Video: Proportional and Nonproportional Relationships

In this video, 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.

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Video Transcript

In this video, we will learn what a proportion is, and about proportional and nonproportional relationships, and how to identify proportionality in real world problems.

Let’s start by recalling a closely linked topic, ratio. Let’s imagine we have a fruit salad recipe which, for two people, requires two apples and four oranges. We could, therefore, write the ratio of apples to oranges as two to four. Now, let’s imagine we wanted to make the fruit salad for four people. We could then simply double the quantities. So, we’d need four apples and eight oranges. In this case, our ratio would be four to eight. We can say that the ratio is the relationship between the parts.

Informally, we can say that proportion is a relationship between a part and a whole. So, if we wanted to take the proportion of apples out of all the fruit, in our first situation this would be two over six. And in our second scenario, the proportion of apples would be four out of 12. We can say in our scenario that the apples and oranges are in proportion. We can say that if we have two quantities 𝐴 and 𝐵, then they’re in proportion when from one situation to another both quantities have been multiplied or divided by the same number. In our fruit salad, we saw that for two people, to get to four people, we multiplied our quantities by two.

We can also think of our proportion relationship in terms of ratios. So, if we have the quantities of 𝐴 and 𝐵 as 𝐴 sub one and 𝐵 sub one in one situation and 𝐴 sub two and 𝐵 sub two in another situation, then we can say that 𝐴 sub one to 𝐵 sub one equals 𝐴 sub two to 𝐵 sub two. In our situation, we have the ratio of apples to oranges was two to four. And that’s equal to the ratio of four to eight. We can say this because four to eight simplifies to the ratio two to four, or that both of them would simplify to the ratio one to two.

We can also write the proportional relationship in the form 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two. And finally, we can also consider proportionality graphically. If we plot 𝐴 versus 𝐵, then there will be a straight line passing through the origin. And finally, when we’re discussing proportion, we must also consider the word rate. A rate is a special type of proportion comparing quantities of a different nature in different units, for example, the price of an item and the quantity. So, in a situation where we paid 40 dollars for eight books, the rate could be written as 40 over eight or five over one, which is essentially five dollars per book.

And we also should be familiar with the term unit rate, which is a rate with a denominator of one. So, now, let’s look at some examples of proportional and nonproportional relationships.

An elevator ascends or goes up at rate of 750 feet per minute. Is the height to which the elevator ascends proportional to the number of minutes it takes to get there?

Let’s start this question by noting that we’re given a rate of 750 feet per minute. We could note this as a fraction of 750 over one. So, let’s take a look at the height that the elevator will ascend for a few different values of the number of minutes. In one minute, we know that the elevator will ascend 750 feet. In two minutes, we’d have two lots of 750 feet, so that’s 1500 feet. In three minutes, we’d have three lots of 750 feet, which is 2250 feet.

In this question, we’re asked if height is proportional to the number of minutes, so let’s recall what it means to be proportional. We can say that two quantities 𝐴 and 𝐵 are in proportion when from one situation to another both quantities have been multiplied or divided by the same number. So, in our first situation, we had 750 over one. That’s 750 feet in one minute. At two minutes, we had the fraction 1500 over two. That’s 1500 feet in two minutes. And at three minutes, we had 2250 feet over three minutes.

We notice that we could get from our first fraction to our second fraction by multiplying the numerator and denominator by two. We can go from our one minute to our three minutes by multiplying our first fraction, 750 over one, by three. Therefore, we can say that our fractions are equal. And therefore, we must have a proportional relationship. So, our answer to the question, is the height proportional to the number of minutes, is yes.

Uptown Pizzeria sells medium pizzas for seven dollars each and charges a three-dollar delivery fee per order. Is the cost of an order proportional to the number of pizzas ordered?

In this question, we’ll go through different scenarios for the cost of pizza and charges. And we’ll verify our answer using a graph. So, let’s start by looking at the cost of an order for different quantities of pizza. So, for one medium pizza, we’re told that the cost is seven dollars and we have a three-dollar delivery charge. So, the total cost of our order would be 10 dollars. For two pizzas, we’d have two times seven dollars. And adding on our three-dollar delivery fee would give us 17 dollars in total. For three pizzas, we’d have three times seven dollars, which is 21 dollars. And adding on our three-dollar delivery would be 24 dollars in total.

