1.4 Ratios and Proportionality
In this chapter we are going to learn abut ratios and proportionality. Before proceeding it would be better if you know how to deal with fractions ( addition, subtraction, multiplication and division). If you want a refresher click on the link on the left side of the page to check out the Integers, Fractions, Decimals chapter.
Let's first try to understand how to use ratios.
Ratios are used to relate one quantity to another. This is particularly important in construction. Before building a house or skyscraper, a design or model of it has to be created. Creating these in an actual size is impossible and impractical; but the drawing or models have to be perfect so than builders can follow the plan and build a safe building. By using ratios, engineers can scale down a huge building to a size that can be put on paper. They can then say accurately that the dimensions drawn on paper are 10 times or 100 times or 600 times... smaller than the actual size. In other they relate the actual or real size to a size that can drawn on paper.
Example 1: Let's assume that we have a drawing of a building that is 15 cm tall on paper. The engineer says that 1 cm on paper represents 1.5 m in real a setting. This ratio can be written as 1cm : 1.5 m or 1 cm : 150 cm. That means the building's actual height is 15 x 1.5 = 22.5 m. For this ratio you must include the proper units or it would not make sense.
Ratios are used in so many different ways and this was just one example.
The colon must be used when expressing a ratio. Let's assume that in your school, for every 100 students there are 10 teachers. Therefore, the ratio of students to teachers is 100 : 10 whereas the ratio of teachers to students is 10 : 100. Take extra care about which number goes on which side of the colon.
If we remember from the fractions section, a fraction is best expressed in its simplified state. The same goes for ratios. When simplifying ratios, if the divide one side of the colon by a number you have to do the samething to the other side. In this case we can divide by 10 to get the simplified ratio. 1 : 10 or 10 : 1
Example 2: Let's look at a slightly more complex ratio. Assume you buy a powdered orange drink. On the packet, the instructions for making it is: mix orange powder, salt and sugar in the following ratio 12 : 2 : 6.
Before trying to make sense of this let's simplify it. All three numbers are divisible by 2 so let's divide by 2.
You get the simplified ratio 6 : 1 : 3. Looking at the instructions again we see that powder comes first followed by salt and sugar. So 6 is for powder, 1 is for salt and 3 is for sugar. Ok so now what? What the ratio means is that for every 6 measurements of powder you add you need to add 1 measurement of salt and 3 measurements of sugar using the same units. If you are using a teaspoon then that means 6 teaspoons powder, 1 teaspoon salt and 3 teaspoons sugar. If you are using cups then 6 cups of powder, 1 cup of salt and 3 cups of sugar.
As you can see, understanding what units are being used is very very important with ratios.
Converting Ratios to Fractions
The cool thing about ratios is that just like decimals they can be converted to fractions. Why would anyone want to convert a ratio to a fraction? There a several reasons why a fraction might be more useful than a ratio.It is particularly useful when you are calculating an unknown value. If we look back at our first example with the ratio 1 cm : 150 cm. Say you want to draw the width of the building, which is about 50 m in real life, on paper. The drawing width is an unknown that needs to be calculated. You are going to have to convert the ratio to a fraction first. In this case converting to a fraction is straight forward because the ratio has only 2 numbers and it is a scale.
You simply set the left hand side value as the numerator and right hind side as the denominator. We also know that the width of the drawn building over the actual width must be equal to 1/150:
We used direct proportion to solve this problem. It is explained in more detail below. It is very important that all units are the same when doing calculations so convert 50 m to cm. You can now do simple algebra to find out what length the actual width should be on paper. If you want a refresher on calculating unknowns, check out the Algebra section.
It is very important to note that sometimes converting a ratio to fraction will give you different answers depending on what values are being compared and whether the values in the ratio are in direct proportion. This is confusing to understand so let's look at an example.
Example 3: Let's assume you and your friend won $184 in a competition. Since you did most of the work, you guys decide to share it in the ratio 5 : 3. So how do we calculate this?
First we obviously convert the ratio to a fraction. In this case, we can't just put 5 in the numerator and 3 in the denominator and say the fraction is 5/3. This is wrong. In this ratio we are not saying that every 5 cm is 3 m or that for every 5 boys there are 3 girls. This is not direct proportion (more details on direct proportion below). Instead the ratio states that the $184 must be divided such that you get 5 parts of it and your friend gets 3 parts. This means that 184 has to be divided to a total of 8 parts (5 parts + 3 parts). Therefore, you get 5/8 of the prize and your friend gets 3/8 of the prize. We got 2 fraction from one ratio in this case.
