Video: GCSE Mathematics Foundation Tier Pack 3 • Paper 3 • Question 5

GCSE Mathematics Foundation Tier Pack 3 • Paper 3 • Question 5

08:18

Video Transcript

The eye colour of 54 people was recorded in the table. We’ve got eye colour and frequency. 21 people had brown eyes, 15 had blue eyes, nine had green eyes, and nine had hazel eyes. Part a) What percentage of the people had blue eyes? Give your answer to the nearest whole percent.

Now percentages are a type of proportion, so we’re trying to find the proportion of people who had blue eyes but express that proportion as a percentage. Now we can express the proportion of people who had blue eyes as a fraction, with the number of people who had blue eyes as the numerator and the total number of people in the sample as the denominator. And 15 people had blue eyes, and the question tells us that there were 54 people altogether. So the proportion with blue eyes is fifteen fifty-fourths, because we want a percentage.

Now when we use fractions to represent proportions, we can think of them as being on a scale of zero to one, zero being none of them and one being all of them. But percentages are on a scale of zero to 100, zero being none of them, 100 percent being all of them. So to convert from the fraction to the percentage, we just need to multiply by 100. We’re scaling up by a factor of 100. And this means that the percentage of people with blue eyes was 15 over 54 times 100. When I put that into my calculator, I get an answer of 27.7777 recurring percent.

Now remember, when you type that into your calculator, you might get an answer like 250 over nine. But if you press the S-to-D button and then press it again, it toggles between different modes of representing that number. So press it once and you get 27.7 recurring. Press it gain and it’ll show you all those sevens recurring.

Okay, so we’ve got our answer as percentage, but we haven’t given it to the nearest whole percent. Now to round that to the nearest whole percent, let’s cover up everything after the decimal point. So we’re thinking 27, but what we need to do is just reveal one more digit, the deciding digit, by sliding back our little colour, and that reveals a seven. If that deciding digit that we just revealed, seven in this case, is five or above, then we’re gonna round the 27 up to 28. If it’s below five, then we’ll leave the 27 as it is, a 27. In this case, seven is five or above. That’s telling us that 27.7 is nearer to 28 than it is to 27. So our answer to part a) is that there are 28 percent of people with blue eyes to the nearest whole percent.

And for part b), we’ve got to draw a pie chart for the data in the table, and we’ve been given a circle with a line to get us started. So let’s add a column to our table so we can work out the angle of the sector for each part of the pie chart.

Now there are a couple of useful facts that we know. We know that, in total, there were 54 people in that group, and we know that the number of degrees in a complete turn is 360. Now we can use the total number of people to help us work out the proportions with the different eye colours.

21 out of 54 have brown eyes, 15 out of 54 have blue eyes, nine out of 54 have green eyes, and nine out of 54 have hazel eyes. So twenty-one fifty-fourths of this 360-degree turn are gonna represent people with brown eyes. Fifteen fifty-fourths are gonna represent people with blue eyes, and so on. So we can show our calculations like this. Twenty-one fifty-fourths of 360 is gonna be the number of degrees representing people with brown eyes. Fifteen fifty-fourths times 360 is gonna be the number of degrees representing people with blue eyes, and so on. And when we put those numbers into our calculator, we see that the sector representing people with brown eyes is gonna have an angle of 140 degrees, the sector representing those with blue eyes is gonna have 100 degrees, and the other two sectors are gonna have angles of 60 degrees.

Now an alternate method for working out those angles is to find the angle for one person and then multiply that by the frequency in each case. So that would mean we’ve got 360 degrees representing a full turn and we’ve got 54 people. So if we do 360 divided by 54, that gives us 20 over three or six and two-thirds of a degree for every person. And then you simply multiply that number by the frequency.

So, for example, for brown-eyed people, there are 21 people, each with six and two-thirds of a degree. That’s 140 degrees. For blue-eyed people, 15 lots of six and two-thirds is 100 degrees, and so on. So we get the same answers. But in this particular case, because we ended up with a number of six and two-thirds of a degree for every person, this method turned up pretty horribly. But in other cases, you might come up with a nice number there, and that’s actually quite a nice, easy way of doing the question.

Okay, let’s clear a bit of space and actually draw the pie chart now. Now first up, I will make a little notch at the centre of the circle there, cause that’s gonna make it easier for us to line up our protractor in the right place. Make sure that the base line between the two zeros on the base of your protractor lines up perfectly with that radius we were given and make sure that the crosshair here lines up with the little notch.

Now we’re gonna do this angle of 140 degrees first. So we start counting from zero at the top, follow the scale round until we find 140, then we can make a little notch at 140. Now it’s important that we follow that scale from zero all the way round then get up to 140, because if we’d have started at 180 and gone to 140, we’d have only gone to a very small angle. In fact, it would’ve been 40 degrees, not 140 degrees.

Then we can line our ruler up carefully with the notch in the middle and our little mark on the outside and draw in our line. So this angle here is 140 degrees. Next, we’re gonna line the protractor up along this line and measure 100 degrees in this direction. So again, line up the base line between the zeros on your protractor and the crosshair here with the middle of the circle. Start counting at zero and count up to 100 degrees, then make our little notch. Then line up our ruler with the centre of the circle on the notch we’ve just made and draw a line.

We’ve now got two sectors: one of 140 degrees and one of 100 degrees. With only two sectors left to mark out, we could either choose this line for our base line or this line for our base line for the protractor. Now I would tend to use this one because it was given to us in the question. If we were slightly inaccurate in the way that we measured this angle and the way that we measured this angle, then this line here could be in the wrong place. So if we mark out one of our sectors from here, we’re more likely to have an accurate 60-degree angle here.

So we line the base line between the zeros of our protractor up again with the radius and the crosshair here up with the centre of the circle. We’re gonna take this last measurement here, 60. Start counting at zero and come back 60 degrees and then make our little notch. For the last time, we can line up our ruler with the middle of the circle and the little notch we made and draw in our line.

So this sector here has an angle of 60 degrees and is going to represent people with hazel eyes. Then it’s just worth checking that that last sector for people with green eyes is in fact 60 degrees. If we start counting at zero and go up to here, we see that yes, it is 60 degrees.

Finally, you must label those sectors correctly. You need to write the eye colours down. You don’t need to write the size of the angle down at the middle there. But it’s worth just checking that they’re right and they match the angles that you calculated in your table.

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