Explainer: Proportional Equations

In this explainer, we will learn how to write an equation to describe the proportional relationship of data in a table or a graph.

Let us recall what it means when two quantities are in a directly proportional relationship.

Directly Proportional Relationship

Two quantities 𝐴 and 𝐡 are directly proportional or in direct proportion 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 𝐡 are in all situations equivalent.

Mathematically, this means that if, in a first situation, quantity 𝐴 is 𝐴 and quantity 𝐡 is 𝐡 and, in another situation, quantity 𝐴 is 𝐴 and quantity 𝐡 is 𝐡, then 𝐴∢𝐡=𝐴∢𝐡 and 𝐴𝐡=𝐴𝐡 when 𝐴 is directly proportional to 𝐡.

So, two proportional quantities are always in the same ratio. Remember that when the quantities are different in nature (expressed with a different unit), we talk about rates.

Let us take an example of the price paid for different numbers of milk bottles, shown in the table below.

Number of Bottles234β‹―π‘₯
Price ($)2.403.604.80⋯𝑦

We can check that the ratio of the price to the number of bottles is the same for the three pairs of values given, namely, that $2.402=$3.603=$4.804=$1.20/.bottlesbottlesbottlesbottle

Hence, we can say that the price is proportional to the number of bottles, and the constant of proportionality is $1.20 per bottle. The constant of proportionality (or unit rate) here is the unit price. We have seen that the ratio of the two quantities in each pair is equal to the constant of proportionality, here the unit price. It means that for any number π‘₯ of bottles (greater than zero), the price 𝑦 is such that 𝑦π‘₯=1.20.

Multiplying both sides of this equation by π‘₯, we find that it is equivalent to 𝑦=1.20π‘₯.

This equation is just saying that the price of a given number of bottles is given by multiplying the unit price by this number of bottles. This relationship is often indicated on a proportionality table or a double-line diagram.

Remember that the constant of proportionality or unit rate of the proportional relationship between quantities 𝐴 and 𝐡 is the value of quantity 𝐴 when quantity 𝐡 is equal to 1. In other words, it is the result of carrying out the division of quantity 𝐴 by quantity 𝐡 for any pair of values we know. It is expressed with the compound unit β€œunit of quantity 𝐴 per unit of quantity 𝐡,” where β€œper” means β€œfor each.”

Looking at the units is very helpful to make sure we have set up the equation correctly. Indeed, as we can see with our example, the units are used consistently: dollarsdollarsnumberofbottlesunitratenumberofbottles(𝑦)=()Γ—(π‘₯).

On the right-hand side of the equation, we have a division of the unit β€œnumber of bottles” by β€œnumber of bottles,” which leads to a number without any unit (the units β€œcancel out”). In terms of unit, we then get dollarsdollars=.

This is consistent.

Let us summarize what we have just found out.

How to Describe a Proportional Relationship with an Equation

A directly proportional relationship between two quantities π‘₯ and 𝑦 is described mathematically with an equation of the form 𝑦=π‘˜π‘₯, where π‘˜ is the constant of proportionality, or unit rate, of this relationship. The unit for π‘˜ is a so-called compound unit, written β€œunit for 𝑦/unit for π‘₯” (the slash is read β€œper,” which means β€œfor each”).

Example 1: Writing an Equation to Describe a Proportional Relationship

In a pasta recipe that serves 4 people, it says to use 440 g of spaghetti.

  • How much spaghetti should be used for 2 people? How much should be used for π‘₯ people?
  • Write an equation in the form 𝑠=π‘šπ‘₯ for the quantity of spaghetti 𝑠 needed for π‘₯ people.

Answer

If 440 g of spaghetti serves 4 people, then we will need half as much to serve 2 people, that is, 220 g.

By doing this, we have assumed that every person eats the same quantity of spaghetti or that we can use an average value of the quantity of spaghetti needed per person.

To find out the quantity needed for π‘₯ people, we need to find the value needed for one person and then multiply this value by the number of people π‘₯. We have the value for 4 people, 440 g, so the value for one person is 440 g divided by 4: 4404=110/.gperson

The quantity that should be used for π‘₯ people is then 110π‘₯.g

If we call 𝑠 the quantity of spaghetti needed for π‘₯ people, we have 𝑠=110π‘₯.

