A mobile phone carrier records the
amount of mobile data downloaded by people in a small town. 250 people downloaded more than 30
gigabytes of data. Work out the number of people who
downloaded between three and 15 gigabytes of data.
The data has been displayed in a
histogram, with the amount of data downloaded on the horizontal axis and the
frequency density on the vertical axis. However, notice that, in this
histogram, the scale on the vertical axis has been left blank. We may need to work out what the
scale is at some point during the question.
We’re asked to work out the number
of people who downloaded between three and 15 gigabytes of data. In a histogram, the frequency of
each group is given by the area of that bar. So to find the requested number of
people, we need to find the area shaded in pink. That’s the area that corresponds to
between three and 15 gigabytes on the horizontal axis.
We can see where three is on the
horizontal axis, as each small square represents one gigabyte of data. So we need to go three small
squares in. Each bar is a rectangle, and the
area of a rectangle is found by multiplying its width by its height. We can work out the width of each
bar. But in order to find their heights,
we do need to work out the scale that has been used on the vertical axis.
We’re told in the question that 250
people downloaded more than 30 gigabytes of data. This means that the combined area
of the final two bars in the histogram is 250. We can work out the widths of these
bars. The first one is 15; that’s the
difference between 45 and 30. And the second is five; that’s the
difference between 50 and 45.
We need to calculate their
heights. Let’s let each small square on the
vertical axis represent 𝑥 units of frequency density. The height of the first bar is five
small squares, so this is five 𝑥. The height of the second bar is 10
small squares, so this is 10𝑥. The area of the first bar is
therefore 15 multiplied by five 𝑥, and the area of the second bar is five
multiplied by 10𝑥.
Remember, the total area of these
two bars is 250, so we can form an equation. 15 multiplied by five is 75, and
five multiplied by 10 is 50. So our equation becomes 75𝑥 plus
50𝑥 is equal to 250. 75 plus 50 is 125, so we have 125𝑥
is equal to 250.
We can solve this equation by
dividing both sides by 125, giving 𝑥 is equal to two. This tells us that each small
square on the frequency density axis represents two units. We can fill in some of the major
values on this scale. 10 squares will represent 20 units,
20 squares will represent 40 units, and so on, all the way up to 100.
Now we know the scale that has been
used on the frequency density axis. We can find the height of each bar,
and therefore we’re able to calculate the areas we want. The first bar has a width of two;
that’s the difference between five and three; and its height is two small squares
above 40. Remember, each small square
represents two units, so the height is 44. The area of this bar is therefore
two multiplied by 44. The second bar has a width of five;
that’s the difference between 10 and five; and its height is one small square below
100. Remember, each small square
represents two units, so the height of the bar is 98. The area of this bar is therefore
five multiplied by 98. The final bar also has a width of
five; that’s the difference between 15 and 10; and its height is one small square
below 80. This means that the height of this
bar is 78. The area of this bar is therefore
given by five multiplied by 78. And we have our full calculation
for the three areas.
The three individual areas are 88,
490, and 390. And adding the values together
gives 968. Remember, the area of each bar
gives the frequency. So the number of people who
downloaded between three and 15 gigabytes of data is 968 people.