2.1.2 Fractions and advanced operations

Achievable ASVAB

2. Mathematics Knowledge

2.1. Foundational math

Fractions and advanced operations

8 min read

Font

Discuss

Feedback

Fractions

Definitions

Fractions: Numbers that represent parts of a whole
Numerator: Top of a fraction
Denominator: Bottom of the fraction
Reciprocal: A fraction flipped upside down
Improper fraction: A fraction whose numerator is greater than or equal to its denominator
Mixed number: A whole number written together with a proper fraction, such as $2 \frac{1}{2}$

You may also see fractions written in different forms, such as improper fractions or mixed numbers. These will be covered in more detail later.

The numerator (top) tells you how many parts you have.
The denominator (bottom) tells you how many equal parts make up the whole.

For example, in the fraction $\frac{3}{4}$ , the whole is divided into $4$ equal parts, and you have $3$ of those parts.

The fraction 1/2. The numerator, 1, is the parts you have; the denominator, 2, is the equal parts in the whole. — Fraction parts

You’ll use a few core fraction skills such as adding, subtracting, multiplying, dividing, working with reciprocals, and converting between fractions, decimals, and percents.

These skills help you solve a wide range of problems, from basic calculations to more complex equations, so it’s important to understand how and when to use each one.

Adding and subtracting fractions

To add or subtract fractions, the denominators must match. The denominator tells you the size of each part, so you can only combine fractions when the parts are the same size.

That means you first need to find a common denominator (a common multiple of the denominators). Then:

Rewrite each fraction with that denominator
Add or subtract the numerators
Keep the denominator the same

Example:

$\frac{1}{2} + \frac{1}{3}$

Start by listing multiples of each denominator:

$2 : 2, 4, 6, 8, 10$

$3 : 3, 6, 9, 12$

The least common denominator is $6$ . To change a fraction without changing its value, multiply the top and bottom by the same number.

To rewrite $\frac{1}{2}$ with denominator $6$ , multiply by $\frac{3}{3}$ . To rewrite $\frac{1}{3}$ with denominator $6$ , multiply by $\frac{2}{2}$ . Multiplying by $\frac{3}{3}$ or $\frac{2}{2}$ doesn’t change the value of the fraction because you’re multiplying by $1$ .

$\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

$\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Now add the numerators and keep the denominator:

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Sidenote

Common denominators

You can always just multiply the denominators by each other for a quick way to get a common denominator. This method might not give you the least common denominator but it will always give you a common denominator.

Knowledge check: Find $\frac{7}{8} - \frac{4}{5}$ .

(spoiler)

$\frac{35}{40} - \frac{32}{40} = \frac{3}{40}$

Multiply fractions

To multiply fractions, multiply straight across. This means you multiply the top numbers together and the bottom numbers together. You do not need a common denominator.

Numerator times numerator
Denominator times denominator

$\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$

Think of it as multiplying two parts of a whole. You’re finding a part of a part.

Example: Cross-cancelling before multiplying

$\frac{4}{9} \times \frac{3}{8}$

Before multiplying straight across, check whether a numerator and the opposite denominator share a common factor - cancelling it first keeps the numbers small. Here, $4$ and $8$ share a factor of $4$ (reducing to $1$ and $2$ ), and $3$ and $9$ share a factor of $3$ (reducing to $1$ and $3$ ):

$\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

Answer: $\frac{1}{6}$

Dividing fractions

Before dividing fractions, it helps to know the reciprocal. The reciprocal is the fraction flipped.

$\frac{3}{4} \to \frac{4}{3}$

To divide fractions, keep the first fraction the same and multiply by the reciprocal of the second fraction (often called keep-change-flip). This works because dividing by a fraction is the same as multiplying by its reciprocal.

Keep, change, flip

Let’s break down how to solve this example. Use the method shown in the graphic above: keep, change, flip.

$\frac{1}{2} \div \frac{3}{4}$

Keep the first fraction the same
Change division to multiplication
Flip the second fraction (take the reciprocal)

$\frac{1}{2} \times \frac{4}{3}$

Now multiply straight across (top times top, bottom times bottom):

$\frac{4}{6}$

Simplify the fraction:

$\frac{2}{3}$

Shortcut: Once you’ve flipped, it’s an ordinary multiplication, so you can cross-cancel the $2$ and $4$ first, as in the multiplication example above, and get $\frac{2}{3}$ directly.

