Fraction Fundamentals and Equivalence

Unit Fractions as Building Blocks

Fractions help us describe parts of a whole. A unit fraction is a fraction with a numerator of 1. Examples are \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), and \(\frac{1}{8}\).

Unit fractions are important because they are the building blocks of all fractions. That means we can make other fractions by putting unit fractions together.

For example, if a whole is cut into 4 equal parts, then one part is \(\frac{1}{4}\). If you have 3 of those parts, you have \(\frac{3}{4}\). So \(\frac{3}{4}\) is made from 3 copies of \(\frac{1}{4}\).

We can write that like this:

This shows that the denominator tells the size of each part, and the numerator tells how many of those parts we have.

Let’s look closely at the two numbers in a fraction:

• Denominator: the bottom number. It tells how many equal parts make 1 whole.
• Numerator: the top number. It tells how many parts we have.

In \(\frac{5}{6}\):

• The denominator is 6, so the whole is split into 6 equal parts.
• Each part is \(\frac{1}{6}\).
• The numerator is 5, so we have 5 of those \(\frac{1}{6}\) parts.

So:

This is the big idea: every fraction is made of unit fractions.

Think about a pizza. If a pizza is cut into 8 equal slices, each slice is \(\frac{1}{8}\) of the pizza. If you eat 2 slices, you ate \(\frac{2}{8}\). That means:

Important idea: The denominator tells the name of the pieces. If the whole is split into 8 equal parts, the pieces are called eighths. If the whole is split into 5 equal parts, the pieces are called fifths.

So:

• \(\frac{1}{2}\) is one half
• \(\frac{1}{3}\) is one third
• \(\frac{1}{4}\) is one fourth
• \(\frac{1}{5}\) is one fifth

When we build a fraction, we keep the same size piece and count how many pieces we have.

Visual idea: Imagine a bar split into 5 equal parts.

• 1 part shaded means \(\frac{1}{5}\)
• 2 parts shaded means \(\frac{2}{5}\)
• 3 parts shaded means \(\frac{3}{5}\)
• 5 parts shaded means \(\frac{5}{5}\), which is 1 whole

This helps us see that fractions grow by adding more unit fractions of the same size.

For example:

Worked Example 1: Build a fraction from unit fractions

Write \(\frac{3}{8}\) as a sum of unit fractions.

Step 1: Look at the denominator. The denominator is 8, so each part is \(\frac{1}{8}\).

Step 2: Look at the numerator. The numerator is 3, so we need 3 parts of size \(\frac{1}{8}\).

Answer:

Worked Example 2: Name the fraction from unit fractions

What fraction is made by adding 4 copies of \(\frac{1}{6}\)?

Step 1: The unit fraction is \(\frac{1}{6}\), so the denominator is 6.

Step 2: There are 4 copies, so the numerator is 4.

Answer:

Worked Example 3: Use a real-world model

A chocolate bar is split into 7 equal pieces. Maya eats 5 pieces. What fraction of the chocolate bar does she eat?

Step 1: Since the bar is split into 7 equal pieces, each piece is \(\frac{1}{7}\).

Step 2: Maya eats 5 pieces, so she eats 5 copies of \(\frac{1}{7}\).

Answer:

So Maya eats \(\frac{5}{7}\) of the chocolate bar.

Worked Example 4: Find the missing fraction

Complete the sentence:

Step 1: Each part is \(\frac{1}{9}\), so the denominator is 9.

Step 2: Count the parts. There are 4 parts, so the numerator is 4.

Answer:

Things to remember

• A unit fraction has a numerator of 1.
• The denominator tells the size of each equal part.
• Any fraction \(\frac{a}{b}\) means \(a\) copies of \(\frac{1}{b}\).
• You can build fractions by adding unit fractions with the same denominator.

Here is the rule in math form:

This means a fraction is made from repeated copies of one unit fraction.

Common mistake to avoid

Do not change the denominator when you count more pieces of the same size. If the pieces are sixths, they stay sixths.

For example, 3 copies of \(\frac{1}{6}\) make \(\frac{3}{6}\), not \(\frac{3}{18}\).

Quick practice to think about

1. Write \(\frac{2}{5}\) as a sum of unit fractions.
2. How many \(\frac{1}{4}\) pieces make \(\frac{3}{4}\)?
3. What fraction is 6 copies of \(\frac{1}{10}\)?
4. If a whole is divided into 3 equal parts, what is one part called?

Answers:

1. \(\frac{1}{5}+\frac{1}{5}\)
2. 3 pieces
3. \(\frac{6}{10}\)
4. \(\frac{1}{3}\), or one third

Summary

Unit fractions are fractions with a 1 on top. They are the small equal parts that build all other fractions. To understand a fraction, first find the unit fraction from the denominator, then count how many of those parts the numerator tells you to use.

