Textbook
1. ACT Math
1.1 ACT math intro
1.2 Pre-algebra
1.2.1 Order of operations
1.2.2 Absolute value
1.2.3 Properties of fractions
1.2.4 Percentages, ratios, proportions, decimals, fractions
1.2.5 Scientific notation
1.2.6 Types of numbers
1.2.7 Basic probability
1.2.8 Mean, median, mode and range
1.2.9 Unit conversions
1.3 Elementary algebra
1.4 Intermediate algebra
1.5 Plane geometry
1.6 Coordinate geometry
1.7 Trigonometry
2. ACT English
3. ACT Reading
4. ACT Science
5. ACT Writing
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1.2.4 Percentages, ratios, proportions, decimals, fractions
Achievable ACT
1. ACT Math
1.2. Pre-algebra

Percentages, ratios, proportions, decimals, fractions

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 ( of is ). Percent literally means “per one hundred.” means or out of .
Ratios Give a numerical rule to compare new data ( shows that for every of my left number I have of my right number).
Proportions Directly relate one figure or number to another. They are often represented as fractions. (Rectangle is twice the size area of rectangle . If we know the area of one, we can find the area of the other.)
Decimals A number smaller than that is attached alongside a whole number (e.g., is ).
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.

Relationships and conversions of operators

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!

Conversion chart for percentages, ratios, proportions, decimals, and fractions

Common questions

Below are some questions you’re likely to see on these topics.

Percentages

What is  of ?

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 in this case).

Go ahead and try it out. This is a very common question found in the earlier portion of the test.

(spoiler)
  1. Convert percentage to decimal.

  1. Multiply the part () by the whole () to get the portion we are looking for.

Ratios

The areas of circles and are expressed as a ratio of . If the area of doubles from to , what will be the area of ?

First, we will need to figure out what the initial area of 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 is doubled.

(spoiler)
  1. Area . If the ratio is initially , then we can find out the pattern to solving for area

    related to is To get from to , we multiply by , so let’s do that with to follow the pattern. . Now makes sense because it can be simplified to if we divide by . This means that the area of circle is .

  2. If is doubled to , we follow the same pattern with . It doubles to ! The new ratio would be , and the area of circle would be .

Proportions

Triangle is similar to triangle and has a length that is twice the size of triangle . If triangle has an area of , what is the area of triangle ?

You’ll want to start by imagining the difference between the two triangles. What would happen if you doubled the base of a triangle?

(spoiler)

Doubling the base of a triangle causes the triangle’s area to double as well.

  • Area of triangle = base height
  • If base , height , area
  • If base , height , area , which is twice the size of the former area

So, the answer to this question is that the area of will be twice the area of , .

Decimals and fractions

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 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 .

Summary

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 (you can think of the percentage as the numerator, and the denominator is always ). 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, , new).

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