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
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
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
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