In this video, we will learn how to write an equation to describe the proportional relationship of data in a table or graph. But first, let’s do a quick review of what it means for two quantities to be in a directly proportional relationship.
When we talk about directly proportional relationships, we’re talking about the ways two quantities relate to each other in any given situation. For example, we might see this kind of relationship in a recipe. If the recipe calls for two cups of sugar and one cup of milk, the recipe has a ratio of sugar to milk of two to one, two cups of sugar for one cup of milk. Using the ratio, we could find out how many cups of sugar we need if we were going to make two batches of cookies. We would multiply the cups of sugar by two, and we would need four cups of sugar. We would also need to multiply the milk by two to keep these ratios in proportion. One times two is two.
When we multiply or divide both of these values by the same number, the result will be directly proportional or in direct proportion to what we started with. If we wanted to write this mathematically, we’ll let 𝐴 represent the sugar and 𝐵 represent the milk. Then 𝐴 one will be the sugar for one batch and 𝐴 two will be sugar for two batches. And we’ll follow the same pattern for the milk. When values are in direct proportion, then the fraction of 𝐴 one over 𝐵 one will be equal to the fraction of 𝐴 two over 𝐵 two. And if we plug in the values we know, that means two over one is equal to four over two, which is a true statement. So we say that two proportional quantities are always in the same ratio.
Remember that when two quantities are different in nature, when we’re talking about two quantities with different units, we talk about rates instead of ratios. Let’s look at one of those examples now.
In this table, we have the price paid for a different number of milk bottles. To find the rates for each of these situations, we would want to find the price paid per bottle. When we’re dealing with rates, we read the division bar as the word “per.” This is price per bottle. If we know that we paid two dollars and 40 cents for two bottles, to find the price per bottle, we would divide the numerator and the denominator by two. When we divide the denominator by two, we’re going from two bottles to one bottle. And when we divide the numerator by two, we’re going from the price for two bottles to the price for one bottle, which is one dollar and 20 cents.
In order for us to determine if these are directly proportional values, we’ll need to check that the price per bottle of buying three bottles and four bottles is the same as buying two bottles. If you pay three dollars and 60 cents for three bottles, we’ll find the unit rate by dividing the numerator and the denominator by three. Three dollars and 60 cents divided by three is one dollar and 20 cents. So far, purchasing two bottles and three bottles have the same rate. So we’ll check for four by dividing numerator and denominator by four. And we see it’s one dollar 20 per bottle. So we can say that yes, there is a proportional price here. And the constant of proportionality is going to be the unit rate.
If we want to know 𝑦, the price of bottles, then we take the constant of proportionality and we multiply it by 𝑥, the number of bottles we’re buying. The price 𝑦 is equal to one dollar and 20 cents times 𝑥, the number of bottles you’re buying. And this, 𝑦 equals 1.20 times 𝑥, is a proportional equation. Putting all this together, we get the general form for proportional equations, 𝑦 equals 𝑘𝑥, where 𝑘 is the constant of proportionality or sometimes called the constant of variation. We might describe this situation as 𝑥 and 𝑦 are directly proportional or 𝑦 varies directly with 𝑥. Or you might see a symbol like this, which says that 𝑦 is directly proportional to 𝑥.
Now, let’s look at some examples using this information.
In a pasta recipe that serves four people, it says to use 440 grams of spaghetti. How much spaghetti should be used for two people? How much should be used for 𝑥 people? Write an equation in the form 𝑠 equals 𝑚𝑥 for the quantity of spaghetti 𝑠 needed for 𝑥 people.
Let’s start with what we know. The recipe says 440 grams will feed four people. Our first question is asking, how much spaghetti should feed two people? We know that two people is half of four people. Four divided by two equals two. And that means if we want to feed half the amount of people, we need half the amount of spaghetti. We would need 440 divided by two, 220 grams, to feed two people. That’s the first part of our question. But we can’t use that same process to find out how much spaghetti you need for 𝑥 people.
In order to find the amount of spaghetti for 𝑥 people, we’ll need the unit rate. We’ll need to know how much spaghetti each person needs. If we know how much two people need, we can divide two by two, which gives us one person. And then we divide the 220 grams that two people need by two, which is 110 grams. Our unit rate is then 110 grams per person. And that means the number of grams for 𝑥 people will be 110 times 𝑥 grams. Or you might write it like this, 110𝑥.
The last part of our questions says, “Write an equation in the form 𝑠 equals 𝑚𝑥 for the quantity of spaghetti 𝑠 needed for 𝑥 people.” The form we’re using is 𝑠 equals 𝑚 times 𝑥. In this case, the 𝑚 represents the constant of variation, which in this case will be the unit rate. We’ll plug in the 110 grams that the unit rate is so that we have 𝑠, the amount of spaghetti needed, will be equal to 110 times 𝑥, the number of people eating.
In this example, we start with a proportional equation and we’re asked to identify the unit rate.
The amount of meat required to feed a captive lion is given by the equation 𝑤 equals nine 𝑑, where 𝑤 is the weight of the meat in kilograms needed to feed a lion for 𝑑 days. What is the unit rate of this proportional relationship?