We recognize that the 110 in the equation is the unit rate: it is 110 g/person. When it is multiplied by a number of people, we get a quantity in g.

Example 2: Identifying a Unit Rate from an Equation

The amount of meat required to feed a captive lion is given by the equation 𝑀=9𝑑, where 𝑀 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?

Answer

We are given the equation 𝑀=9𝑑, which describes the proportional relationship between the weight of meet needed to feed a lion (𝑀) and the number of days𝑑 he can then be fed with this amount of meat. Recall that the unit rate is the multiplying constant π‘˜ in the general equation 𝑦=π‘˜π‘₯. By comparing with 𝑀=9𝑑, we see that the constant is 9. Now, we need to think about the unit of π‘˜. If we look at our equation in terms of units, it gives kgunitfornumberofdays(𝑀)=π‘˜(π‘˜)Γ—(𝑑).

By dividing both sides of the β€œunit equation” by the unit β€œnumber of days,” we find that unitforkgnumberofdaysπ‘˜(π‘˜)=(𝑀)(𝑑).

We see that π‘˜ is expressed as kilogram per day. Hence, the unit rate of this proportional relationship is 9 kg/day.

Example 3: Identifying a Unit Rate from an Equation

The quantity of paint π‘ž needed to cover a certain wall area π‘Ž is given by π‘ž=0.1π‘Ž, where π‘ž is expressed in liters and π‘Ž in square meters. The quantity of paint is thus proportional to the wall area. What is the unit rate of this proportional relationship?

Answer

Recall that the unit rate is the multiplying constant π‘˜ in the general equation 𝑦=π‘˜π‘₯, which describes the proportional relationship between 𝑦 and π‘₯, and is expressed in β€œunit for 𝑦/unit for π‘₯.” Here, we have π‘ž=0.1π‘Ž.

Hence, we see that the unit rate is 0.1; however, we need to find its unit. For this, we need to look at the units of π‘ž and π‘Ž. The quantity of paint π‘ž is expressed in liters and the area π‘Ž in square meters. Therefore, the unit of π‘˜ is liters per square meter.

Our answer is that the unit rate is 0.1 L/m2.

Example 4: Writing an Equation to Describe a Proportional Relationship from a Table

The table shows how many pages of a book Daniel has read at different times.

Time (Minutes) 12 28 36 48 60
Number of Pages 9 21 27 36 45
  • Is Daniel reading at a constant speed? Why?
    1. No, because the number of pages read is not proportional to the reading time.
    2. Yes, because the number of pages read is proportional to the reading time.
  • What is the constant of proportionality (or unit rate)? What does it represent?
    1. 9 pages, the first number of pages
    2. 45 pages, the last number of pages
    3. 60 pages per minute, the total reading time
    4. 0.75 pages per minute, the time needed to read a page
    5. 0.75 pages per minute, the reading speed
  • Write an equation in the form 𝑝=π‘šπ‘‘ for the number of pages read 𝑝 in 𝑑 minutes.
    1. 𝑝=43𝑑
    2. 𝑝=0.75+𝑑
    3. 𝑝=43+𝑑
    4. 𝑝=9+𝑑
    5. 𝑝=0.75𝑑

Answer

If Daniel is reading at a constant speed, then the number of pages he reads is proportional to the reading time. This means that this number of pages is then given by the time multiplied by a certain constant, which is called constant of proportionality or unit rate.

To decide using the values given in the table whether Daniel is reading at a constant speed, we need to check if the values given in each column are in the same ratio. The first ratio is 9 pages in 12 minutes. This can be reduced to 3 pages read every 4 minutes. The second ratio is 21 pages read in 28 minutes. As 21=3Γ—7, we need to check if 28=4Γ—7. This is true, so 9∢12=21∢28 or 912=2128.

We can check in the same way that all the pairs of values given in the table are in the same ratio.