Building Non-Unit Fractions

Fractions help us show parts of a whole. Sometimes we talk about one part, like one-fourth. Sometimes we talk about several equal parts together, like three-fourths.

This lesson is about building non-unit fractions. A unit fraction is a fraction with 1 on top, such as \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{5}\). A non-unit fraction has a number greater than 1 on top, such as \(\frac{2}{3}\), \(\frac{4}{5}\), or \(\frac{7}{8}\).

The big idea is this: a non-unit fraction is made by putting together copies of a unit fraction.

For example, \(\frac{3}{4}\) means 3 copies of \(\frac{1}{4}\).

We can write that like this:

This is called iterating a unit fraction. That means repeating the same unit fraction again and again.

Let’s look at the parts of a fraction:

• The bottom number tells how many equal parts the whole is split into.
• The top number tells how many of those equal parts we have.

In \(\frac{5}{6}\):

• the 6 means the whole is split into 6 equal parts,
• the 5 means we have 5 of those sixths.

So:

How to build a non-unit fraction

1. Look at the denominator, the bottom number. It tells the size of each piece.
2. Make the unit fraction using that denominator.
3. Look at the numerator, the top number. It tells how many times to use that unit fraction.
4. Put those unit fractions together.

For example, to build \(\frac{4}{7}\):

• The denominator is 7, so the unit fraction is \(\frac{1}{7}\).
• The numerator is 4, so we need 4 copies of \(\frac{1}{7}\).

Thinking with pictures

Imagine a strip split into 5 equal parts. Each part is \(\frac{1}{5}\). If you shade 3 of the 5 parts, you have \(\frac{3}{5}\).

You can think:

This shows that non-unit fractions are made from equal-size pieces. The pieces must be equal. If the parts are not equal, it does not show a correct fraction of the whole.

Worked Example 1

Build the fraction \(\frac{2}{6}\).

Step 1: The denominator is 6, so the unit fraction is \(\frac{1}{6}\).

Step 2: The numerator is 2, so we need 2 copies of \(\frac{1}{6}\).

Answer: \(\frac{2}{6}\) is built from two one-sixths.

Worked Example 2

Build the fraction \(\frac{5}{8}\).

Step 1: The denominator is 8, so each piece is \(\frac{1}{8}\).

Step 2: The numerator is 5, so we combine 5 pieces of size \(\frac{1}{8}\).

Answer: \(\frac{5}{8}\) means five one-eighths.

Worked Example 3

A rectangle is split into 4 equal parts. 3 parts are shaded. What fraction is shaded, and how is it built from unit fractions?

There are 4 equal parts, so each part is \(\frac{1}{4}\).

Since 3 parts are shaded, the shaded fraction is \(\frac{3}{4}\).

Built from unit fractions:

Answer: The shaded fraction is \(\frac{3}{4}\), which is three one-fourths.

Worked Example 4

Which fraction is shown by 6 copies of \(\frac{1}{10}\)?

If we put together 6 unit fractions of \(\frac{1}{10}\), we get:

Answer: 6 copies of \(\frac{1}{10}\) make \(\frac{6}{10}\).

Important ideas to remember

• A unit fraction has 1 as the numerator.
• A non-unit fraction is made from more than one unit fraction.
• The denominator tells the size of each equal part.
• The numerator tells how many parts are being counted.
• \(\frac{a}{b}\) means \(a\) copies of \(\frac{1}{b}\).

For example:

Common mistakes to avoid

• Do not use parts that are different sizes. Fraction parts must be equal.
• Do not mix up the numerator and denominator.
• Do not think \(\frac{3}{5}\) means 3 wholes and 5 parts. It means 3 parts out of 5 equal parts.

Try this way of thinking:

• \(\frac{2}{3}\) means two one-thirds.
• \(\frac{7}{8}\) means seven one-eighths.
• \(\frac{3}{10}\) means three one-tenths.

When you say a fraction out loud, it can help you understand it. For example, “three-fourths” means 3 copies of “one-fourth.”

Summary

Non-unit fractions are built from unit fractions. To understand a fraction like \(\frac{3}{4}\), think of it as 3 copies of \(\frac{1}{4}\).

If you know the denominator, you know the size of each part. If you know the numerator, you know how many of those parts to count. That is how we build and understand non-unit fractions.

Fractions on a Number Line

A number line shows numbers in order from left to right. The farther right you move, the greater the number is.

Fractions can be shown on a number line too. This helps us see that a fraction is not just part of a shape or part of a set. A fraction is also a number with its own place on the line.

For example, \(\frac{1}{2}\) is the number halfway between \(0\) and \(1\). On a number line, it has one exact spot.

1. Start with whole numbers

On a number line, whole numbers are marked at equal distances:

$$0 \qquad 1 \qquad 2 \qquad 3$$

The space from one whole number to the next is called an interval. To place fractions, we divide an interval into equal parts.

