Math Lesson 4.1.4 - Part : Whole vs Part : Part Ratios

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Welcome to our Math lesson on Part : Whole vs Part : Part Ratios, this is the fourth lesson of our suite of math lessons covering the topic of Ratios, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Part : Whole vs Part : Part Ratios

We will use another example to illustrate the difference between these two types of ratios.

Example 7:

A company has 120 employees where 45 are male and 3/5 of female employees have attended university.

What part of the staff are female?
How many women from the staff have attended university?
What part of the staff are female employees who have attended university?
What is the maximum number of groups that can be formed by preserving the same gender and education structure of the whole staff?

Solution 7

The number of female workers in the company is 120 - 45 = 75. This number makes
Female∶Total = 75∶120
= 75/120
= 75 ÷ 15/120 ÷ 15
= 5/8 of the staff
We have 3 out of 5 women from the staff who have attended university. Thus, the total number of female staff N_fu who have attended university is
N_fu = 3/5 of 75
= 3/5 × 75
= 3 × 75/5
= 45 women
The above number divided by the total gives the part of the staff that are both female employee and have attended university. Thus,
R(N_fu ) = N_fu/N_tot
= 45/120
= 45 ÷ 15/120 ÷ 15
= 3/8 of the staff
The number of male employees is also 45. This number makes 3/8 of the staff, as found above. Therefore, a group of 8 people must contain 5 women (where 3 of them have attended university) and 3 men. This is the smallest group we can form by preserving the original structure of the company staff. Therefore, the maximum number of groups possible is
N_groups = 120 employees/8 employees/group
= 15 groups

Scaling Up Ratios

We can use the ratio between two or more quantities given in the simplest form to find other equivalent ratios, this can help us find the number of elements in a group. We saw this property when explaining ratios expressed in a number line, where we could find other ratios besides the required one by looking which numbers were aligned vertically. This time however, we can use not two horizontal separate axes placed one under the other but two perpendicular axes instead, which allows us to draw a straight line which gives all possible relationships between the quantities involved through a straight line drawn from the origin.

The advantage of this method known as scaling up ratios consists in the fact that we can obtain a larger number of possible combinations between the quantities involved, which follow the rule given in the ratio. However, it also has a disadvantage: we cannot include more than three quantities in the calculations, as the maximum number of axes we can use is three (the space is 3D).

From algebra, it is known that if two quantities (we often call them "variables", as the values of one quantity affect the corresponding values of the other) are in a constant ratio, the line that shows all possible relations of the two quantities (we call it the "graph" of relation), is linear. The best feature of this method consists on the need for only two corresponding values; one for each quantity. These values act like coordinates of the graph, expressed by number pairs. Then, we can draw a straight line that connects the origin and the given point expressed by the above values (coordinates). This line can be extended further in that direction to obtain other pairs of numbers.

It is better to have the ratio expressed in this form plotted in a millimetre paper, as in the example below.

Example 8

A factory produces hats and shirts in the ratio 3:4.

Plot a graph to show this relationship
From the graph, find the number of shirts produced for every 21 hats
Again, use the graph to find the number of hats produced for every 24 shirts

Solution 8

We express the hats on the horizontal axis and shirts on the vertical one - all this in a millimetre paper, as the one shown below. Since for every 3 hats we have 4 shirts produced, then for every 30 hats there are 40 shirts produced. In this way, we use the pair (30, 40) as a reference for our graph (which starts from the origin). Then, we continue further in that direction. $Math Tutorials: Ratios Example$
If we look at the graph carefully, we notice that for 21 hats there are 28 shirts produced. This is because the point (21, 28) is a point of the graph.
Again, by observing the graph carefully, we notice that for 24 shirts, there are 18 hats produced. This is because the point (18, 24) is also a point of the graph. $Math Tutorials: Ratios Example$ As you see, if we have the graph of a ratio plotted, no further calculations are necessary; we can find all values needed by looking at the graph.

More Ratios Lessons and Learning Resources

Ratio and Proportion Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
4.1	Ratios
Lesson ID	Math Lesson Title	Lesson	Video Lesson
4.1.1	Definition of a Ratio
4.1.2	Ratio in a Number Line
4.1.3	Finding the Fraction of One Quantity Out of the Total
4.1.4	Part : Whole vs Part : Part Ratios
4.1.5	Finding the New Ratio When One Quantity Changes
4.1.6	Dividing a Given Ratio

Whats next?

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Continuing learning ratio and proportion - read our next math tutorial: Rates. Applications of Ratios and Rates in Practice

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