The diagram below represents 200 people who visited a water park one Saturday. Of these 200 people, the ratio of adults to senior citizens was three to two. Part a) Show that 36 senior citizens went to the park that Saturday.
There is also a part b which we’ll look at in a moment. So this data has been represented in a pie chart with sections that represent children, adults, and then senior citizens. Key information given in the question is that there were 200 people in total and within these 200 people, the ratio of adults to senior citizens was three to two. Looking at the pie chart, we’re also given that the angle for the sector which represents children is 198 degrees.
Now, we can’t work out the angles for the individual sectors for adult and senior citizens without measuring with a protractor. And we were not told that this pie chart is drawn accurately. But we can work out the sum of the angles for these two sectors because the sum of angles in a full turn is 360 degrees. The angle for these two sectors combined then is 360 degrees minus 198 degrees. That’s 162 degrees.
Now, there are two ways that we could continue from here. Firstly, we could work out how many of the 200 people are represented by an angle of 162 degrees on the pie chart. The fraction of the pie chart taken up by adults and senior citizens together is 162 over 360. We can then multiply this by the total of 200 to work out how many of the 200 people are represented on these two sections. We can use a calculator for this and it gives 90.
So we now know the total number of adults and senior citizens is 90 and we need to divide this in the ratio three to two to work out how many adults and how many senior citizens there are. First, we find the sum of three and two which is five. So there are five equal parts in this ratio in total.
To work out the value of one part, we can divide the total by five which gives 18. To find the value of two parts, we need to multiply the value of one part by two, 18 by two which gives 36. As two parts in this ratio represents the number of senior citizens, we’ve shown that there are indeed 36 senior citizens who went to the water park on Saturday.
The other method we could have followed would be to divide that angle of 162 degrees in the ratio three to two to give the angle representing adults and the angle representing senior citizens. In the same way that we divided 90 in the ratio three to two, we would first divide 162 by five — that’s the total number of parts — to see that one part is equal to 32.4. We would then multiply this by two to see that two parts are equivalent to 64.8 degrees. So the angle on our pie chart for senior citizens is 64.8 degrees.
We could then work out the number of people represented by this angle in the same way as we worked out the number of people represented by an angle of 162 degrees in our previous method. The fraction of the pie chart taken up by the sector for senior citizens would be 64.8 over 360 and then multiplying this by 200 would calculate the number of senior citizens. This calculation would also give 36. So either method could be used.
Now, I’m going to delete some of the working out on the screen in order to make space for the next part of the question. But you are going to need some of the numbers that we’ve worked out.
Part b) of the question says, “Draw a bar chart to represent the number of children, adults, and senior citizens who went to the park that Saturday.”
A bar chart displays a category on the horizontal axis against the frequency on the vertical axis. The categories for our bar chart are going to be child, adult, and senior citizen. So we can go ahead and add those labels to the horizontal axis.
Now, the bars need to be evenly spaced and they need to be equal width. So you can see that each of the bars I’ve started drawing are one large square in width and there’s a gap of one large square between them.
We already know from part a of the question that there are 36 senior citizens. So we have this frequency. But what about the number of adults and the number of children? Well, remember in part a, we used ratio in order to show that the number of senior citizens was 36. Two parts of the ratio were worth 36 and one part was worth 18.
Adults were represented by three parts of this ratio. So to work out the number of adults, we need to multiply 18 by three, which gives 54. Finally, to work out the number of children, we recall that the total was 200. So if we subtract 54 and 36 — that’s the number of adults and senior citizens — from this total, we’ll be left with the number of children — that’s 110.
So now we’ve worked out the frequencies for the vertical axis on our bar chart, we need to consider what scale we should use. Our scale needs to start at zero and it needs to go up as far as the highest frequency which is 110. Now, if we look at this scale, we see that that are seven large squares. So if we choose each large square to represent 20, then our vertical axis will go up as far as 140 — that’s 20 multiplied by seven.
So we start at zero and we fill in our scale at multiples of 20. Now, within each large square, there are 10 small squares. So 10 small squares represents 20, which means that one small square will represent two people. So we’ve worked out the scale for our vertical axis. We found that there were 110 children. So this will be halfway between 100 and 120 on the vertical axis or you can think of it as being five small squares above 100. So the bar for children will be here.
We found that there were 54 adults. Now, 54 is six below 60. So that’s three small squares below 60. We can fill in the top of the bar for adults here. And finally, for senior citizens, there are 36 senior citizens; that’s four below 40. So the top of the bar will be two small squares below 40. So we have our three bars. We can shade them in if we wish, although this isn’t essential. We also make sure we have a label on our vertical axis to show that it represents the frequency.
So in part a of the question, we showed that there were 36 senior citizens who visited the water park on Saturday and in part b we’ve calculated the frequency for children, adults, and senior citizens and then drawn a bar chart to represent this data.