Simplifying fractions

After solving a fraction problem, you usually simplify your answer. This means writing the fraction in its simplest form.

Fractions can be written in different ways while representing the same value. This is why we simplify. For example, $\frac{1}{2}$ is the same as $\frac{2}{4}$ or $\frac{4}{8}$ .

In most math work, you write the final answer in simplest (most reduced) form. That means the numerator and denominator have no common factor other than $1$ .

To simplify a fraction:

Find the greatest common factor of the numerator and denominator.
Divide both by that factor.

Example:

If we have the fraction $\frac{4}{10}$ , we want to see if the fraction can be reduced by looking at the factors for both the numerator and the denominator.

$4 : 1, 2, 4$

$10 : 1, 2, 5, 10$

Since both numbers share the common factor of $2$ , divide both the numerator and denominator by $2$ .

$\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}$

Answer: $\frac{2}{5}$

Mixed numbers

A mixed number combines a whole number and a fraction, like $2 \frac{1}{2}$ . Before you add, subtract, multiply, or divide mixed numbers, it helps to convert them to improper fractions.

Converting a mixed number to an improper fraction:

Multiply the whole number by the denominator.
Add the numerator to that product.
Place the result over the original denominator.

$2 \frac{1}{2} = \frac{( 2 \times 2 ) + 1}{2} = \frac{5}{2}$

Converting an improper fraction to a mixed number: Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the same denominator.

$\frac{5}{2} = 2 remainder 1 = 2 \frac{1}{2}$

Example: Multiplying mixed numbers

Multiply $2 \frac{1}{2} \times 1 \frac{1}{3}$ .

First, convert both mixed numbers to improper fractions:

$2 \frac{1}{2} = \frac{5}{2} 1 \frac{1}{3} = \frac{4}{3}$

Multiply straight across:

$\frac{5}{2} \times \frac{4}{3} = \frac{20}{6}$

Simplify, then convert back to a mixed number:

$\frac{20}{6} = \frac{10}{3} = 3 \frac{1}{3}$

Answer: $3 \frac{1}{3}$

Converting fractions

Fractions can also be written as decimals and percents. To convert a fraction to a percent, first write it as a decimal by dividing the numerator by the denominator. Then multiply by 100.

Example:

Convert $\frac{1}{4}$ to a percent:

First, divide the numerator by the denominator to get a decimal:

$\frac{1}{4} = 1 \div 4 = 0.25$

Next, convert the decimal to a percent by multiplying by $100$ :

$0.25 \times 100 = 25%$

So,

$\frac{1}{4} = 0.25 = 25%$

No-calculator tip: If dividing the numerator by the denominator means long division, simplify the fraction first and try rewriting it over $10$ or $100$ . For example, $\frac{4}{16}$ simplifies to $\frac{1}{4}$ , which is $\frac{25}{100}$ , or $25%$ . It also helps to memorize a few common fraction-percent pairs: $\frac{1}{4} = 25%$ , $\frac{1}{8} = 12.5%$ , and $\frac{1}{3} = 33 \frac{1}{3} %$ .

Knowledge check: Convert $\frac{6}{20}$ to a percent

(spoiler)

Simplify first: $\frac{6}{20} = \frac{3}{10} = \frac{30}{100} = 30%$

Fraction Basics

Numerator (top) = parts you have; denominator (bottom) = equal parts in whole
Reciprocal = fraction flipped upside down
Improper fraction: numerator ≥ denominator; mixed number: whole number + proper fraction (e.g., $2 \frac{1}{2}$ )

Adding & Subtracting Fractions

Denominators must match before combining
Find common denominator (multiply denominators together as quick method, though not always least common denominator), rewrite fractions, then add/subtract numerators only
Keep denominator unchanged in final answer

Multiplying & Dividing Fractions

Multiply: multiply numerators together and denominators together (no common denominator needed); cross-cancel shared factors first to simplify
Divide: use “keep, change, flip” — keep first fraction, change ÷ to ×, flip (reciprocal) the second fraction, then multiply straight across

Simplifying Fractions

Reduce fraction so numerator/denominator share no common factor other than $1$
Method: find greatest common factor, divide both numerator and denominator by it
Example: $\frac{4}{10} = \frac{2}{5}$

