Video: Pack 1 • Paper 2 • Question 11

Pack 1 • Paper 2 • Question 11

08:31

Video Transcript

The following pie charts give information about the ages of people visiting two beaches, one in Cornwall and one in South Wales. The number of people at each beach is proportional to the area of each pie chart. What proportion of the total number of beach visitors were between the ages of 31 to 50? Give your answer as a percentage to two significant figures. And note that the pie charts are drawn to scale.

Now with these questions, when the figures are drawn to scale, it’s important that we print out the page of the correct size. Or else our measurements will be different and we’ll get a different answer. When we print them out, we need to make sure that we use scale 100 percent or actual size on our printer settings. Don’t use scale to fit or a booklet format or multiple pages per page.

We’re gonna need to take some measurements from these pie charts. Firstly, we’re going to need to measure the radius of each circle. And secondly, we’re going to need to measure the size of this angle for the sector for the 31 to 50 year olds on South Wales pie chart and also this angle for the sector of the 31 to 50 year olds in the Cornwall pie chart. And we’re going to need to use that information in conjunction with the fact that we know that the number of people at each beach is proportional to the area of each pie chart. And we then ask what proportion of the total number of beach visitors were between the ages of 31 to 50.

First thing, let’s look at the radius of the pie chart for the South Wales area. We need to line up the zero on our ruler with the center of the circle. And we need to measure this distance really carefully. So look directly above and look down on your ruler so that you don’t get any parallax error.

And we can see here that the measurement is 2.4 centimeters. And for Cornwall, again carefully line up the ruler so that the zero is at the center of the pie chart. And then measure this radius here, which takes us up to exactly three centimeters.

Now we know the radii of each of those pie charts. Let’s go on to measure the angles. Well, for this, we’re going to need to use our protractor. And we’re gonna need to lay it down very carefully so that the base line on our protractor lines up perfectly with one of the arms of the sector that we’re interested in, for 31 to 50 year olds. But we also need to line it up very carefully so that the center of the crosshairs lines up perfectly with the center of the pie chart. Then identify where the other arm of the sector that you’re interested in is. And here it is. And we then got to measure the size of this angle here.

Now there are two different sets of numbers on the scale on our protractor. And we need to start counting from zero and work our way all around the scale. And in this case, that takes us up to 90 degrees. And we find that, for South Wales, for 31- to 50-year-olds, that sector of the pie chart has an angle of 90 degrees. And again, for Cornwall, we need to line up the base line on our protractor with one of the arms of the sector that we’re looking for and the crosshairs in the middle of the protractor here with the center of the pie chart.

We then identify where the arms of the sector are. And in this case, we’ve got to measure the size of this angle here between them. Again, there are two sets of numbers on the scale. We’re interested in starting counting from zero. And we measure the size of the angle going round here, which in this case takes us up to 60 degrees.

So now we have all our measurements, let’s transfer them back to our original question. So for South Wales then, the radius of the pie chart was 2.4 centimeters and the angle of the sector that we’re interested in was 90 degrees. For Cornwall, the radius was three centimeters and the angle of the sector we’re interested in is 60 degrees.

Now we can set off and find the proportion of the total number of beach visitors that were between the ages of 31 to 50. So let’s work out the area of the whole pie chart for the South Wales. Now that’s a circle. And the formula for the area of a circle is the area is equal to 𝜋 times the radius squared. And in this case, that’s 𝜋 times 2.4 squared. And when I square 2.4, I get 5.76.

Now I’m gonna leave this in terms of 𝜋 because that’s the most accurate way I can leave it. So the area of the pie chart for South Wales is 5.76 times 𝜋 squared centimeters. And now we need to work out the area of the sector that we’re interested in for 31- to 50-year-olds.

Now remember that this sector is only 90 out of 360 degrees of a full circle. So the proportion of the whole circle taken up by that sector is ninety three hundred and sixtieths. And that cancels to a quarter. So the area of that sector is one-quarter of 5.76𝜋 square centimeters, which is 1.44𝜋 square centimeters. So let’s just make a note of those over here and go on to work out the equivalent values for Cornwall.

Well, again, to work out the total area of the Cornwall pie chart, it’s gonna be 𝜋 times the radius squared. But this time, the radius is three centimeters, not 2.4. So that’s gonna be 𝜋 times three squared. So the total area of the Cornwall pie chart is nine 𝜋 square centimeters. And again, I’m gonna leave my answer in terms of 𝜋 for maximum accuracy.

And now to work out the area of the sector for the 31- to 50-year-olds, we’re looking for what proportion of a whole turn is this angle here. It’s 60 out of 360. So the area of the sector we’re interested in the Cornwall pie chart is sixty three hundred and sixtieths of nine 𝜋 square centimeters. This time, that proportion, sixty to three hundred and sixtieths, cancels down to a sixth. So the total area of that sector is a sixth of nine 𝜋 square centimeters, which is 1.5𝜋 square centimeters.

Now we can calculate the proportion of all the visitors through 31 to 50 years old. We just simply need the number of 31- to 50-year-olds over the total number of visitors that visit those beaches.

Now we don’t have the actual number of 31- to 50-year-olds or the total number of visitors to those beaches. So how are we gonna work out the proportion? Well, remember, in the original question, we were told that the number of people recorded at each beach is proportional to the area of each pie chart. So instead of using those numbers, we’re going to use the corresponding areas.

So for the number of 31- to 50-year-olds in South Wales, we’re gonna use 1.44𝜋. And for Cornwall, we’re gonna use 1.5𝜋. Then for the total number of visitors, we’re gonna use the corresponding total area. So that’s 5.76𝜋 for South Wales and nine 𝜋 for Cornwall. Then 1.44𝜋 plus 1.5𝜋 is 2.94𝜋. And 5.76𝜋 plus nine 𝜋 is equal to 14.76𝜋. So the proportion is 2.94𝜋 over 14.76𝜋.

And the more eagle-eyed amongst you will notice that the 𝜋s are going to cancel if I divide top and bottom by 𝜋. So I’m left with just 2.94 divided by 14.76, which my calculator tells me is 0.199186 and so on.

Now we’re nearly there. But we’re not quite there. We’ve got to give our answer as a percentage, not just a straight proportion. And we’ve got to round it to two significant figures. Now to convert my proportion into a percentage, I need to multiply it by 100. So that’s 19.9186 and so on percent.

And finally, to round to two significant figures, the one is my first significant figure. The nine is my second significant figure. But I need to decide whether I need to round up or whether I can leave it at 19. So we look at the next digit. And that’s a nine, which is five or above. So I’m gonna ha- need to round up. Now rounding 19 up, it becomes 20. So our answer is 20 percent to two significant figures.

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