to the previous section).
The content in this section builds on the content that was discussed in the previous section. Before working through this section, make sure that you have worked through the previous section (click here to be taken back to the previous section).
Example 1
9 machines work at the same rate. It takes these 9 machine 14 hours to complete a task.
After working for 5 hours, 3 of these machines break down. The remaining machines continue working until the task has been completed.
How many hours in total does it take for the machines to complete the task?
Like the previous question, the first step is to work out the total number of machine-hours. We work this out by multiplying the number of machines (9) by the number of hours (14).
9 machines work at the same rate. It takes these 9 machine 14 hours to complete a task.
After working for 5 hours, 3 of these machines break down. The remaining machines continue working until the task has been completed.
How many hours in total does it take for the machines to complete the task?
Like the previous question, the first step is to work out the total number of machine-hours. We work this out by multiplying the number of machines (9) by the number of hours (14).
This calculation tells us that it takes 126 machine-hours to complete the task. The next part of the question is best answered by using a timeline starting from 0 and ending at an undefined point.
We were told at the start of the question that after 5 hours 3 of the machines break. This means that for the first 5 hours we have 9 machines and for the remaining hours needed to complete the task, we only have 6 machines (9 – 3 = 6). I am now going to work out how many machine-hours are contributed by the 9 machines working for 5 hours. I am able to do this by multiplying the number of machines (9) by the number of hours (5), which gives us 45 machine-hours. I have added this information to the diagram below. mh on the timeline below stands for machine-hours.
From the diagram above, you can see that we need to have 126 machine-hours in total and after 5 hours, 45 machine-hours have been contributed. We now need to find out how many machine-hours are left, and we do this by taking the number of hours contributed (45) from the number of hours required (126), which tells us that we still need to have 81 machine-hours (126 – 45 = 81).
For the rest of the time needed to complete the job, there will only be 6 machines because 3 of the 9 machines broke down. Therefore, we need to have 81 machine-hours from the remaining 6 machines for the job to be completed. We are able to work out how long it will take for the 6 machines to produce 81 machine-hours of work by dividing the number of machine-hours required (81) by the number of machines (6).
The calculation above tells us that it will take 13.5 hours for the remaining 6 machines to complete the job. This information has been added to the timeline below.
We now need to give our answer to the question. The question asks us to find the total number of hours needed to complete the job. According to the timeline above, it will take 18.5 hours to complete the job.
Example 2
All printers at a company work at the same rate. It takes 20 minutes for 6 printers to print a certain number of photos.
The company starts printing the photos with the 6 printers. After 7 minutes, the company add 2 more printers. How long will it take in total for all of the photos to be printed? Give your answer in minutes and seconds.
Like all of the previous questions, the first step is to work out the number of machine-minutes it takes to print out all of the photos. We can do by multiplying the number of machines (6) by the number of minutes (20).
All printers at a company work at the same rate. It takes 20 minutes for 6 printers to print a certain number of photos.
The company starts printing the photos with the 6 printers. After 7 minutes, the company add 2 more printers. How long will it take in total for all of the photos to be printed? Give your answer in minutes and seconds.
Like all of the previous questions, the first step is to work out the number of machine-minutes it takes to print out all of the photos. We can do by multiplying the number of machines (6) by the number of minutes (20).
This tells us that it takes 120 machine-minutes to print all of the photos. I am now going to have a timeline that starts from 0 and end with an undefined time.
The question tells us that there are 6 machines at the start and then after 7 minutes, 2 more machines are added. This means that we have 6 machines for the first 7 minutes and 8 machines for the remaining minutes until the job is completed.
I am now going to work out how many machine-minutes are contributed by the 6 machines in the first 7 minutes. We do this by multiplying the number of machines (6) by the number of minutes (7), which means that 42 machine-minutes are contributed in the first 7 minutes. I have added this information to the timeline below. mm means machine-minutes.
I am now going to work out how many machine-minutes are contributed by the 6 machines in the first 7 minutes. We do this by multiplying the number of machines (6) by the number of minutes (7), which means that 42 machine-minutes are contributed in the first 7 minutes. I have added this information to the timeline below. mm means machine-minutes.
We need to have a total of 120 machine-minutes in order for all of the photos to be printed. We have just found out that during the first 7 minutes, 42 machine-minutes have been contributed. We can find out how many machine-minutes are left by taking the number of machine-minutes that have been contributed (42) from the total number of machine-minutes required (120). This tells us that in order for the job to be completed, 78 more machine-minutes are required (120 – 42 = 78).
These 78 machine-minutes will be contributed by the 8 machines (the 6 existing machines and the two additional machines that have been added). We can find out how long it takes for these 8 machines to contribute 78 machine-minutes by dividing the number of machine-minutes (78) by the number of machines (8), which is 9.75 minutes. This information has been added to the timeline below.
The question asks us to work out how long it takes in total for the photos to be printed. We can find this out by adding the minutes from the 6 machines (7 minutes) and the minutes from the 8 machines (9.75 minutes) together. This tells us that it takes 16.75 minutes to print all of the photos (7 + 9.75).
The question asks us to give our answer in minutes and seconds. The 16 minutes is fine, but we need to convert the .75 from minutes to second. One minute is 60 seconds, so we convert minutes to second by multiplying by 60. Therefore, we multiply 0.75 by 60 to get 45 seconds.
This means that it takes 16 minutes and 45 seconds for the printers to print the required number of photos.