It is not terribly important that you memorize each of these characteristics, but it will help you later in this chapter when we discuss the relationships between them. If you can easily identify which operator (percentage, ratio, proportion, decimal, fraction) we are working with, then you will have a much easier time distinguishing their relationships.
Operator | Relationship |
---|---|
Percentages | Express a certain portion of a whole number ($50%$ of $20$ is $10$). Percent literally means “per one hundred.” $50%$ means $50/100$ or $50$ out of $100$. |
Ratios | Give a numerical rule to compare new data ($1:2$ shows that for every $1$ of my left number I have $2$ of my right number). |
Proportions | Directly relate one figure or number to another. They are often represented as fractions. (Rectangle $A$ is twice the size area of rectangle $B$. If we know the area of one, we can find the area of the other.) |
Decimals | A number smaller than $1$ that is attached alongside a whole number (e.g., $2.5$ is $2+0.5$). |
Fractions | A way to express a part of a whole without using decimals. This allows for the use of variables in the scope of non-whole numbers. |
In the following lattice, you can match up an operator column with an operator row to see an example of the relationship that they share. They do not all need to make immediate sense, as we will explore their relationships in example problems later. The gray blocks represent a same-operator relationship (i.e., you don’t need to convert between the same type), so there is nothing special to note there!
Below are some questions you’re likely to see on these topics.
What is $40%$ of $200$?
For this question, we will need to know how to use percentages. Refer to our definition: it’s a part of a whole. In order to use it, we will need to convert it to a decimal or fraction. Then, we will multiply that portion by the whole (which is $200$ in this case).
Go ahead and try it out. This is a very common question found in the earlier portion of the test.
$40%/100=0.40$
$0.40∗200=80$
The areas of circles $A$ and $B$ are expressed as a ratio of $3:7$. If the area of $A$ doubles from $6$ to $12$, what will be the area of $B$?
First, we will need to figure out what the initial area of $B$ is according to the ratio given, then we can follow the pattern to see what the new area will be. Alternatively, we can skip the first step (if we are comfortable with ratios) and apply the ratio after area $A$ is doubled.
Area $A=6$. If the ratio is initially $3:7$, then we can find out the pattern to solving for area $B$
$3:7$ related to $A:B$ is $6:?$ To get from $3$ to $6$, we multiply by $2$, so let’s do that with $7$ to follow the pattern. $7∗2=14$. Now $6:14$ makes sense because it can be simplified to $3:7$ if we divide by $2$. This means that the area of circle $B$ is $14$.
If $6$ is doubled to $12$, we follow the same pattern with $14$. It doubles to $28$! The new ratio would be $12:28$, and the area of circle $B$ would be $28$.
Triangle $A$ is similar to triangle $B$ and has a length that is twice the size of triangle $B$. If triangle $A$ has an area of $10$, what is the area of triangle $B$?
You’ll want to start by imagining the difference between the two triangles. What would happen if you doubled the base of a triangle?
Doubling the base of a triangle causes the triangle’s area to double as well.
So, the answer to this question is that the area of $B$ will be twice the area of $A$, $20$.
Oftentimes decimals and fractions are used casually throughout the test. It is important that you are familiar with these kinds of numbers. We will explore the type of question that converts between the two:
What is $256/23$ expressed as a decimal?
Hopefully this looks easy to you! Not because it is easy math, but because you can put it into your calculator! There are questions like this that can easily be solved using your calculator—you just have to recognize them as they come. If you put this into your calculator, you should get an answer of $11.13$.
The biggest thing that you need to take away from this chapter is that all five of these operators are essentially the same thing: they relate two numbers together. Fractions relate the numerator (top number) to the denominator (bottom number). Decimals are a numerical representation of fractions. Percentages relate a number to $100$ (you can think of the percentage as the numerator, and the denominator is always $100$). Ratios relate the left number to the right number (like sideways fractions). Proportions relate one object or distance to another. Most of the time, these are relating a part (numerator, top number, original) to a whole (denominator, $100$, new).
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