Explainer: Proportional and Nonproportional Relationships

In this explainer, we will learn how to recognize ratios that are in proportion, find an unknown term in a proportion, and identify proportionality in real-world problems.

The word “relationship” means that we are looking at how two different quantities are connected. For instance, in which way does the price of an item relate to its amount? In general (that is, when there is no special offer), the price of an item is proportional to its amount. We will see how this type of relationship is mathematically described.

Let us first recall the notion of ratio between two quantities.

Ratios

Ratios are used to compare two numbers or quantities. They describe a relationship between these two quantities. For instance, if a fruit salad was made with 4 apples and 6 pears, then the ratio of apples to pears in the recipe is 46.

Let us take another example. A publisher wants to publish a children’s book with a ratio of illustrated pages to text pages of 34.

We see that the book could have 7,14,21,28, pages, with 3,6,9,12, illustrated pages and 4,8,12,16, text pages. (Have you noticed that the total number of pages is a multiple of 7?) The corresponding ratios of illustrated pages to text pages are then 34, 68, 912, 1216, and so on.

The two quantities we are looking at here (the number of illustrated pages and the number of text pages), and so their relationship, can be represented in various ways, as shown here.

All these ratios are equivalent: they all describe the relationship “there are 3 illustrated pages for every four text pages.” And, indeed, when they are expressed in their simplest form, they are all written as 34.

When in all situations the ratios of one quantity to the other are equivalent, then we say that both quantities are proportional or are in proportion. We can say as well that the relationship between the two quantities is a proportional relationship.

In our example with the books, the number of illustrated pages and the number of text pages are proportional. If one quantity is doubled, then the other is doubled as well, and if one quantity is halved, then the other is halved as well.

Let us summarize what a proportional relationship is.

Proportional Relationship

Two quantities 𝐴 and 𝐵 are proportional, or in proportion, when from one situation to another, both quantities have been multiplied (or divided) by a same number. It follows that the ratios of quantity 𝐴 to quantity 𝐵 are in all situations equivalent.

Mathematically, it means that if in a first situation, quantity 𝐴 is 𝐴 and quantity 𝐵 is 𝐵, and, in another situation, quantity 𝐴 is 𝐴 and quantity 𝐵 is 𝐵, then 𝐴𝐵=𝐴𝐵 and 𝐴𝐵=𝐴𝐵 when 𝐴 is proportional to 𝐵.

Let us see with an example the different mathematical expressions for equivalent part-to-part ratios.

Example 1: Expressing Mathematically Two Equivalent Part-to-Part Ratios

A4 paper has a width of 21 cm and a length of 29.7 cm. A3 paper is made out of two A4 sheets, as shown in the figure.

Which of the following mathematical sentences prove that the width-to-length ratio in A3 is (almost exactly) the same as in A4?

  1. 29.7×29.72142
  2. 29.729.742×21
  3. 29.7×4229.721
  4. 29.7×2142×29.7
  5. 2129.729.742

Answer

The width-to-length ratio of A4 is 2129.7.

The length of A3 is 2×21=42cm, and its width is 29.7 cm. Hence, its width-to-length ratio is 29.742.

The two ratios are almost equivalent (which means it is approximately the same ratio) if both sides of the first ratio are multiplied by the same number to give the two sides of the second ratio.

For the first side of the ratio, we have 21×29.721=29.7. So, for the second side, we have 29.7×29.72142.

Therefore, sentence A is correct.

We can also multiply both sides of the previous equation by 21, which gives 29.7×29.742×21.

So, sentence B is correct.

Now, dividing both sides of the previous equation by 29.7×42, we get 29.7422129.7.

Hence, sentence E is also correct.

The correct answers, then, are A, B, and E.

So far, we have talked about ratios that compare two quantities of the same nature, often two parts of the same whole. Proportions are ratios that compare a part to a whole. For instance, if we consider 8 girls (a part) out of 20 students (the whole), we say that the proportion of girls in this group is 820. The ratio is then written as a fraction, and it can be expressed in its simplest form as 25. We can now imagine a school with one class of 20 students with 8 girls, another class of 15 students with 6 girls, and one class of 30 students with 12 girls. In the three classes, the proportions of girls are 820, 615, and 1230. All these fractions are equivalent to 25, and we say that the three classes have the same proportion of girls.

Let us see now with the next example how to identify equivalent proportions in a context.

Example 2: Equivalent Part-to-Whole Ratios (Proportions) in a Context

The table gives the amount five families in a neighborhood gave to charities last year along with their annual income.

Family 1Family 2Family 3Family 4Family 5
Amount Given to Charities ($)450650400540550
Annual Income ($)45,00065,00032,00036,00055,000

For these five families, is the amount given to charities last year proportional to their income?

Answer

We need to check here whether, for all families, the fraction of their income spent on charity donation is the same. We have the following:

  • Family 1: 45045,000=1100,
  • Family 2: 65065,000=1100,
  • Family 3: 40032,000=4320=180,
  • Family 4: 54036,000=543,600=271,800=3200,
  • Family 5: 55055,000=1100.

