### Video Transcript

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, letโs 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, weโre 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, weโll let ๐ด represent the sugar and ๐ต represent the milk. Then ๐ด one will be the sugar for one batch and ๐ด two will be sugar for two batches. And weโll follow the same pattern for the milk. When values are in direct proportion, then the fraction of ๐ด one over ๐ต one will be equal to the fraction of ๐ด two over ๐ต 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 weโre talking about two quantities with different units, we talk about rates instead of ratios. Letโs 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 weโre dealing with rates, we read the division bar as the word โper.โ 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, weโre going from two bottles to one bottle. And when we divide the numerator by two, weโre 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, weโll 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, weโll 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 weโll check for four by dividing numerator and denominator by four. And we see itโs 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 ๐ฆ, the price of bottles, then we take the constant of proportionality and we multiply it by ๐ฅ, the number of bottles weโre buying. The price ๐ฆ is equal to one dollar and 20 cents times ๐ฅ, the number of bottles youโre buying. And this, ๐ฆ equals 1.20 times ๐ฅ, is a proportional equation. Putting all this together, we get the general form for proportional equations, ๐ฆ equals ๐๐ฅ, where ๐ is the constant of proportionality or sometimes called the constant of variation. We might describe this situation as ๐ฅ and ๐ฆ are directly proportional or ๐ฆ varies directly with ๐ฅ. Or you might see a symbol like this, which says that ๐ฆ is directly proportional to ๐ฅ.

Now, letโs 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 ๐ฅ people? Write an equation in the form ๐ equals ๐๐ฅ for the quantity of spaghetti ๐ needed for ๐ฅ people.

Letโs 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. Thatโs the first part of our question. But we canโt use that same process to find out how much spaghetti you need for ๐ฅ people.

In order to find the amount of spaghetti for ๐ฅ people, weโll need the unit rate. Weโll 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 ๐ฅ people will be 110 times ๐ฅ grams. Or you might write it like this, 110๐ฅ.

The last part of our questions says, โWrite an equation in the form ๐ equals ๐๐ฅ for the quantity of spaghetti ๐ needed for ๐ฅ people.โ The form weโre using is ๐ equals ๐ times ๐ฅ. In this case, the ๐ represents the constant of variation, which in this case will be the unit rate. Weโll plug in the 110 grams that the unit rate is so that we have ๐ , the amount of spaghetti needed, will be equal to 110 times ๐ฅ, the number of people eating.

In this example, we start with a proportional equation and weโre asked to identify the unit rate.

The amount of meat required to feed a captive lion is given by the equation ๐ค equals nine ๐, where ๐ค is the weight of the meat in kilograms needed to feed a lion for ๐ days. What is the unit rate of this proportional relationship?

Weโre given the proportional relationship ๐ค equals nine ๐. When we think about the general form of a proportional relationship, ๐ฆ equals ๐ times ๐ฅ, ๐ is called the constant of variation. And when weโre dealing with problems that have rates, the ๐-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 ๐. Which tells us that, in one day, the lion eats nine kilograms of meat and makes the unit rate nine kilograms per day.

Now, letโs look at a case where weโre 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 ๐ equals ๐ times ๐ก for the number of pages read ๐ in ๐ก minutes.

Letโs 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 theyโre 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, weโll 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, โIs Daniel reading at a constant speed?โ, 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 Danielโs reading speed.

And finally, we want to write an equation in the form ๐ equals ๐ times ๐ก, where ๐ is the number of pages read in ๐ก minutes. We recognize that this is a similar form to ๐ฆ equals ๐๐ฅ. Itโs 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 ๐ก, youโll find out ๐, the number of pages heโs 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?

Letโs think about what we know. For five dollars and 28 cents, Michael sold three dozen eggs. On our graph, the ๐ฅ-axis represents the number of dozens of eggs and the ๐ฆ-axis represents the cost. If we know that he sold three dozen for five dollars and 28 cents, letโs try to plot this point on the graph. Point ๐ด is located at three along the ๐ฅ-axis, but itโs located at 15.84 on the ๐ฆ-axis. This point is saying three dozen eggs would cost 15 dollars and 84 cents. And that means we can eliminate ๐ด.

Notice that point ๐ต is located at about eight on the ๐ฆ-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, weโre thinking that this second line is the line weโre 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 ๐ต, itโs located at 4.5 along the ๐ฅ-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 weโre 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 ๐ต represents the rate Michael sold the eggs for.

Letโs review what weโve learned about proportional equations. Two quantities ๐ฆ and ๐ฅ are directly proportional when the ratio of ๐ฆ to ๐ฅ is constant. A directly proportional relationship is described mathematically with an equation in the form ๐ฆ equals ๐๐ฅ, where ๐ is the constant of proportionality, or unit rate, of the relationship. The constant of proportionality has a compound unit, the unit of ๐ฆ per unit of ๐ฅ, for example miles per gallon or pages per minute.