Ratios and Proportions
A ratio is used to compare two quantities. Ratios are typically written in one of three ways – as fractions, as numbers separated by a colon, or as words – 2/3, 2 : 3, or 2 to 3.
While we may not realize it, we use ratios every day. For example, when we talk about a vehicle’s rate of travel being 45 mph, we are talking about the ratio of distance traveled to time traveled.
Just as a fraction represents a part of something out of a whole (written as: a part/the whole ),
a ratio can be expressed as either: a part/the whole OR a part/another part
Because ratios compare values, they can either compare individual pieces to one another or an individual piece to the whole.
e.g. If John has 5 white socks and 3 black socks,
- the ratio of white socks to black socks is 5 to 3
- the ratio of white socks to all socks is 5 to 8 (Why 8? Because there are 5 white and 3 black socks, so together they make 5+3 = 8 socks total.)
There are three types of problems involving ratios:
- Reducing ratios – Just as fractions can be reduced, so too can ratios by dividing the numerator and denominator by the same amount.
Julia has a fruit basket. 25 of them are apples and 15 of them are oranges. What is the ratio of apples to oranges in her basket?
Right now, the ratio is 25:15. But they have the greatest common divisor of 5, so this ratio can be reduced.
Now, 25/5 = 5 and 15/5 = 3
Simply put, 25 : 15 = 25/15 = 5/3 . So the fruits have a ratio of 5:3.
If a ratio is given to you in any form other than a fraction, set it up as a fraction immediately, the first term going on top, the second on the bottom.
2. Increasing Ratios – Just like you can reduce ratios, you can also do the opposite and increase them. In order to do so, you must multiply each piece of the ratio by the same amount.
Suppose, in an ice cream the ration of sugar to cream is 2:3
And so on.
3. Finding the whole – If you are given a ratio comparing two parts (piece : another piece), and you are told to find the whole amount, simply add all the pieces together.
It may help you to think of this like an algebra problem wherein each side of the ratio is a certain multiple of x. Because each side of the ratio must always be divided or multiplied by the same amount to keep the ratio consistent, we can think of each side as having the same variable attached to it.
Let us look at an example problem to understand this more clearly:
Mike has a basket of eggs. There are two different kinds of eggs in the basket–white and brown. The brown eggs are in a ratio of 2:3 to the white eggs. What is NOT a possible number of eggs that Mike can have in the basket?
Solution – In order to find out how many eggs he has total, we must add the two pieces together.
Brown eggs : White eggs = 2 : 3
This means that we can represent 2 : 3 as
2X : 3X
Why? Because each side must change at the same rate. And in this case, our rate is x.
So if you were asked to find the total amount, you would add the pieces together.
2X + 3X = 5X
The total amount is 5X. Therefore, the total MUST be divisible by 5.
This means that the total number of eggs in the basket has to either be 5 or any multiple of 5. Why? Because 2:3 is the most reduced form of the ratio of eggs in the basket. This means he could have:
And so forth. We don’t know exactly how many eggs he has, but we know that it must be a multiple of 5.
This means our answer is C (12). There is no possible way that he can have 12 eggs in the basket.
Note that ratios can be used to compare more than two quantities as well.
A proportion is a mathematical statement equating two ratios. Two ratios are said to be equal if, when written in fractional form, the fractions are equivalent fractions. You can compare and solve ratios using cross multiplications. If the cross products are equal, the ratios are equal.
When solving a proportion question, you’ll want to rewrite the proportion in fractional format. Let’s take a look at an example:
Note, in the above example, you could have also solved this problem by observation if you recognized that 8/16 is simply 1/2. One-half of 4 is 2. Observations such as this will provide significant time savings on the exam, but on more difficult problems, it’s important to know the basics.
Relationships (or Variations) deal with explaining, in mathematical language, how one quantity changes with respect to one or more other quantities.
There are two primary types of variation: direct variation and inverse relationship.
Two variables are said to be directly proportional if they vary in such a way that one of the variables is a constant multiple of the other, or equivalently if they have a constant ratio.
For example two variables x and y are directly proportional if y = kx, where k is a constant.
In real life we come across multiple examples of direct proportionality:
- A vehicle moving at a constant speed: Distance traveled is proportional to the time
- Distance on the map: Distance between two points on the map is proportional to the actual distance between them
- Shadow on the ground: Length of the shadow is proportional to the height of the tower
- Cake Recipe: Ingredients in the recipe are in proportion to the weight of the cake
Let us look at an example problem:
If it takes 3 buckets of paint to cover 100 square feet, how many buckets of paint will be needed to cover 500 square feet?
Problems involving direct variation can be solved using proportions. Let us use the information provided in the problem to set up our proportion:
When we say “y varies directly as x,” we could also write: . In the above equation, k is called the constant of variation.
In our paint example, the number of buckets of paint varies directly with the square footage that will be covered. The constant of variation is 3/100. You would then have:
When two variables or quantities change in opposite directions, you have inverse variation. Let’s take a look at an example:
The time it takes to paint a house varies with the number of people doing the work.
In this example, the time required to paint the house varies inversely with the number of people painting. This means the more people painting the house, the less total time it will take to paint. When we say “y varies inversely with x,” we can express this as:
Once again, k is the constant of variation. We can find k by rearranging the formula as k = xy.
Thus, k is simply the product of the known values for the two variables. Let’s consider the following example:
A particular hotel has a custodial staff of 12 employees, and they can typically clean all of the hotel rooms in 6 hours. If four members of the custodial staff are not at work today, how long will it take the remaining custodians to clean all of the hotel rooms?
Now, the total time taken to complete the job is inversely proportional to the number of workers.
The constant of variation, k, is simply 6 12 = 72.
We want to know the numbers of hours it will take the remaining custodians to do the job. Since 4 workers are absent, that leaves 8 workers (12 – 4 = 8).
We can thus solve the problem as detailed below:
The correct answer is 9 hours.
Following are some key points about direct and inverse proportionality:
1.) When you are looking for direct proportionality, divide. See if the ratios are same for various pairs of values.
2.) When you are looking for inverse proportionality, multiply. See if the products are same for various pairs of values.
3.) The constant of proportionality is the ratio of two quantities in case of direct proportion and the product of two quantities in case of inverse proportion.
4.) If x is directly proportional to y and y is directly proportional to z, x will also be directly proportional to z.
5.) If x is directly proportional to y and y is inversely proportional to z, x will also be inversely proportional to z.
6.) If x is inversely proportional to y and y is inversely proportional to z, x will be directly proportional to z.