2. The denominator tells how many equal parts

In a fraction like \(\frac{3}{4}\):

• The denominator, \(4\), tells how many equal parts the whole is split into.
• The numerator, \(3\), tells how many of those parts we count.

So to place \(\frac{3}{4}\) on a number line, look at the space from \(0\) to \(1\), divide it into 4 equal parts, and count 3 parts from 0.

3. Unit fractions come first

A unit fraction has a 1 on top, like \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{5}\).

Unit fractions are helpful because they show the size of one equal part.

• \(\frac{1}{2}\) means 1 out of 2 equal parts
• \(\frac{1}{3}\) means 1 out of 3 equal parts
• \(\frac{1}{4}\) means 1 out of 4 equal parts

If you know where \(\frac{1}{4}\) is, then you can find:

• \(\frac{2}{4}\): two jumps of \(\frac{1}{4}\)
• \(\frac{3}{4}\): three jumps of \(\frac{1}{4}\)
• \(\frac{4}{4}\): four jumps of \(\frac{1}{4}\), which equals \(1\)

4. Fractions between 0 and 1

To graph a fraction between \(0\) and \(1\), use these steps:

1. Look at the denominator.
2. Split the space from \(0\) to \(1\) into that many equal parts.
3. Count the number of parts named by the numerator.
4. Place the point there.

Worked Example 1: Plot \(\frac{2}{3}\)

Step 1: The denominator is \(3\), so divide the space from \(0\) to \(1\) into 3 equal parts.

Step 2: The numerator is \(2\), so count 2 parts from 0.

That point is \(\frac{2}{3}\).

We can think of the marks as:

$$0 \quad \frac{1}{3} \quad \frac{2}{3} \quad 1$$

5. Fractions greater than 1

Fractions do not stop at \(1\). Some fractions are greater than 1, such as \(\frac{5}{4}\) or \(\frac{7}{3}\).

To place these on a number line, keep making equal parts past \(1\).

For fourth grade, it helps to count by the unit fraction.

Worked Example 2: Plot \(\frac{5}{4}\)

The denominator is \(4\), so each whole must be divided into 4 equal parts.

Count by fourths:

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

So \(\frac{5}{4}\) is one fourth past 1.

It is placed between \(1\) and \(2\), at the first fourth mark after \(1\).

6. Fractions that name the same point

Sometimes different fractions land on the same place on the number line. These are called equivalent fractions.

For example:

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

On a number line, both fractions are at the same point halfway between \(0\) and \(1\).

This is an important idea: if two fractions are equivalent, they have the same value and the same location on the number line.

Worked Example 3: Show that \(\frac{1}{2}\) and \(\frac{2}{4}\) are at the same point

First, divide the space from \(0\) to \(1\) into 2 equal parts. The middle point is \(\frac{1}{2}\).

Now divide the same space from \(0\) to \(1\) into 4 equal parts. Two fourths, \(\frac{2}{4}\), lands exactly at the middle too.

So:

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

7. Connecting area models, sets, and number lines

You may have seen fractions in pictures before, such as:

• shaded parts of a rectangle
• pieces of a circle
• a set of objects

Those models show parts of a whole. A number line shows the amount as a number from 0 upward.

For example, if a rectangle has 3 out of 4 equal parts shaded, that is \(\frac{3}{4}\). On a number line, \(\frac{3}{4}\) is the point three fourths of the way from \(0\) to \(1\).

So the picture model and the number line both show the same fraction in different ways.

8. Compare fractions on a number line

A fraction farther to the right is greater. A fraction farther to the left is smaller.

For example, on the same number line:

$$\frac{1}{4} < \frac{3}{4}$$

because \(\frac{1}{4}\) is to the left of \(\frac{3}{4}\).

Also:

$$\frac{5}{3} > 1$$

because \(\frac{5}{3}\) is to the right of \(1\).

Worked Example 4: Plot and compare \(\frac{3}{6}\) and \(\frac{5}{6}\)

Both fractions have denominator 6, so divide the space from \(0\) to \(1\) into 6 equal parts.

Place \(\frac{3}{6}\) at the third mark from 0. Place \(\frac{5}{6}\) at the fifth mark from 0.

Since \(\frac{5}{6}\) is farther right, it is greater.

So:

$$\frac{3}{6} < \frac{5}{6}$$

Also, \(\frac{3}{6}\) is the same as \(\frac{1}{2}\), so it is halfway between \(0\) and \(1\).

9. Tips for success

• Always make equal parts. Unequal parts give the wrong location.
• Read the denominator first. It tells how many parts to split each whole into.
• Then count the numerator. Count that many parts from 0.
• Remember that \(1\) can be written as a fraction. For example, \(\frac{4}{4}=1\) and \(\frac{3}{3}=1\).
• Equivalent fractions share a point. For example, \(\frac{2}{3}\) and \(\frac{4}{6}\) would land at the same place.

