In all of the examples in the previous sections, the machines and individuals have all worked at the same rate. We are now going to have an example whereby the individuals completing a task work at different rates.

**Example 1**

The next example will involve two people working at different rates.

Jeremy and Richard are translating a document. If they were translating the document on their own, it would take Jeremy 10 days and it would take Richard 8 days.

Jeremy starts to translate the document. On the fourth day, Jeremy joins Richard and they translate the document together.

How many days will it take to translate the whole document?

The first step in answering questions like this is to work out the rate of translating per day and I am going to work the rate out as a percentage.

We are told at the start of the question that it would take Jeremy 10 days to translate 100% of the document. We work the percentage of the document that he translates per day by dividing 100% by 10 (the number of days), which gives us 10% per day.

We now do the same for Richard. The question tells us that it takes him 8 days to translate 100% of the document. We work out the percentage translated per day by dividing 100% by 8, which is 12.5% per day.

I am now going to answer this question by using a table. The table will have 4 rows that are days, Jeremy, Richard and total. The table is shown below.

The question tells us that Jeremy translates the document for the first 3 days by his self. We have worked out that Jeremy translates 10% of the document per day. Therefore, I am going to put 10% in his row for the first 3 days and an x in Richards row because he doesn’t start translating the document until day 4. I will then work out the totals after each day. The first 3 days are filled in for the table below.

From the table we can see that after 3 days, 30% of the document has been translated.

From day 4, both Jeremy and Richard are translating the document; Jeremy translates the document at a rate of 10% per day and Richard translates the document at a rate of 12.5% per day. The figures for day 4 are shown in the table below.

From day 4, both Jeremy and Richard are translating the document; Jeremy translates the document at a rate of 10% per day and Richard translates the document at a rate of 12.5% per day. The figures for day 4 are shown in the table below.

We now continue the process until the total percentage is 100% or above 100%. This is because the document won’t be fully translated until the percentage is 100% or above.

From the table above, we can see that the first day where the total percentage of the document is above 100% is day 7. Therefore, it takes 7 days for the document to be translated.