We’re given the proportional relationship 𝑤 equals nine 𝑑. When we think about the general form of a proportional relationship, 𝑦 equals 𝑘 times 𝑥, 𝑘 is called the constant of variation. And when we’re dealing with problems that have rates, the 𝑘-value is the unit rate. Now, almost immediately, we recognize that this constant is nine. But we now need to think carefully about what kind of units that this nine will be in. If we remember that our unit rate is how much of something per one unit of something else. In this case, the unit rate is how many kilograms a lion eats in one day, some kilograms per day.
To find that out, we can plug in one for the number of days 𝑑. Which tells us that, in one day, the lion eats nine kilograms of meat and makes the unit rate nine kilograms per day.
Now, let’s look at a case where we’re given a table of data to process.
The table shows how many pages of a book Daniel has read at different times. Is Daniel reading at a constant speed? Why? What is the constant of proportionality, the unit rate? And what does it represent? Write an equation in the form 𝑝 equals 𝑚 times 𝑡 for the number of pages read 𝑝 in 𝑡 minutes.
Let’s start at the beginning. Is Daniel reading at a constant speed? If Daniel is reading at a constant speed, then the number of pages he reads is proportional to the reading time. Which would mean that there is some constant you can multiply the reading time by to get the number of pages that Daniel reads. To find out if this is true, we need to check each column and find out if they’re all in the same ratio.
The first ratio is nine pages in 12 minutes. We could simplify this ratio by dividing the numerator and the denominator by three, which tells us that Daniel reads three pages every four minutes. In order for the ratio to be constant, the next column should also be in a ratio of three pages every four minutes. It is 21 pages every 28 minutes. And if we divide the numerator and the denominator by seven, we see that this ratio is, again, three pages in four minutes.
To confirm this is true, we’ll need to check the other three columns though. The third column is 27 pages in 36 minutes. If we notice that three times nine equals 27 and four times nine equals 36, we confirm that the third column is in this ratio. The fourth column is three times 12, which is 36 pages, and four times 12, which is 48 minutes. And the fifth column is equal to three pages times 15 over four minutes times 15, 45 pages in 60 minutes.
The answer to the first part of the question, “Is Daniel reading at a constant speed?”, is yes. Because the number of pages read is proportional to the reading time.
Now, we want to know what the constant of proportionality, what the unit rate is and what it represents. Daniel can read three pages in four minutes. If we want to write this as an amount that he can read per minute, four divided by four is one. But remember that if we divide the denominator by four, we need to divide the numerator by four. And so, we need the numerator to be three divided by four. Well, we could write it as a fraction of three-fourths a page per minute or as a decimal, 0.75 of a page per minute. Daniel reads three-fourths of a page per minute. We would say then that the unit rate is 0.75 parts of a page per minute. This value represents Daniel’s reading speed.
And finally, we want to write an equation in the form 𝑝 equals 𝑚 times 𝑡, where 𝑝 is the number of pages read in 𝑡 minutes. We recognize that this is a similar form to 𝑦 equals 𝑘𝑥. It’s the form of a proportional equation. And in this general form, the constant is our unit rate. In this case, the unit rate is 0.75. Since Daniel can read three-fourths of a page per minute, if you plug in the number of minutes he reads for 𝑡, you’ll find out 𝑝, the number of pages he’s read.
In this example, we want to use a unit rate and identify it on a graph.
Michael sold three dozen eggs for five dollars and 28 cents. Which of the following points in the graph represents the rate at which Michael sold the eggs?
Let’s think about what we know. For five dollars and 28 cents, Michael sold three dozen eggs. On our graph, the 𝑥-axis represents the number of dozens of eggs and the 𝑦-axis represents the cost. If we know that he sold three dozen for five dollars and 28 cents, let’s try to plot this point on the graph. Point 𝐴 is located at three along the 𝑥-axis, but it’s located at 15.84 on the 𝑦-axis. This point is saying three dozen eggs would cost 15 dollars and 84 cents. And that means we can eliminate 𝐴.
Notice that point 𝐵 is located at about eight on the 𝑦-axis, 7.92. We know that when there are three dozen eggs, there should be a cost of five dollars and 28 cents, which should just be a little bit below eight. At this point, we’re thinking that this second line is the line we’re interested in. But how can we check this mathematically? We can do this by finding a unit rate. The unit rate would be the cost per dozen. Three divided by three is one. And so, when we divide five dollars and 28 cents by three, we find the price per dozen. 5.28 divided by three is 1.76. The cost per dozen is one dollar and 76 cents.
If we look at point 𝐵, it’s located at 4.5 along the 𝑥-axis. That represents four and a half dozen. If we want to check this coordinate, we can calculate the cost of four and a half dozen using the unit rate. If one dozen costs a dollar 76, we multiply the numerator and the denominator by four and a half. 1.76 times 4.5 is 7.92. And so, what we’re saying is, at the rate that Michael sold three dozen eggs for 5.28, you could buy four and a half dozen for seven dollars and 92 cents. Point 𝐵 represents the rate Michael sold the eggs for.
Let’s review what we’ve learned about proportional equations. Two quantities 𝑦 and 𝑥 are directly proportional when the ratio of 𝑦 to 𝑥 is constant. A directly proportional relationship is described mathematically with an equation in the form 𝑦 equals 𝑘𝑥, where 𝑘 is the constant of proportionality, or unit rate, of the relationship. The constant of proportionality has a compound unit, the unit of 𝑦 per unit of 𝑥, for example miles per gallon or pages per minute.