10. Quick check ideas

Ask yourself these questions when you plot a fraction:

• How many equal parts should each whole have?
• Did I count the correct number of parts from 0?
• Is my fraction less than 1, equal to 1, or greater than 1?
• Does my point make sense on the number line?

Summary

A fraction is a number, and it has an exact place on a number line.

To graph a fraction, divide each whole into equal parts using the denominator, then count parts using the numerator.

Fractions can be between 0 and 1, equal to 1, or greater than 1. Equivalent fractions, like \(\frac{1}{2}\) and \(\frac{2}{4}\), land on the same point.

Partitioning Continuous Models means splitting one whole shape or one whole length into equal parts.

In 4th grade, a continuous model is something that is connected all the way through, like a rectangle, circle, strip, or number line. It is not a group of separate objects. When we partition a continuous model, we divide it into parts that are all the same size.

This is very important for fractions. A fraction tells us how many equal parts of a whole we have. If the parts are not equal, then the fraction is not shown correctly.

For example, when a shape is cut into 4 equal parts, each part is called one fourth, or \(\frac{1}{4}\). If the 4 parts are different sizes, they are not fourths.

Why Equal Parts Matter

Fractions are fair shares. Equal parts make the shares fair.

• If 1 pizza is split into 2 equal pieces, each piece is \(\frac{1}{2}\).
• If 1 candy bar is split into 4 equal pieces, each piece is \(\frac{1}{4}\).
• If a number line from 0 to 1 is split into 3 equal lengths, each length is \(\frac{1}{3}\).

When we partition, we ask: Are all the parts equal? If yes, then the model shows fractions correctly.

Continuous Models You May See

There are different kinds of continuous models.

• Shapes like rectangles, squares, and circles
• Bars or strips like a ribbon or a fraction bar
• Number lines that show equal lengths between numbers

Even though these models look different, the rule is the same: partition into equal parts.

How to Partition a Shape

1. Start with one whole.
2. Decide how many equal parts you need.
3. Draw lines to split the whole into that many parts.
4. Check that every part is the same size.

If a rectangle is partitioned into 3 equal strips, each strip is \(\frac{1}{3}\) of the rectangle.

If a circle is partitioned into 6 equal slices, each slice is \(\frac{1}{6}\) of the circle.

How to Partition a Number Line

A number line is a continuous model too. The space between 0 and 1 is one whole length.

To partition a number line:

1. Find the interval you are working with, like from 0 to 1.
2. Decide how many equal parts to make.
3. Place marks so the lengths between marks are equal.
4. Name the points with fractions.

If the distance from 0 to 1 is split into 4 equal parts, the points are:

\(\frac{1}{4}, \frac{2}{4}, \frac{3}{4}, 1\)

Each jump is the same length, so each jump is one fourth.

Important Idea: The Denominator

The denominator tells how many equal parts the whole is split into.

• In \(\frac{1}{2}\), the whole is split into 2 equal parts.
• In \(\frac{1}{3}\), the whole is split into 3 equal parts.
• In \(\frac{1}{5}\), the whole is split into 5 equal parts.

The denominator does not just mean the number of pieces. It means the number of equal pieces.

Worked Example 1: Partitioning a Rectangle

A rectangle needs to be partitioned into 4 equal parts. What does each part represent?

Step 1: We start with 1 whole rectangle.

Step 2: We split it into 4 equal sections.

Step 3: Since there are 4 equal parts, each part is \(\frac{1}{4}\).

Answer: Each part represents one fourth, or \(\frac{1}{4}\).

If 3 of those parts are shaded, the shaded amount is:

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

Worked Example 2: Is This Correct Partitioning?

A circle is cut into 3 pieces, but one piece is much bigger than the other two. Does each piece represent \(\frac{1}{3}\)?

No. The pieces are not equal in size.

To show thirds, the circle must be cut into 3 equal parts.

Answer: This is not a correct model for \(\frac{1}{3}\).

Worked Example 3: Partitioning a Number Line

Partition the number line from 0 to 1 into 5 equal parts. What fraction names each point?

Step 1: The whole interval is from 0 to 1.

Step 2: We divide that distance into 5 equal lengths.

Step 3: Each jump is \(\frac{1}{5}\).

The points are:

• First mark: \(\frac{1}{5}\)
• Second mark: \(\frac{2}{5}\)
• Third mark: \(\frac{3}{5}\)
• Fourth mark: \(\frac{4}{5}\)
• End point: \(\frac{5}{5}=1\)

Answer: The number line is partitioned into fifths.

Worked Example 4: Finding the Mistake

A strip is divided by 3 lines, making 4 parts. A student says, “Each part is \(\frac{1}{4}\).” Is the student always correct?

Not always.

Making 4 parts does not automatically mean fourths. The 4 parts must be equal in size.