In this question, we’re asked if the cost of the order is proportional to the number of pizzas ordered. We can recall that if we have two quantities 𝐴 and 𝐵, then they are in proportion when from one situation to another both quantities have been multiplied by the same number. We could also consider this as 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two, where 𝐴 sub one and 𝐵 sub one are the quantities of 𝐴 and 𝐵 in one situation and 𝐴 sub two and 𝐵 sub two are the quantities of 𝐴 and 𝐵 in another situation.

So, if we take our situation with one pizza, we could say that the cost per pizza would be 10 over one, which is equivalent to 10 dollars per pizza. In our second situation, the cost per pizza would be 17 dollars over two, which is 8 dollars 50 per pizza. In our third situation, we’d have 24 dollars for the order divided by three pizzas, so that’s eight dollars per pizza. So, in order to have a proportional relationship, we would need to check if our fractions 10 over one, 17 over two, and 24 over three are equal. And no, they’re not equal. So, we can say that the cost of an order and the number of pizzas are not proportional.

Let’s have a look at verifying this using a graph. We can plot the number of pizzas versus the cost of the order. Using the values we calculated earlier that one pizza would have a total cost of the order of 10 dollars, two pizzas would have an order cost of 17 dollars, and three pizzas would have an order cost of 24 dollars, we can plot these and draw a line through them. Here, we have a straight line which doesn’t pass through the origin. This would indicate a nonproportional linear relationship.

In fact, if we look at the point where it crosses the 𝑦-axis, we can see that this would be at the coordinate zero, three, which is the slightly bizarre situation of ordering zero pizzas and getting charged three dollars for delivery. If we have a graph of two proportional quantities, then we would have a straight-line graph which passes through the origin. As we don’t have this here, then this confirms our original answer that the cost of an order and the number of pizzas ordered are not proportional.

In the next example, we’ll take a closer look at the graph of a proportional relationship and understand the different aspects of it.

Hannah works as a baby sitter. The proportional relationship between the number of hours she works and the total amount of money she earns is shown in the graph. Which of the following statements is not true? Option A) the point 𝑄 shows that Hannah would earn 60 dollars if she worked four hours. Option B) the unit rate of this proportional relationship is 15 dollars per hour. Option C) any point of coordinates 𝑥, 𝑦 on this graph shows that Hannah would earn 𝑦 dollars if she worked 𝑥 hours. Option D) if Hannah worked for 10 hours, she would earn 150 dollars. Option E) if Hannah worked for four hours, she would earn 15 dollars.

So, here, we have the graph of a proportional relationship between the number of hours that Hannah works and the total amount of money she earns. We can confirm that it is a proportional relationship because it’s a straight-line graph and it passes through the origin. The point zero, zero would be the situation where Hannah works zero hours and gets paid zero money. So, let’s have a look at the statements and decide if they are true or false.

Let’s start with statement A, the point 𝑄 shows that Hannah would earn 60 dollars if she worked four hours. So, if we look at four on our 𝑥-axis, we can see that point 𝑄 also has the same 𝑥-coordinate. And in fact, the 𝑦-coordinate would be 60, indicating 60 dollars. So, here, we would have Hannah works for four hours and earns 60 dollars. So, this is point 𝑄 and indicates that statement A is true.

Looking at statement B, we have the term unit rate. We can recall that a unit rate is a proportion with different quantities which has a denominator of one. In our case, our proportion will be the dollars, or the money earned, over the number of hours. To find the unit rate, we need to have how many dollars for one hour. Using our graph to help us then, if we look at one hour on the 𝑥-axis, this will be the point one, 15, which is equivalent to Hannah earning 15 dollars in one hour. And so, our statement B is true. She earns 15 dollars per hour.