12=4Γ—328=4Γ—736=4Γ—948=4Γ—1260=4Γ—15
9=3Γ—321=3Γ—727=3Γ—936=3Γ—1245=3Γ—15

The ratio of pages read to time in minutes is constant; we can say that the number of pages read is proportional to the reading time. Therefore, the reading speed is constant since it is the proportionality constant or unit rate of this relationship.

So, the answer is β€œyes,” because the number of pages read is proportional to the reading time.

We have just seen that the ratio of pages read to time in minutes is 3 pages for 4 minutes. It can be written as 34pagesminutes. This ratio is the unit rate of this proportional relationship. It is the constant that is multiplied by the time in minutes to give the numberofpages read. We can express it as a decimal by dividing 3 by 4.

We find it is 0.75 pages/minute.

Observe that the unit is consistent: we have divided 3 pages by 4minutes, so we get 0.75 (which is 3 divided by 4) pages divided by minute, which is written pages/minute (and is read β€œpages per minute”).

The reading speed is the unit rate, it represents the number of pages read in a time unit, here a minute.

We see that, for any time 𝑑 expressed in minutes, the number of pages read 𝑝 is given by multiplying this time by the unit rate, 0.75 pages/minute. The corresponding equation is 𝑝=0.75𝑑.

Example 5: Writing an Equation to Describe a Proportional Relationship

Scarlett paints one garden chair in 12minutes. Write an equation for the number of chairs 𝑐 that she could paint in β„Ž hours.

Answer

If Scarlett works at a constant rate, then the number of chairs 𝑐 she can paint is proportional to the time β„Ž she spends painting. To write the corresponding equation, we need to find the unit rate of this proportional relationship, that is, the number of chairs she paints in one hour. We know that she paints one chair in 12 minutes. As an hour is made up of 60 minutes, 12 minutes is 1260 of an hour.

Using a double-line diagram, we visualize that the number of chairs that Scarlett paints in one hour is given by multiplying 1 (the number of chairs she paints in 12 minutes) by 6012.

And we find that 1Γ·6012=5 (as 12 minutes is indeed one-fifth of an hour). The unit rate is therefore 5 chairs per hour and is nothing else than the ratio of the number of chairs to be painted to the time needed to paint them. Using the pair of values we are given, we find it is indeed 1Γ·1260=1Γ—6012=5/.chairsh

Now that we have found the unit rate, it is easy to write an equation for the number of chairs 𝑐 she can paint in β„Ž hours since it is given by multiplying any time expressed in hours by the unit rate expressed in β€œnumber of chairs per hour.” Hence, we have 𝑐=5β„Ž.

Note that a proportional relationship can be taken in the reverse way, that is, by swapping the two quantities. In our last example, this would be working out the time needed to paint a given number of chairs. The unit rate would then be the time needed to paint one chair. We know it is 12 minutes, which, expressed in hours, is 15 of an hour. Hence, the unit rate is 15 of an hour per chair. And the corresponding equation linking β„Ž and 𝑐 is β„Ž=15𝑐.

Both equations are actually one and the same equation. One gets β„Ž=15𝑐 from 𝑐=5β„Ž simply by dividing by the unit rate 5 chairs/hour.

Let us have a quick look at the units in both equations here. In the first one, we have numberofchairsnumberofchairshourshours(𝑐)=Γ—(β„Ž) and in the second one hourshoursnumberofchairsnumberofchairs(β„Ž)=Γ—(𝑐).

Key Points

  • Two quantities 𝑦 and π‘₯ are directly proportional or in direct proportion when, from one situation to another, both quantities have been multiplied (or divided) by the same number.
  • This means that the ratio of 𝑦 to π‘₯ is constant: 𝑦π‘₯=π‘˜, where π‘˜ is a constant called the constant of proportionality, or unit rate, of the proportional relationship between 𝑦 and π‘₯
  • A directly proportional relationship between two quantities π‘₯ and 𝑦 is described mathematically with an equation of the form 𝑦=π‘˜π‘₯, where π‘˜ is the constant of proportionality, or unit rate, of this relationship.
  • The constant of proportionality has a compound unit: unit of 𝑦 per unit of π‘₯, for instance, kilometers per hour, dollars per kilogram, minutes per page, and so on.

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