Mixed Numbers

Convert mixed → improper: multiply whole number by denominator, add numerator, place over same denominator (e.g., $2 \frac{1}{2} = \frac{5}{2}$ )
Convert improper → mixed: divide numerator by denominator; quotient = whole number, remainder = new numerator
Convert to improper fractions before multiplying/dividing mixed numbers, then simplify and convert back if needed

Converting Fractions to Decimals/Percents

Divide numerator by denominator to get decimal, then multiply by $100$ for percent
Shortcut: simplify fraction first, try rewriting over $10$ or $100$ to avoid long division
Memorize key conversions: $\frac{1}{4} = 25%$ , $\frac{1}{8} = 12.5%$ , $\frac{1}{3} = 33 \frac{1}{3} %$

Next | 2.1.3 Core math concepts

Fractions and advanced operations

Fractions

Definitions

Fractions: Numbers that represent parts of a whole
Numerator: Top of a fraction
Denominator: Bottom of the fraction
Reciprocal: A fraction flipped upside down
Improper fraction: A fraction whose numerator is greater than or equal to its denominator
Mixed number: A whole number written together with a proper fraction, such as $2 \frac{1}{2}$

You may also see fractions written in different forms, such as improper fractions or mixed numbers. These will be covered in more detail later.

The numerator (top) tells you how many parts you have.
The denominator (bottom) tells you how many equal parts make up the whole.

For example, in the fraction $\frac{3}{4}$ , the whole is divided into $4$ equal parts, and you have $3$ of those parts.

You’ll use a few core fraction skills such as adding, subtracting, multiplying, dividing, working with reciprocals, and converting between fractions, decimals, and percents.

These skills help you solve a wide range of problems, from basic calculations to more complex equations, so it’s important to understand how and when to use each one.

Adding and subtracting fractions

To add or subtract fractions, the denominators must match. The denominator tells you the size of each part, so you can only combine fractions when the parts are the same size.

That means you first need to find a common denominator (a common multiple of the denominators). Then:

Rewrite each fraction with that denominator
Add or subtract the numerators
Keep the denominator the same

Example:

$\frac{1}{2} + \frac{1}{3}$

Start by listing multiples of each denominator:

$2 : 2, 4, 6, 8, 10$

$3 : 3, 6, 9, 12$

The least common denominator is $6$ . To change a fraction without changing its value, multiply the top and bottom by the same number.

To rewrite $\frac{1}{2}$ with denominator $6$ , multiply by $\frac{3}{3}$ . To rewrite $\frac{1}{3}$ with denominator $6$ , multiply by $\frac{2}{2}$ . Multiplying by $\frac{3}{3}$ or $\frac{2}{2}$ doesn’t change the value of the fraction because you’re multiplying by $1$ .

$\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

$\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Now add the numerators and keep the denominator:

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Sidenote

Common denominators

Knowledge check: Find $\frac{7}{8} - \frac{4}{5}$ .

(spoiler)

$\frac{35}{40} - \frac{32}{40} = \frac{3}{40}$

Multiply fractions

To multiply fractions, multiply straight across. This means you multiply the top numbers together and the bottom numbers together. You do not need a common denominator.

Numerator times numerator
Denominator times denominator

$\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$

Think of it as multiplying two parts of a whole. You’re finding a part of a part.

Example: Cross-cancelling before multiplying

$\frac{4}{9} \times \frac{3}{8}$

Before multiplying straight across, check whether a numerator and the opposite denominator share a common factor - cancelling it first keeps the numbers small. Here, $4$ and $8$ share a factor of $4$ (reducing to $1$ and $2$ ), and $3$ and $9$ share a factor of $3$ (reducing to $1$ and $3$ ):

$\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

Answer: $\frac{1}{6}$

Dividing fractions

Before dividing fractions, it helps to know the reciprocal. The reciprocal is the fraction flipped.

$\frac{3}{4} \to \frac{4}{3}$

Keep, change, flip

Let’s break down how to solve this example. Use the method shown in the graphic above: keep, change, flip.

$\frac{1}{2} \div \frac{3}{4}$

Keep the first fraction the same
Change division to multiplication
Flip the second fraction (take the reciprocal)

$\frac{1}{2} \times \frac{4}{3}$

Now multiply straight across (top times top, bottom times bottom):

$\frac{4}{6}$

Simplify the fraction:

$\frac{2}{3}$

Shortcut: Once you’ve flipped, it’s an ordinary multiplication, so you can cross-cancel the $2$ and $4$ first, as in the multiplication example above, and get $\frac{2}{3}$ directly.

