Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 3 • Question 25

Thomas is unpacking a delivery of 200 boxes from a lorry into a warehouse. There are three different box sizes. The number of boxes is in the ratio small : medium : large = 12 : 5 : 3. Using a forklift, each trip from the lorry to the warehouse and back takes 𝑥 minutes. Thomas can only transport one type of box per trip. In one trip he can transport 24 small boxes, 10 medium boxes, or 3 large boxes. It takes him one and half hours to unload the lorry and return to his truck. Work out the value of 𝑥.

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Video Transcript

Thomas is unpacking a delivery of 200 boxes from a lorry into a warehouse. There are three different box sizes. The number of boxes is in the ratio small to medium to large is equal to 12 to five to three. Using a forklift, each trip from the lorry to the warehouse and back takes 𝑥 minutes. Thomas can only transport one type of box per trip. In one trip, he can transport 24 small boxes, 10 medium boxes, or three large boxes. It takes him one and half hours to unload the lorry and return to his truck. Work out the value of 𝑥.

Let’s break this question down into three parts. We’ll begin by using the information about the number of boxes and the ratio in which they shared to work out how many of each type of box there are. We’ll then use the information about the number of boxes he can transport per trip to work out how many trips are going to be required for each type of box.

And we’re looking to work out the value of 𝑥. That’s the time taken for each trip from the lorry to the warehouse and back and that’s in minutes. So our final step is going to be to work out how long one trip takes Let’s begin by working out how many of each type of box there are.

We know there are 200 boxes and they split in a ratio small to medium to large, 12 to five to three. There are two ways that we can divide 200 into the ratio of 12 to five to three. Both methods begin with us working out the total number of parts in our ratio that’s 12 plus five plus three which is 20.

Our first method is to then share 200 boxes into these 20 parts. That’s 200 divided by 20 which is equal to 10. We then multiply each part of our ratio by this number 10. So for the number of small boxes, that’s 12 multiplied by 10. For the number of medium boxes, it’s five multiplied by 10. And for the number of large boxes, it’s three multiplied by 10. And we can see that we have 120 small boxes, 50 medium boxes, and 30 large boxes.

And we’ll briefly consider the other method for sharing 200 in the ratio 12 to five to three. We know that the total number of parts is 20. And out of these, 12 of these parts represent the number of small boxes. So that means twelve twentieths of 200 of the boxes are small. And in turn, it means five twentieths of the 200 boxes are medium. And three twentieths of the 200 boxes are large.

And since we commonly replaced the word “of” with the multiplication symbol, we can calculate the number of small boxes by working out twelve twentieths multiplied by 200. For medium boxes, that’s five twentieths multiplied by 200 and for large boxes, it’s three twentieths multiplied by 200. And once again, we get the same values as before: 120 small boxes, 50 medium boxes, and 30 large boxes.

And if we want to check our answer is correct, we could add 120, 50, and 30 and check that it gives us 200 the starting number, which indeed does. So we’ve completed part one. We’ve worked out how many small boxes, medium boxes, and large boxes there are.

Next, we’re going to work out how many trips it takes for each type of box. Thomas can transport 24 small boxes in one trip. We can work out the number of trips is going to take him to transport 120 boxes by dividing 120 by 24. That’s five and that means he needs five trips to transport all the boxes.

We’ll repeat this process for the medium boxes. It’s 10 boxes per trip. And there are 50 boxes. So we’re going to work out 50 divided by 10 which once again is five. So we need five trips to transport all the medium boxes. For the large boxes, he can transport three per trip. And there are 30 in total. So we work out 30 divided by three which is 10. He needs 10 trips to transport all the large boxes. Five plus five plus 10 is 20. He’s going to take 20 trips in total.

And we’ve completed step two. We’ve worked out the number of trips it will take him to transport each type of box. And for part three, we’re going to use this information to work out how long one trip is. Remember that’s the value of 𝑥.

And 𝑥 is in minutes. So we’ll need to take that into account as we work through this final step. It takes him a total of one and a half hours to unload the lorry and return to his truck. That’s the same as 90 minutes. We worked out that he completes 20 trips during that time. So if we divide 90 by 20, it will tell us the time taken for one trip.

90 divided by 20 is 4.5. And we can see that the value of 𝑥 which represents the time taken to complete a trip from the lorry to the warehouse and back is 4.5 or four and a half minutes.

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