We see that, for families 1, 2, and 5, the fraction of their annual income spent on charity donation is the same (1100), but families 3 and 4 spent another fraction of their income on charity donation, namely, 180 and 3200. So, the answer is no: for these five families, the amount given to charities is not proportional to their income.

Here, we only needed to identify whether the five proportions of income spent on charity donation are the same or not. However, we often want to be able to compare proportions and that is why we usually express them as percentages. We would get the following:

  • Family 1: 1%,
  • Family 2: 1%,
  • Family 3: ×1,000100=1.25%,
  • Family 4: 3÷2200÷2=1.5%,
  • Family 5: 1%,

Finally, rates are ratios that compare two quantities of different nature (expressed in different units), for instance, when we compare the price of an item and its quantity.

Let us say that 3 books cost $9. The ratio of the price to the number of books will be expressed as the quotient 93, which is 3. This 3 means that one book costs $3. We often say $3 per book. This is the coefficient of proportionality, or unit rate (here unit price), of the proportional relationship between the price and the number of books. If the price for 5 books is $15, then the ratio of the price to the number of books is 155, which is 3 as well. Here, the price is proportional to the number of books—namely, the price is $3 times the number of books. This proportional relationship can be represented on a double number line. In a proportional relationship between two quantities, all pairs of values of the two quantities are vertically aligned on the double number line.

Let us see with the next example how to use a double-line diagram to help us identify whether two quantities are proportional.

Example 3: Identifying Proportional Relationships

Determine whether the quantities in the following rates are proportional: $40 for 4 bracelets and $36 for 3 bracelets.

  1. Not proportional
  2. Proportional

Answer

The two quantities we are looking at are the price and the number of bracelets one gets for this price.

In the first situation given, the price is $40 and the number of bracelets is 4. The ratio price/number of bracelets is therefore 404=10. (This means that one bracelet costs $10.)

In the second situation given, the price is $36, and the number of bracelets is 3. The ratio price/number of bracelets is therefore 363=12. (This means that one bracelet costs $12.)

The two ratios (or rates, because the cost and number of bracelets are different in nature) are not the same, so the two quantities are not proportional.

We can visualize that these two quantities are not in a proportional relationship by using a double line diagram. For the first situation, I align 4 bracelets with $40. Then, I place 3 on the number line for the number of bracelets and $36 on the price number line. We see that 3 and 36 are not aligned.

In the previous examples, we have identified whether two quantities are in a proportional relationship by determining whether the ratio of one quantity to the other (be it a part-to-part ratio, part-to-whole ratio, or rate) is constant. Now, we are going to look at two questions where a relationship between two quantities is described and, from this description, we will decide whether it is a proportional relationship or not.

Example 4: Identifying a Proportional Relationship from a Description

A website is selling e-books. It offers a discount of $0.50 for every referral you make. Is the amount of money saved by that offer proportional to the number of referrals you make?

Answer

It is said that one gets a discount of $0.50 for every referral one makes. This “$0.50 for every referral” is a coefficient of proportionality: it shows that, for any number, 𝑛, of referrals, the discount will be 𝑛×$0.50. So, the answer is yes: the amount of money saved (it is the amount of the discount) is proportional to the number of referrals.

Example 5: Identifying a Nonproportional Relationship from a Description

Uptown Pizzeria sells medium pizzas for $7 each and charges a $3 delivery fee per order. Is the cost of an order proportional to the number of pizzas ordered?

Answer

It is said here that each pizza costs $7 but there is also a delivery fee on top of the price for the pizzas. It means that

  • for one pizza, the order costs 1×7+3=10;
  • for two pizzas, the order costs 2×7+3=17;
  • for three pizzas, the order costs 3×7+3=24;

and so on.

The cost of the order is not proportional to the number of pizzas since the cost for two pizzas is not twice as much as the cost for one pizza, and the cost for three pizzas is not thrice as much as for one pizza.

This can be shown on a double-line diagram as well. In this diagram, we aligned 1 pizza with its price ($10).

The price and the number of pizzas are not aligned for two and three pizzas, showing that we have here three different rates: $10 per pizza, $8.50 per pizza, and $8 per pizza.

Hence, the cost of an order is not proportional to the number of pizzas.

Let us summarize what we have learned so far.

Key Points

  1. Two quantities are said to be proportional when, from one situation to another, both quantities have been multiplied (or divided) by the same number. It follows that the ratios of quantity 𝐴 to quantity 𝐵 are in all situations equivalent.
  2. When we know the values of two different quantities in at least two situations, we can identify whether the quantities are proportional in these situations. For this, we work out the ratio of one of the quantities to the other in each situation. The quantities are proportional if the ratios are equivalent (that is, they have the same simplest form).
  3. How the two quantities in a ratio relate to each other can be visualized in different ways.

Ratios that compare two parts of a whole:

Ratios that compare a part and a whole (called a proportion):

Ratios that compare two quantities of different nature (called a rate):

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