If the strip is split into 4 equal parts, then each part is \(\frac{1}{4}\).

If the parts are unequal, then the strip does not show fourths.

Answer: The student is correct only if the 4 parts are equal.

Tips for Checking Equal Partitioning

• Count the parts.
• Check whether all parts are the same size.
• On a number line, check whether the spaces between marks are the same length.
• Remember: equal parts, not just any parts.

Common Mistakes

• Counting pieces that are not equal: 4 unequal pieces are not fourths.
• Forgetting the whole: You must know what one whole is before naming fractions.
• Uneven marks on a number line: Fraction marks must be evenly spaced.

Practice Thinking

Ask yourself these questions when you see a fraction model:

• What is the whole?
• How many equal parts is the whole split into?
• What fraction is one part?
• Are the parts really equal?

Summary

Partitioning continuous models means dividing a connected whole, like a shape or number line, into equal parts.

Equal parts are the key to understanding fractions. If a whole is partitioned into \(n\) equal parts, each part is \(\frac{1}{n}\).

Whether you are working with rectangles, circles, bars, or number lines, always check that the parts are the same size. That is how you know the fraction model is correct.

Fractions of a Discrete Set

A fraction can tell us about part of a whole object, like part of a pizza. But a fraction can also tell us about part of a group of separate objects.

A group of separate objects is called a discrete set. For example, 12 crayons, 8 apples, or 20 buttons are all discrete sets because we can count each object one by one.

When we find a fraction of a discrete set, we are finding how many objects are in that part of the group.

Example idea: If you want to find \(\frac{1}{4}\) of 12 stars, you are finding 1 out of 4 equal parts of the 12 stars.

How to find a fraction of a set

1. Look at the denominator. The denominator is the bottom number. It tells how many equal groups to make.
2. Split the set into equal groups. Share the objects equally.
3. Look at the numerator. The numerator is the top number. It tells how many of those equal groups to count.

You can think of it like this:

Important idea: The objects must be split into equal groups. Fractions are fair shares.

Worked Example 1: Finding \(\frac{1}{2}\) of a set

Find \(\frac{1}{2}\) of 10 cookies.

Step 1: The denominator is 2, so split 10 cookies into 2 equal groups.

$$10 \div 2 = 5$$

Each group has 5 cookies.

Step 2: The numerator is 1, so take 1 group.

So, \(\frac{1}{2}\) of 10 is 5.

Worked Example 2: Finding \(\frac{1}{4}\) of a set

Find \(\frac{1}{4}\) of 12 pencils.

Step 1: The denominator is 4, so split 12 pencils into 4 equal groups.

$$12 \div 4 = 3$$

Each group has 3 pencils.

Step 2: The numerator is 1, so take 1 group.

So, \(\frac{1}{4}\) of 12 is 3.

Worked Example 3: Finding \(\frac{2}{3}\) of a set

Find \(\frac{2}{3}\) of 15 marbles.

Step 1: The denominator is 3, so split 15 marbles into 3 equal groups.

$$15 \div 3 = 5$$

Each group has 5 marbles.

Step 2: The numerator is 2, so take 2 groups.

$$5 + 5 = 10$$

So, \(\frac{2}{3}\) of 15 is 10.

You can also think of it as:

$$\frac{1}{3} \text{ of } 15 = 5$$

$$\frac{2}{3} \text{ of } 15 = 2 \times 5 = 10$$

Worked Example 4: Finding \(\frac{3}{5}\) of a set

Find \(\frac{3}{5}\) of 20 stickers.

Step 1: The denominator is 5, so split 20 stickers into 5 equal groups.

$$20 \div 5 = 4$$

Each group has 4 stickers.

Step 2: The numerator is 3, so take 3 groups.

$$4 + 4 + 4 = 12$$

So, \(\frac{3}{5}\) of 20 is 12.

A helpful pattern

To find a fraction of a set:

• Divide by the denominator.
• Multiply by the numerator.

For example, to find \(\frac{3}{4}\) of 16:

$$16 \div 4 = 4$$

$$4 \times 3 = 12$$

So, \(\frac{3}{4}\) of 16 is 12.

Why this works

The denominator tells how many equal parts the whole set is split into. The numerator tells how many of those parts we want.

So first we find the size of one part, then we count the number of parts we need.

Let’s compare some fractions of the same set

Suppose we have 18 toy cars.

• \(\frac{1}{3}\) of 18: $$18 \div 3 = 6$$
• \(\frac{2}{3}\) of 18: $$6 \times 2 = 12$$
• \(\frac{3}{3}\) of 18: all 3 groups, so the answer is 18

This shows that bigger numerators mean more groups are counted, if the denominator stays the same.

Be careful!

• Do not just add the numerator and denominator.
• Do not forget to make equal groups.
• Always divide first by the denominator.
• Then count or multiply by the numerator.

