In this explainer, we will learn how to find an unknown term in a proportion and solve word problems on proportions involving fractions, mixed numbers, and decimals.
The word “relationship” in proportional and nonproportional relationships means that we are looking at how two different quantities are connected. For instance, in which way does the price of identical items relate to the number of items? In general (e.g., when there is no special offer), the price of identical items is directly proportional to the number of items. We will see how this type of relationship is mathematically described.
Let us first recall the notion of ratio between two quantities.
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 oranges, then the ratio of apples to oranges in the recipe is .
In this example of the apples to oranges ratio, we could image making a fruit salad with half the quantities of the previous one, that is, two apples and three oranges. Or we could use three times “two apples and three oranges,” that is, six apples and nine oranges.
The three ratios of apples to oranges (, , and ) are all equivalent; they all describe the same relationship between the number of apples and the number of oranges. This relationship can be described with the words “there are two apples for every three oranges.”
This equivalency can be illustrated with the shown diagrams.
When in all situations the ratios of one quantity to the other are equivalent, then we say that both quantities are directly proportional or are in direct proportion. We can say as well that the relationship between the two quantities is a directly proportional relationship.
In our example with the fruit salad, the number of apples and the number of oranges are directly 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.
Note that when talking about direct proportions, the word “direct” or “directly” is often forgotten. Do remember that when only “proportion” or “proportional” is written, then the word “direct” or “directly” is implicit.
Ratios are written as fractions (or quotients) for proportions and rates. Ratios are often used for part-to-part ratios, while proportions are part-to-whole ratios. Rates are ratios that compare two quantities of different nature (expressed in different units), for instance, when we compare the prices of identical items and the number of items.
Let us imagine that 3 given books cost $9. The ratio of the price to the number of books will be expressed as the quotient , which is three. This three means that one book costs $3. We often say $3 per book. This $3/book is then the unit rate of the proportional relationship between the number of books and the price.
If the price is directly proportional to the number of books, then the price for 5 books is given by five times the price per book; that is, . We see that the ratio of the price to the number of books has not changed.
This directly proportional relationship can be represented on a double number line. All pairs of values (number of books/price) are vertically aligned on the double-line diagram.
Let us summarize what a directly proportional relationship is.
Directly Proportional Relationship: Definition of a Proportion
Two quantities and are directly proportional, or in direct proportion, when ratios of quantity to quantity are in all situations equivalent.
Mathematically, this means that if in a given situation quantity is and quantity is and in another situation quantity is and quantity is , then and when is directly proportional to .
It follows that a proportion can also refer to the above equality which shows that quantities and are directly proportional. We say that the ratios and form a proportion.
Let us now describe how to identify a proportional relationship.
How to Identify a Proportional Relationship
When we know the values of two different quantities in at least two situations, we can identify if the quantities are proportional in these situations or not. 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 (i.e., if they have the same simplest form).
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)
Let us see with one example how to identify a directly proportional relationship between two quantities.
Example 1: Identifying a Proportional Relationship from Two Pairs of Values
Determine which of the following pairs of ratios forms a proportion.
- $7 for 5 cookies; $63 for 45 cookies
- $7 for 5 cookies; $63 for 44 cookies
- $7 for 5 cookies; $63 for 47 cookies
- $7 for 5 cookies; $63 for 46 cookies
In all the four options, the price for five cookies is $7, but the number of cookies that one gets with $63 is different.
To determine for which option the price is proportional to the number of cookies, we can first work out the price per cookie when five cookies cost $7, and then we can find how many cookies can be bought at this unit price with $63.
The price per cookie is the ratio of the price to the number of cookies:
Let us find now how many cookies can be bought with $63 at this price. For this, we need to find how many times there are $1.40 in $63:
We have found that, with $7 for 5 cookies and $63 for 45 cookies, the price of cookies is proportional to the number of cookies. We have indeed
We could also have found that $63 is nine times as much as $7. If the price of cookies is proportional to the number of cookies, then one should get nine times as many cookies with $63 as with $7; that is,
The previous example can be illustrated with the double-line diagram. The price of 63 dollars is aligned with 45 cookies, showing that 45 cookies for $63 are sold at the same unit price as 5 cookies for $7.
Example 2: Identifying Which Ratio Forms a Proportion with a given Ratio
Olivia wants to enlarge a photo that is 4 inches by 6 inches. Which of the following is proportional to the original photo?
- 24 in by 36 in
- 8 in by 10 in
- 18 in by 24 in
- 20 in by 24 in
- 16 in by 20 in
When enlarging a photo, it is important to keep the width-to-height ratio (called aspect ratio) constant; otherwise, the photo will not look as the original.
The aspect of the original ratio here is . We need to find which of the given ratios is equivalent to , whose simplest form is .
We can express all the given ratios in their simplest form and compare with .
We find that
- can be simplified by 12: ,
- can be simplified by 2: ,
- can be simplified by 6: ,
- can be simplified by 4: , and
- can be simplified by 4: .
Hence, the dimensions 24 inches by 36 inches are proportional to the original dimensions of Olivia’s photo.
Example 3: Finding a Missing Value Working Out the Unit Rate
Isabella earned $87.12 in a week for working 18 hours. If she works 25 hours the following week, determine the amount of money she will earn.
We know that Isabella earned $87.12 in 18 hours. As her salary is proportional to the number of hours, we can find the unit rate of this proportional relationship, that is, how much she is paid for one hour and then multiply the unit rate by 25 hours to find how much she will earn when working 25 hours.
To find the unit rate, we divide her salary for 18 hours ($87.12) by 18 hours. We find:
In 25 hours, she will earn 25 times the unit rate:
If she works 25 hours, she will earn $121.
- Two quantities and are directly proportional, or in direct proportion, when ratios of quantity to quantity are in all situations equivalent.
- This is described mathematically with and , where , and , are values of quantities and in two different situations.
- The equation shows that the ratios and form a proportion.
- When identifying whether two quantities are proportional, we need to check whether the ratios of one quantity to the other are or would be equal in all possibly known situations.