# Question Video: Solving Word Problem Involving Proportions

It took 3 people 2 1/3 hours to paint a large room. How long would it take 6 people to paint the same room assuming that they all work at the same rate?

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### Video Transcript

It took three people two and one-third hours to paint a large room. How long would it take six people to paint the same room assuming that they all work at the same rate?

Let’s start by considering this as a ratio of the people to the time taken. So for three people, taking two and third hours to paint a room, we could write this as the ratio three to two and a third. We need to work out for six people how long will it take to paint the same room. It would be very easy to think we must simply multiply by two. However, we need to think carefully about the problem that’s presented. If we were painting a room and we asked someone to help us, would we expect it to take a longer time or a shorter time? We’d would expect that more people would mean that the time taken would be less. Here, we have an example of an indirect proportion. In an indirect proportion, as one variable increases, the other variable decreases. Here, as the number of people increases, the time taken would actually decrease.

So to answer this question, let’s take the variable 𝑡 to represent the time taken. We can write that three times two and a third equals six times 𝑡. The left-hand side of our equation represents the total time taken by our three people spending two and a third hours each painting the room. We can simplify our equation by writing three times seven over three equals six 𝑡, since our mixed member fraction two and a third becomes seven over three as an improper fraction. Since our threes on the left-hand side will cancel, we then have seven equals six 𝑡. To find 𝑡 then, we can divide both sides of our equation by six giving us seven over six. We could also write this as one and one-sixth. So our final answer then is that it takes six people one and one-sixth hours to paint the room.

Let’s check our answer by considering this indirect proportion. We know that it takes three people two and a third hours to paint the room. We know that we’re considering six people and we must multiply three by two to get six. The time taken by three people is two and a third hours. And we’ve established that we don’t multiply this value by two. However, since there is an indirect proportion, the values are still proportional in some way. The inverse of multiplying by two would be dividing by two, which is equivalent to multiplying by a half. So when we divide two and a third by two, we would get one and one-sixth hours, which would confirm our original answer of one and a sixth hours.