Let’s have a look then at statement C, any point of coordinates 𝑥, 𝑦 on this graph shows that Hannah would earn 𝑦 dollars if she worked 𝑥 hours. Well, let’s look at the coordinate zero, zero. This would be when she works zero hours and gets zero dollars. At point one, 15, she works one hour and gets paid 15 dollars. Equally, the coordinate two, 30 means she works two hours getting paid 30 dollars. And the coordinate three, 45 means she works three hours and gets paid 45 dollars. So, for any coordinate 𝑥, 𝑦, this means she works 𝑥 hours and gets paid 𝑦 dollars, which is equivalent to statement C. So, it must be true as well.

Looking at statement D, if Hannah worked for 10 hours, she would earn 150 dollars. So, let’s see if we can use the graph. If we look on our 𝑥-axis where the number of hours equals 10, we can see that the line does not go through this point. But we can use another piece of information to work out the value. And that is that Hannah earns 15 dollars per hour. We can do this by writing the statement that 15 over one is equal to what over 10.

We can write this because we know that if quantities 𝐴 and 𝐵 are in proportion, then 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two, where 𝐴 sub one and 𝐵 sub one are the quantities of 𝐴 and 𝐵 in a situation and 𝐴 sub two and 𝐵 sub two are the quantities of 𝐴 and 𝐵 in a different situation. So, returning to our calculation then, we can see that if we take the denominator in our fraction 15 over one and multiply it by 10, this would give us 10. So, we can also multiply our numerator by 10, which gives 150 over 10. So, in 10 hours, Hannah would earn 150 dollars. Therefore, our statement D is true.

Looking at our final statement E then, if Hannah worked for four hours, she would earn 15 dollars. So, if we look at four hours on our 𝑥-axis, we can see that this would be at the point four, 60 on the line. This means that in four hours, Hannah earns 60 dollars. We could also consider that when Hannah earns 15 dollars, she will have done one hour of work. Using either of these approaches, we could say that this statement that if Hannah worked for four hours, she would earn 15 dollars is definitely not true. And so, option E is the statement which is no true.

In the next question, we’re going to investigate the proportion or nonproportion in a geometrical shape and its area.

Is the length of one side of the given figure proportional to its area?

So, let’s have a look at the shape in this question. We can see that there are four right angles and two sides labeled the same length. So, we must have a square. We’re asked if the length of one side is proportional to the area. So, let’s recall how to find the area of a square. And that is that the area of a square is equal to the length times the length, or the length squared. So, the area of our square is 𝑆 squared.

Let’s recall proportion. If we have two quantities 𝐴 and 𝐵 which are proportional, then that means from one situation to another, both quantities have been multiplied by the same number. We know that in one situation, the area of our square is equal to 𝑆 squared. Let’s imagine another situation where we double the length of our sides. In this case, the area of our second square, or square two, would be equal to two 𝑆 times two 𝑆, which is four 𝑆 squared. We can note that the area of our first square, which we could call square one, was equal to 𝑆 squared. So, the area of square two is equal to four times the area of square one.

Now, let’s imagine another situation where we multiply the length of our square by three. So, in this case, the area of square three would be equal to three 𝑆 times three 𝑆, which is nine 𝑆 squared. And given that our first square was equal to 𝑆 squared, then this means that the area of square three is equal to nine times the area of square one. So, if we consider these values as fractions of the length over the area, in the first situation we have the length 𝑆 over 𝑆 squared. We then double the lengths, so the fraction would be two 𝑆 over the area of four 𝑆 squared. And in our final situation, we had three 𝑆 as the length over the area of nine 𝑆 squared.

These two quantities would be proportional if we can say that they are multiplied by the same number. However, going from the first fraction to the second fraction would mean the numerator was multiplied by two and the denominator was multiplied by four. We can also see that from the first fraction to the third fraction, we multiplied the numerator by three and the denominator by nine, which means that these have not been multiplied by the same number. So, the answer to the question, is the length of one side of this figure proportional to its area, is no.

So, to summarise the key points of this video, we have quantities 𝐴 and 𝐵 are in proportion if they are multiplied or divided by the same number. The graph of a proportional relationship is a straight line which passes through the origin. And finally, nonproportional relationships do not have quantities which are multiplied by the same number. They may have a linear relationship which shows as a straight line on a graph, but these straight lines would not pass through the origin.

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