Simplifying fractions

After solving a fraction problem, you usually simplify your answer. This means writing the fraction in its simplest form.

Fractions can be written in different ways while representing the same value. This is why we simplify. For example, $\frac{1}{2}$ is the same as $\frac{2}{4}$ or $\frac{4}{8}$ .

In most math work, you write the final answer in simplest (most reduced) form. That means the numerator and denominator have no common factor other than $1$ .

To simplify a fraction:

Find the greatest common factor of the numerator and denominator.
Divide both by that factor.

Example:

If we have the fraction $\frac{4}{10}$ , we want to see if the fraction can be reduced by looking at the factors for both the numerator and the denominator.

$4 : 1, 2, 4$

$10 : 1, 2, 5, 10$

Since both numbers share the common factor of $2$ , divide both the numerator and denominator by $2$ .

$\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}$

Answer: $\frac{2}{5}$

Mixed numbers

A mixed number combines a whole number and a fraction, like $2 \frac{1}{2}$ . Before you add, subtract, multiply, or divide mixed numbers, it helps to convert them to improper fractions.

Converting a mixed number to an improper fraction:

Multiply the whole number by the denominator.
Add the numerator to that product.
Place the result over the original denominator.

$2 \frac{1}{2} = \frac{( 2 \times 2 ) + 1}{2} = \frac{5}{2}$

$\frac{5}{2} = 2 remainder 1 = 2 \frac{1}{2}$

Example: Multiplying mixed numbers

Multiply $2 \frac{1}{2} \times 1 \frac{1}{3}$ .

First, convert both mixed numbers to improper fractions:

$2 \frac{1}{2} = \frac{5}{2} 1 \frac{1}{3} = \frac{4}{3}$

Multiply straight across:

$\frac{5}{2} \times \frac{4}{3} = \frac{20}{6}$

Simplify, then convert back to a mixed number:

$\frac{20}{6} = \frac{10}{3} = 3 \frac{1}{3}$

Answer: $3 \frac{1}{3}$

Converting fractions

Fractions can also be written as decimals and percents. To convert a fraction to a percent, first write it as a decimal by dividing the numerator by the denominator. Then multiply by 100.

Example:

Convert $\frac{1}{4}$ to a percent:

First, divide the numerator by the denominator to get a decimal:

$\frac{1}{4} = 1 \div 4 = 0.25$

Next, convert the decimal to a percent by multiplying by $100$ :

$0.25 \times 100 = 25%$

So,

$\frac{1}{4} = 0.25 = 25%$

Knowledge check: Convert $\frac{6}{20}$ to a percent

(spoiler)

Simplify first: $\frac{6}{20} = \frac{3}{10} = \frac{30}{100} = 30%$

Achievable ASVAB

2. Mathematics Knowledge

2.1. Foundational math

Fractions and advanced operations

8 min read

Font

Discuss

Feedback

Fractions

Definitions

Fractions: Numbers that represent parts of a whole
Numerator: Top of a fraction
Denominator: Bottom of the fraction
Reciprocal: A fraction flipped upside down
Improper fraction: A fraction whose numerator is greater than or equal to its denominator
Mixed number: A whole number written together with a proper fraction, such as $2 \frac{1}{2}$

You may also see fractions written in different forms, such as improper fractions or mixed numbers. These will be covered in more detail later.

The numerator (top) tells you how many parts you have.
The denominator (bottom) tells you how many equal parts make up the whole.

For example, in the fraction $\frac{3}{4}$ , the whole is divided into $4$ equal parts, and you have $3$ of those parts.

You’ll use a few core fraction skills such as adding, subtracting, multiplying, dividing, working with reciprocals, and converting between fractions, decimals, and percents.

These skills help you solve a wide range of problems, from basic calculations to more complex equations, so it’s important to understand how and when to use each one.

Adding and subtracting fractions

To add or subtract fractions, the denominators must match. The denominator tells you the size of each part, so you can only combine fractions when the parts are the same size.

That means you first need to find a common denominator (a common multiple of the denominators). Then:

Rewrite each fraction with that denominator
Add or subtract the numerators
Keep the denominator the same

Example:

$\frac{1}{2} + \frac{1}{3}$

Start by listing multiples of each denominator:

$2 : 2, 4, 6, 8, 10$

$3 : 3, 6, 9, 12$

The least common denominator is $6$ . To change a fraction without changing its value, multiply the top and bottom by the same number.

