Percentages, ratios, proportions, decimals, fractions | Pre-algebra | ACT Math

You don’t need to memorize every detail in this table, but you will use these characteristics later when we connect the ideas. A helpful first step is to identify which operator you’re working with (percentage, ratio, proportion, decimal, fraction). Once you know the operator, it’s much easier to choose the right relationship or conversion.

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.

Relationships and conversions of operators

In the following lattice, match an operator in the column with an operator in the row to see an example of the relationship between them. You don’t need every box to make sense right away - we’ll work through these relationships in example problems later.

The gray blocks show same-operator relationships (meaning no conversion is needed), so there’s nothing special to note in those boxes.

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

Common questions

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

Percentages

What is $40%$ of $200$ ?

This question asks you to use a percentage as a part of a whole. A common method is to convert the percentage to a decimal or fraction, then multiply by the whole (which is $200$ here).

(spoiler)

Convert percentage to decimal.

$40%/100 = 0.40$

Multiply the part ( $0.40$ ) by the whole ( $200$ ) to get the portion.

$0.40 * 200 = 80$

Ratios

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 $B$ ?

Use the ratio to connect $A$ and $B$ . You can first find the original area of $B$ and then apply the same change, or (if you’re comfortable) apply the ratio after $A$ doubles.

(spoiler)

Area $A = 6$ . If the ratio is initially $3 : 7$ , then we can find 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 $28$ .

Proportions

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 $B$ ?

Start by thinking about what “twice the length” means for area. If you double a triangle’s base while keeping the height the same, what happens to the area?

(spoiler)

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

Area of triangle = $(\frac{1}{2})$ base $*$ height
If base $= 1$ , height $= 1$ , area $= \frac{1}{2}$
If base $= 2$ , height $= 1$ , area $= 1$ , which is twice the size of the former area

So, the answer to this question is that the area of $B$ will be twice the area of $A$ , $20$ .

Decimals and fractions

Decimals and fractions show up throughout the test, so you should be comfortable working with both. Here’s a common conversion question:

What is $256/23$ expressed as a decimal?

This is a good example of a problem you can solve directly with a calculator. Enter $256/23$ , and you should get $11.13$ .

Summary

The main idea to take from this chapter is that all five of these operators describe relationships between numbers.

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 relate a part (numerator, top number, original) to a whole (denominator, $100$ , new).

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.

Relationships and conversions of operators

The gray blocks show same-operator relationships (meaning no conversion is needed), so there’s nothing special to note in those boxes.

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

Common questions

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

Percentages

What is $40%$ of $200$ ?

This question asks you to use a percentage as a part of a whole. A common method is to convert the percentage to a decimal or fraction, then multiply by the whole (which is $200$ here).

(spoiler)

Convert percentage to decimal.

$40%/100 = 0.40$

Multiply the part ( $0.40$ ) by the whole ( $200$ ) to get the portion.

$0.40 * 200 = 80$

Ratios

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 $B$ ?

Use the ratio to connect $A$ and $B$ . You can first find the original area of $B$ and then apply the same change, or (if you’re comfortable) apply the ratio after $A$ doubles.

(spoiler)

Area $A = 6$ . If the ratio is initially $3 : 7$ , then we can find 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 $28$ .

Proportions

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 $B$ ?

Start by thinking about what “twice the length” means for area. If you double a triangle’s base while keeping the height the same, what happens to the area?

(spoiler)

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

Area of triangle = $(\frac{1}{2})$ base $*$ height
If base $= 1$ , height $= 1$ , area $= \frac{1}{2}$
If base $= 2$ , height $= 1$ , area $= 1$ , which is twice the size of the former area

So, the answer to this question is that the area of $B$ will be twice the area of $A$ , $20$ .

Decimals and fractions

Decimals and fractions show up throughout the test, so you should be comfortable working with both. Here’s a common conversion question:

What is $256/23$ expressed as a decimal?

This is a good example of a problem you can solve directly with a calculator. Enter $256/23$ , and you should get $11.13$ .

Summary

The main idea to take from this chapter is that all five of these operators describe relationships between numbers.