Check your thinking

If you need to find \(\frac{2}{4}\) of 8 oranges:

1. Split 8 into 4 equal groups: $$8 \div 4 = 2$$
2. Take 2 groups: $$2 + 2 = 4$$

So, \(\frac{2}{4}\) of 8 is 4.

That makes sense because \(\frac{2}{4}\) is the same amount as half, and half of 8 is 4.

Try these ideas in your head

• \(\frac{1}{5}\) of 25 is 5, because $$25 \div 5 = 5$$
• \(\frac{2}{5}\) of 25 is 10, because $$5 \times 2 = 10$$
• \(\frac{4}{5}\) of 25 is 20, because $$5 \times 4 = 20$$

Summary

A fraction of a discrete set means part of a group of counted objects.

To find it, use the denominator to split the set into equal groups. Then use the numerator to tell how many groups to count.

Remember:

If the groups are equal, your fraction answer will make sense.

Visualizing Equivalent Fractions means showing that two different-looking fractions can name the same amount.

For example, \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent fractions. They look different, but they cover the same space and land at the same point on a number line.

In this lesson, you will learn how to see equivalent fractions using pictures and number lines. This helps fractions make sense, instead of feeling like just numbers to memorize.

First, remember what a fraction means.

• The denominator tells how many equal parts the whole is split into.
• The numerator tells how many of those parts we have.

In \(\frac{3}{4}\), the whole is split into 4 equal parts, and 3 parts are chosen.

Equivalent fractions are fractions that are equal in value, even if the numbers are different.

Here are some equivalent fractions:

• \(\frac{1}{2} = \frac{2}{4}\)
• \(\frac{2}{3} = \frac{4}{6}\)
• \(\frac{3}{4} = \frac{6}{8}\)

They are equal because they show the same amount of the same whole.

One way to visualize equivalent fractions is with shape models.

Imagine one rectangle split into 2 equal parts. If 1 part is shaded, that shows \(\frac{1}{2}\).

Now imagine the same-sized rectangle split into 4 equal parts. If 2 parts are shaded, that shows \(\frac{2}{4}\).

Even though the second rectangle has more pieces, the shaded amount is still the same. So:

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

This works because each half can be split into 2 smaller equal pieces. Then 1 shaded half becomes 2 shaded fourths.

You can also use circles, strips, or squares. The important idea is that the wholes must be the the same size.

If the wholes are not the same size, you cannot compare the fractions fairly.

Another way to visualize equivalent fractions is on a number line.

A number line shows where fractions are located between 0 and 1.

If two fractions are equivalent, they land on the same point on the number line.

For example, mark 0 and 1 on a number line.

If you split the space from 0 to 1 into 2 equal parts, the middle point is \(\frac{1}{2}\).

If you split that same space from 0 to 1 into 4 equal parts, the second mark is \(\frac{2}{4}\).

Both fractions are at the same location. That shows:

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

Parallel number lines can help you compare fractions even more clearly.

Think of two number lines stacked one above the other.

• The top number line is divided into 2 equal parts.
• The bottom number line is divided into 4 equal parts.

On the top line, \(\frac{1}{2}\) is the middle point.

On the bottom line, \(\frac{2}{4}\) is also the middle point.

When the points line up straight above and below each other, the fractions are equivalent.

Why do equivalent fractions happen?

Equivalent fractions happen when we split each part into smaller equal parts, but the total amount stays the same.

For example, start with \(\frac{1}{3}\). If each third is split into 2 equal pieces, the whole now has 6 equal pieces. The shaded 1 third becomes 2 sixths.

So:

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

You are not changing the amount. You are only changing how the whole is partitioned.

A helpful pattern is this:

If you multiply the numerator and denominator by the same number, you get an equivalent fraction.

For example:

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

And:

$$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$$

This matches what we see in pictures and on number lines.

Worked Example 1: Using a shape model

Are \(\frac{1}{2}\) and \(\frac{2}{4}\) equivalent?

1. Draw one rectangle split into 2 equal parts. Shade 1 part.
2. Draw another same-sized rectangle split into 4 equal parts. Shade 2 parts.
3. Compare the shaded areas.

The shaded amounts are the same, so the fractions are equivalent.

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

Worked Example 2: Using a number line

Are \(\frac{2}{3}\) and \(\frac{4}{6}\) equivalent?

1. Draw a number line from 0 to 1.
2. Split one line into 3 equal parts and mark \(\frac{2}{3}\).
3. Split another line into 6 equal parts and mark \(\frac{4}{6}\).
4. See if the points line up.

They land at the same point, so they are equivalent.

$$\frac{2}{3} = \frac{4}{6}$$

Worked Example 3: Looking carefully at a non-example

Are \(\frac{1}{2}\) and \(\frac{1}{4}\) equivalent?

Let’s compare them.

• \(\frac{1}{2}\) means 1 out of 2 equal parts.
• \(\frac{1}{4}\) means 1 out of 4 equal parts.