To rewrite $\frac{1}{2}$ with denominator $6$ , multiply by $\frac{3}{3}$ . To rewrite $\frac{1}{3}$ with denominator $6$ , multiply by $\frac{2}{2}$ . Multiplying by $\frac{3}{3}$ or $\frac{2}{2}$ doesn’t change the value of the fraction because you’re multiplying by $1$ .

$\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

$\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Now add the numerators and keep the denominator:

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Sidenote

Common denominators

Knowledge check: Find $\frac{7}{8} - \frac{4}{5}$ .

(spoiler)

$\frac{35}{40} - \frac{32}{40} = \frac{3}{40}$

Multiply fractions

To multiply fractions, multiply straight across. This means you multiply the top numbers together and the bottom numbers together. You do not need a common denominator.

Numerator times numerator
Denominator times denominator

$\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$

Think of it as multiplying two parts of a whole. You’re finding a part of a part.

Example: Cross-cancelling before multiplying

$\frac{4}{9} \times \frac{3}{8}$

Before multiplying straight across, check whether a numerator and the opposite denominator share a common factor - cancelling it first keeps the numbers small. Here, $4$ and $8$ share a factor of $4$ (reducing to $1$ and $2$ ), and $3$ and $9$ share a factor of $3$ (reducing to $1$ and $3$ ):

$\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

Answer: $\frac{1}{6}$

Dividing fractions

Before dividing fractions, it helps to know the reciprocal. The reciprocal is the fraction flipped.

$\frac{3}{4} \to \frac{4}{3}$

Keep, change, flip

Let’s break down how to solve this example. Use the method shown in the graphic above: keep, change, flip.

$\frac{1}{2} \div \frac{3}{4}$

Keep the first fraction the same
Change division to multiplication
Flip the second fraction (take the reciprocal)

$\frac{1}{2} \times \frac{4}{3}$

Now multiply straight across (top times top, bottom times bottom):

$\frac{4}{6}$

Simplify the fraction:

$\frac{2}{3}$

Shortcut: Once you’ve flipped, it’s an ordinary multiplication, so you can cross-cancel the $2$ and $4$ first, as in the multiplication example above, and get $\frac{2}{3}$ directly.

Simplifying fractions

After solving a fraction problem, you usually simplify your answer. This means writing the fraction in its simplest form.

Fractions can be written in different ways while representing the same value. This is why we simplify. For example, $\frac{1}{2}$ is the same as $\frac{2}{4}$ or $\frac{4}{8}$ .

In most math work, you write the final answer in simplest (most reduced) form. That means the numerator and denominator have no common factor other than $1$ .

To simplify a fraction:

Find the greatest common factor of the numerator and denominator.
Divide both by that factor.

Example:

If we have the fraction $\frac{4}{10}$ , we want to see if the fraction can be reduced by looking at the factors for both the numerator and the denominator.

$4 : 1, 2, 4$

$10 : 1, 2, 5, 10$

Since both numbers share the common factor of $2$ , divide both the numerator and denominator by $2$ .

$\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}$

Answer: $\frac{2}{5}$

Mixed numbers

A mixed number combines a whole number and a fraction, like $2 \frac{1}{2}$ . Before you add, subtract, multiply, or divide mixed numbers, it helps to convert them to improper fractions.

Converting a mixed number to an improper fraction:

Multiply the whole number by the denominator.
Add the numerator to that product.
Place the result over the original denominator.

$2 \frac{1}{2} = \frac{( 2 \times 2 ) + 1}{2} = \frac{5}{2}$

$\frac{5}{2} = 2 remainder 1 = 2 \frac{1}{2}$

Example: Multiplying mixed numbers

Multiply $2 \frac{1}{2} \times 1 \frac{1}{3}$ .

First, convert both mixed numbers to improper fractions:

$2 \frac{1}{2} = \frac{5}{2} 1 \frac{1}{3} = \frac{4}{3}$

Multiply straight across:

$\frac{5}{2} \times \frac{4}{3} = \frac{20}{6}$

Simplify, then convert back to a mixed number:

$\frac{20}{6} = \frac{10}{3} = 3 \frac{1}{3}$

Answer: $3 \frac{1}{3}$

Converting fractions

Fractions can also be written as decimals and percents. To convert a fraction to a percent, first write it as a decimal by dividing the numerator by the denominator. Then multiply by 100.