You can double check to make sure that your fractions are correct by seeing if they add up to 1. After that you multiply 5/8 by $184 to calculate how much you earn and multiply 3/8 by $184 to calculate how much your friend should get. You can double check again by seeing if what you get + what your friend gets = $184.
Let's look at one more example:
Example 4: Assume you are to paint your room wall a new color. So you buy one liter of concentrated paint that is to be diluted when painting the wall. According to the instructions on the container each wall must be applied with multiple coats of paint. This means that you going to have take a specific amount of paint from the concentrated solution and dilute it in 1 liter of water for each coat. It also says for best results, one liter of their paint makes for 4 coats if mixed and applied in the following ratio 4 : 5 : 6 : 5. How are we going to calculate this?
There are four numbers in this ratios which means we can make 4 different fractions. If we add the number we see that there 20 parts:
Always remember to simplify the fractions because it makes further calculation easier. Now that we know the fractions, let's calculate the amount of concentrated painted we take for each coat:
Therefore, for the 1st coat you take 200 ml of concentrated painted and mix it in 1 liter of water. After this dries you apply the 2nd coat by using 250 ml of concentrated paint and mixing it in 1 liter of water. To make sure your answers are correct you can double check by either adding the fractions together to see if it equals 1 or by adding the calculated amounts and checking if they equal 1 liter.
There two types of proportionalities we will discuss in this section. Direct proportionality and inverse proportionality.
We discussed briefly about direct proportionality in the ratios section in Example 1. Let's discuss it in a bit more detail here.
Proportionality is a property used to describe two quantities that are related in a way where their ratio is constant.
Now this is confusing to understand just by reading the definition. It basically just means that if you change one of the quantities by a certain factor then the other quantity also changes by that same factor. So let's look at Example 1 again.
The ratio is 1 cm : 1.5 m. The building height is 15 cm on paper. To find the actual size, h, of the building we are going to have to say the 15 cm and h are directly proportional. This means that the ratio of 15 cm : h m must be equal to the constant ratio 1 cm : 1.5 m.
Therefore, by using the process of direct proportionality we were able to calculate the actual size of the building to be 22.5 m.
Let's look at another example:
A car travels at a fixed speed for 15 minutes. It traveled a total distance of 8 km. How long will it take to travel 12 km at the same speed?
Since the car is travelling at the same speed we can use direct proportionality here. Which means we can say that the ratio of distance : time is constant or that 8 : 15 is the constant ratio. The unknown ratio is 12 : x.
Since we know the ratios are directly proportional we equate them together and calculate what x is:
It will take 22.5 minutes to travel 12 km. It is important to note that if these quantities were not directly proportional we could not have equated them together to find x.
Let's look at one last example:
A recipe for 6 cupcakes requires you to use 100 g of sugar. Since your friends are coming over you decide to make 10 cupcakes. How much sugar do you need?
The two ratios you need to use to solve this are 6 cupcakes : 100 g and 10 cupcakes : x g.
By using direct proportion we equate the two ratios and assume that the unknown ratio = the constant ratio(6:10).
Always remember to simplify your fractions.
Inverse proportionality states that the product (means multiplication) of two quantities is constant unlike in direct proportion where ratios are constant. So if we have two quantities a and b then the product is a x b. Let's look at an example to understand this.
Every morning you ride your bicycle to school from your house. Let's say you ride at a constant speed of 280 m/min and it takes you 15 min to reach school. You don't like to be late and you always prepare for the worst case scenario. You want to know the safest speed at which you need to ride to get to school in 10 minutes in case you had to leave home 5 minutes late.
To solve this we need to use inverse proportion. Why? This is because we know that when we increase our speed then the time it takes to travel a distance will decrease. So, increase speed means decrease time. If we were using direct proportion then it would have been increase speed increase time or decrease speed decrease time which is not correct.
Since this in in inverse proportionality, the product of speed and time must be a constant:
The constant is 4200. This means to get to school in 10 minutes, the new speed x 10 minutes must be equal to the constant:
You would have to ride at 420 m/min or 25.2 km/h to reach school in 10 minutes.
Assuming the two quantities are a and b:
For Direct Proportionality : Ratio is constant --- a : b = constant
Inverse Proportionality : Product is constant --- a x b = constant