On a shape model, half of a rectangle is larger than one fourth of the same rectangle.

On a number line, \(\frac{1}{2}\) is farther from 0 than \(\frac{1}{4}\).

So these fractions are not equivalent.

$$\frac{1}{2} \ne \frac{1}{4}$$

Worked Example 4: Building a new equivalent fraction

Find an equivalent fraction for \(\frac{3}{4}\).

We can split each fourth into 2 equal smaller parts. Then the whole has 8 equal parts instead of 4.

The 3 shaded fourths become 6 shaded eighths.

So:

$$\frac{3}{4} = \frac{6}{8}$$

On parallel number lines, \(\frac{3}{4}\) on fourths and \(\frac{6}{8}\) on eighths land at the same point.

How to check if fractions are equivalent

• Use the same-sized whole.
• Draw models and compare shaded parts.
• Place both fractions on number lines from 0 to 1.
• See whether they cover the same amount or land at the same point.

Important things to remember

• Equivalent fractions look different but mean the same amount.
• The whole must be the same size when comparing fractions.
• More pieces does not always mean more amount.
• If both the numerator and denominator change in the same way, the fraction can stay equal.

Quick practice ideas

• Draw \(\frac{1}{3}\) and \(\frac{2}{6}\). Do they shade the same amount?
• Mark \(\frac{2}{4}\) and \(\frac{1}{2}\) on number lines. Do they land at the same spot?
• Draw \(\frac{4}{8}\) and decide which simpler fraction it matches.

Summary

Equivalent fractions are fractions that name the same amount. You can visualize them with shape models by shading the same-sized whole in different ways. You can also use number lines, where equivalent fractions land on the same point. When you understand the pictures, the numbers make much more sense.

Generating Equivalent Fractions via Multiplication

Fractions can look different but still name the same amount. These are called equivalent fractions.

For example, if you cut a sandwich into 2 equal parts and eat 1 part, you ate \(\frac{1}{2}\). If you cut that same sandwich into 4 equal parts, eating 2 parts is \(\frac{2}{4}\). The pieces are smaller, but the total amount eaten is still the same.

That means:

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

In this lesson, you will learn how to make equivalent fractions by multiplying.

What does equivalent mean?

Equivalent means equal in value. Two fractions are equivalent if they are the same size, even if the numbers in the fraction are different.

Here are some equivalent fractions:

• \(\frac{1}{2}=\frac{2}{4}\)
• \(\frac{1}{3}=\frac{2}{6}\)
• \(\frac{3}{4}=\frac{6}{8}\)

The rule for generating equivalent fractions

To make an equivalent fraction, multiply the numerator and the denominator by the same whole number.

Remember:

• The numerator is the top number.
• The denominator is the bottom number.

If we start with \(\frac{a}{b}\), then multiplying both parts by the same number gives:

$$\frac{a}{b}=\frac{a\times n}{b\times n}$$

This works because you are really multiplying the fraction by \(1\).

For example, if you multiply by \(\frac{2}{2}\), that equals \(1\), because:

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

And multiplying by 1 does not change the value. So:

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

Since \(\frac{2}{2}=1\), the value stays the same. Only the way it looks changes.

Why does this make sense?

Imagine 1 out of 2 equal parts is shaded. That is \(\frac{1}{2}\).

If each of those 2 parts is split into 2 smaller equal parts, there are now 4 equal parts total. The shaded amount also splits into 2 smaller parts. So now 2 out of 4 parts are shaded.

The total shaded amount did not change. So:

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

You made more pieces, but you did not make more of the whole.

Steps to generate an equivalent fraction

1. Look at the fraction.
2. Choose a whole number to multiply by, such as 2, 3, or 4.
3. Multiply the numerator by that number.
4. Multiply the denominator by that same number.
5. Write the new fraction.

Worked Example 1

Make an equivalent fraction for \(\frac{1}{4}\) by multiplying by 2.

Multiply the numerator and denominator by 2:

$$\frac{1\times 2}{4\times 2}=\frac{2}{8}$$

So, \(\frac{1}{4}=\frac{2}{8}\).

Check the idea: 1 out of 4 equal parts is the same amount as 2 out of 8 equal parts.

Worked Example 2

Make an equivalent fraction for \(\frac{2}{3}\) by multiplying by 3.

Multiply both numbers by 3:

$$\frac{2\times 3}{3\times 3}=\frac{6}{9}$$

So, \(\frac{2}{3}=\frac{6}{9}\).

Important: We multiplied both the top and bottom by the same number. If we do not use the same number, the fraction will change value.

Worked Example 3

Write two equivalent fractions for \(\frac{3}{5}\).

First, multiply by 2:

$$\frac{3\times 2}{5\times 2}=\frac{6}{10}$$

Next, multiply by 4:

$$\frac{3\times 4}{5\times 4}=\frac{12}{20}$$

So, two equivalent fractions are \(\frac{6}{10}\) and \(\frac{12}{20}\).