Example:

Convert $\frac{1}{4}$ to a percent:

First, divide the numerator by the denominator to get a decimal:

$\frac{1}{4} = 1 \div 4 = 0.25$

Next, convert the decimal to a percent by multiplying by $100$ :

$0.25 \times 100 = 25%$

So,

$\frac{1}{4} = 0.25 = 25%$

Knowledge check: Convert $\frac{6}{20}$ to a percent

(spoiler)

Simplify first: $\frac{6}{20} = \frac{3}{10} = \frac{30}{100} = 30%$

Fraction Basics

Numerator (top) = parts you have; denominator (bottom) = equal parts in whole
Reciprocal = fraction flipped upside down
Improper fraction: numerator ≥ denominator; mixed number: whole number + proper fraction (e.g., $2 \frac{1}{2}$ )

Adding & Subtracting Fractions

Denominators must match before combining
Find common denominator (multiply denominators together as quick method, though not always least common denominator), rewrite fractions, then add/subtract numerators only
Keep denominator unchanged in final answer

Multiplying & Dividing Fractions

Multiply: multiply numerators together and denominators together (no common denominator needed); cross-cancel shared factors first to simplify
Divide: use “keep, change, flip” — keep first fraction, change ÷ to ×, flip (reciprocal) the second fraction, then multiply straight across

Simplifying Fractions

Reduce fraction so numerator/denominator share no common factor other than $1$
Method: find greatest common factor, divide both numerator and denominator by it
Example: $\frac{4}{10} = \frac{2}{5}$

Mixed Numbers

Convert mixed → improper: multiply whole number by denominator, add numerator, place over same denominator (e.g., $2 \frac{1}{2} = \frac{5}{2}$ )
Convert improper → mixed: divide numerator by denominator; quotient = whole number, remainder = new numerator
Convert to improper fractions before multiplying/dividing mixed numbers, then simplify and convert back if needed

Converting Fractions to Decimals/Percents

Divide numerator by denominator to get decimal, then multiply by $100$ for percent
Shortcut: simplify fraction first, try rewriting over $10$ or $100$ to avoid long division
Memorize key conversions: $\frac{1}{4} = 25%$ , $\frac{1}{8} = 12.5%$ , $\frac{1}{3} = 33 \frac{1}{3} %$

Next | 2.1.3 Core math concepts

Fractions and advanced operations

Fractions

Definitions

Fractions: Numbers that represent parts of a whole
Numerator: Top of a fraction
Denominator: Bottom of the fraction
Reciprocal: A fraction flipped upside down
Improper fraction: A fraction whose numerator is greater than or equal to its denominator
Mixed number: A whole number written together with a proper fraction, such as $2 \frac{1}{2}$

You may also see fractions written in different forms, such as improper fractions or mixed numbers. These will be covered in more detail later.

The numerator (top) tells you how many parts you have.
The denominator (bottom) tells you how many equal parts make up the whole.

For example, in the fraction $\frac{3}{4}$ , the whole is divided into $4$ equal parts, and you have $3$ of those parts.

You’ll use a few core fraction skills such as adding, subtracting, multiplying, dividing, working with reciprocals, and converting between fractions, decimals, and percents.

These skills help you solve a wide range of problems, from basic calculations to more complex equations, so it’s important to understand how and when to use each one.

Adding and subtracting fractions

To add or subtract fractions, the denominators must match. The denominator tells you the size of each part, so you can only combine fractions when the parts are the same size.

That means you first need to find a common denominator (a common multiple of the denominators). Then:

Rewrite each fraction with that denominator
Add or subtract the numerators
Keep the denominator the same

Example:

$\frac{1}{2} + \frac{1}{3}$

Start by listing multiples of each denominator:

$2 : 2, 4, 6, 8, 10$

$3 : 3, 6, 9, 12$

The least common denominator is $6$ . To change a fraction without changing its value, multiply the top and bottom by the same number.

To rewrite $\frac{1}{2}$ with denominator $6$ , multiply by $\frac{3}{3}$ . To rewrite $\frac{1}{3}$ with denominator $6$ , multiply by $\frac{2}{2}$ . Multiplying by $\frac{3}{3}$ or $\frac{2}{2}$ doesn’t change the value of the fraction because you’re multiplying by $1$ .

$\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

$\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Now add the numerators and keep the denominator:

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Sidenote

Common denominators

Knowledge check: Find $\frac{7}{8} - \frac{4}{5}$ .

(spoiler)

$\frac{35}{40} - \frac{32}{40} = \frac{3}{40}$

Multiply fractions

To multiply fractions, multiply straight across. This means you multiply the top numbers together and the bottom numbers together. You do not need a common denominator.

Numerator times numerator
Denominator times denominator

$\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$

Think of it as multiplying two parts of a whole. You’re finding a part of a part.

Example: Cross-cancelling before multiplying

$\frac{4}{9} \times \frac{3}{8}$

Before multiplying straight across, check whether a numerator and the opposite denominator share a common factor - cancelling it first keeps the numbers small. Here, $4$ and $8$ share a factor of $4$ (reducing to $1$ and $2$ ), and $3$ and $9$ share a factor of $3$ (reducing to $1$ and $3$ ):

$\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

Answer: $\frac{1}{6}$

Dividing fractions

Before dividing fractions, it helps to know the reciprocal. The reciprocal is the fraction flipped.

$\frac{3}{4} \to \frac{4}{3}$

Keep, change, flip

Let’s break down how to solve this example. Use the method shown in the graphic above: keep, change, flip.

$\frac{1}{2} \div \frac{3}{4}$

Keep the first fraction the same
Change division to multiplication
Flip the second fraction (take the reciprocal)

$\frac{1}{2} \times \frac{4}{3}$

Now multiply straight across (top times top, bottom times bottom):

$\frac{4}{6}$

Simplify the fraction:

$\frac{2}{3}$

Shortcut: Once you’ve flipped, it’s an ordinary multiplication, so you can cross-cancel the $2$ and $4$ first, as in the multiplication example above, and get $\frac{2}{3}$ directly.

Simplifying fractions

After solving a fraction problem, you usually simplify your answer. This means writing the fraction in its simplest form.

Fractions can be written in different ways while representing the same value. This is why we simplify. For example, $\frac{1}{2}$ is the same as $\frac{2}{4}$ or $\frac{4}{8}$ .

In most math work, you write the final answer in simplest (most reduced) form. That means the numerator and denominator have no common factor other than $1$ .

To simplify a fraction:

Find the greatest common factor of the numerator and denominator.
Divide both by that factor.

Example:

If we have the fraction $\frac{4}{10}$ , we want to see if the fraction can be reduced by looking at the factors for both the numerator and the denominator.

$4 : 1, 2, 4$

$10 : 1, 2, 5, 10$

Since both numbers share the common factor of $2$ , divide both the numerator and denominator by $2$ .

$\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}$

Answer: $\frac{2}{5}$

Mixed numbers

A mixed number combines a whole number and a fraction, like $2 \frac{1}{2}$ . Before you add, subtract, multiply, or divide mixed numbers, it helps to convert them to improper fractions.

Converting a mixed number to an improper fraction:

Multiply the whole number by the denominator.
Add the numerator to that product.
Place the result over the original denominator.

$2 \frac{1}{2} = \frac{( 2 \times 2 ) + 1}{2} = \frac{5}{2}$

$\frac{5}{2} = 2 remainder 1 = 2 \frac{1}{2}$

Example: Multiplying mixed numbers

Multiply $2 \frac{1}{2} \times 1 \frac{1}{3}$ .

First, convert both mixed numbers to improper fractions:

$2 \frac{1}{2} = \frac{5}{2} 1 \frac{1}{3} = \frac{4}{3}$

Multiply straight across:

$\frac{5}{2} \times \frac{4}{3} = \frac{20}{6}$

Simplify, then convert back to a mixed number:

$\frac{20}{6} = \frac{10}{3} = 3 \frac{1}{3}$

Answer: $3 \frac{1}{3}$

Converting fractions

Fractions can also be written as decimals and percents. To convert a fraction to a percent, first write it as a decimal by dividing the numerator by the denominator. Then multiply by 100.

Example:

Convert $\frac{1}{4}$ to a percent:

First, divide the numerator by the denominator to get a decimal:

$\frac{1}{4} = 1 \div 4 = 0.25$

Next, convert the decimal to a percent by multiplying by $100$ :

$0.25 \times 100 = 25%$

So,

$\frac{1}{4} = 0.25 = 25%$

Knowledge check: Convert $\frac{6}{20}$ to a percent

(spoiler)

Simplify first: $\frac{6}{20} = \frac{3}{10} = \frac{30}{100} = 30%$