That means:

$$\frac{3}{5}=\frac{6}{10}=\frac{12}{20}$$

Worked Example 4

Fill in the missing number:

$$\frac{4}{7}=\frac{\square}{21}$$

Ask: What happened to the denominator?

\(7\) became \(21\). Since \(7\times 3=21\), we multiply by 3.

Now multiply the numerator by 3 too:

$$4\times 3=12$$

So the missing number is 12, and:

$$\frac{4}{7}=\frac{12}{21}$$

What to remember

• Equivalent fractions name the same amount.
• To generate an equivalent fraction, multiply the numerator and denominator by the same whole number.
• This works because you are multiplying by a fraction equal to \(1\), like \(\frac{2}{2}\) or \(\frac{3}{3}\).
• The numbers change, but the value stays the same.

Common mistake to avoid

Do not multiply only the numerator or only the denominator.

For example, starting with \(\frac{1}{2}\):

• Correct: \(\frac{1\times 2}{2\times 2}=\frac{2}{4}\)
• Incorrect: changing \(\frac{1}{2}\) to \(\frac{1}{4}\)

\(\frac{1}{2}\) and \(\frac{1}{4}\) are not equal. One-half is larger than one-fourth.

Try thinking like this

When you see a fraction, you can ask:

• “What number can I multiply the top and bottom by?”
• “Did I use the same number for both?”
• “Does my new fraction name the same amount?”

Quick practice ideas

• \(\frac{1}{5}\) multiplied by 2 gives \(\frac{2}{10}\).
• \(\frac{2}{7}\) multiplied by 4 gives \(\frac{8}{28}\).
• \(\frac{5}{6}\) multiplied by 2 gives \(\frac{10}{12}\).

Summary

Equivalent fractions are fractions that are equal in value. You can generate them by multiplying the numerator and denominator by the same whole number. This is like multiplying by 1, so the amount does not change.

Generating Equivalent Fractions via Division

Fractions can look different but still mean the same amount. These are called equivalent fractions.

For example, \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{4}{8}\) are all equal. They name the same part of a whole.

In this lesson, we will learn how to make equivalent fractions by dividing. This helps us write a fraction in a simpler form.

What does dividing a fraction mean here?

A fraction has two parts:

• the numerator on top
• the denominator on the bottom

To make an equivalent fraction by division, we divide the numerator and denominator by the same number.

If we do that, the fraction keeps the same value.

For example:

Since we divided both parts by 2, the fraction is still equal. So \(\frac{6}{8}\) and \(\frac{3}{4}\) are equivalent fractions.

Important rule: You must divide the top and bottom by the same whole number.

If you divide only one part, the value changes, and it is no longer an equivalent fraction.

How do we know what number to divide by?

Look for a number that goes into both the numerator and the denominator evenly. This is called a common factor.

Here are some common factors students often use:

• 2
• 3
• 4
• 5
• 6
• 10

For example, in \(\frac{12}{16}\):

• 12 and 16 can both be divided by 2
• 12 and 16 can both be divided by 4

That means we can simplify the fraction.

Worked Example 1

Simplify \(\frac{8}{12}\).

First, find a number that divides both 8 and 12. Both can be divided by 4.

Now divide both parts by 4:

So, \(\frac{8}{12} = \frac{2}{3}\).

This means \(\frac{8}{12}\) and \(\frac{2}{3}\) are equivalent fractions.

Worked Example 2

Simplify \(\frac{10}{15}\).

10 and 15 can both be divided by 5.

Divide top and bottom by 5:

So, \(\frac{10}{15} = \frac{2}{3}\).

Worked Example 3

Simplify \(\frac{18}{24}\).

Both 18 and 24 can be divided by 6.

Divide both by 6:

So, \(\frac{18}{24} = \frac{3}{4}\).

We could also divide by 2 first:

Then divide again by 3:

This shows that sometimes a fraction can be simplified in more than one step.

Worked Example 4

Simplify \(\frac{14}{21}\).

14 and 21 can both be divided by 7.

Divide both parts by 7:

So, \(\frac{14}{21} = \frac{2}{3}\).

When is a fraction in simplest form?

A fraction is in simplest form when the numerator and denominator do not have any common factor greater than 1.

For example:

• \(\frac{2}{3}\) is in simplest form because 2 and 3 cannot both be divided by the same whole number greater than 1.
• \(\frac{3}{4}\) is in simplest form for the same reason.

How to simplify a fraction step by step

1. Look at the numerator and denominator.
2. Find a number that divides both evenly.
3. Divide the numerator and denominator by that same number.
4. Check if the new fraction can be simplified again.
5. Stop when you cannot divide both by the same number greater than 1.

Example with two steps

Simplify \(\frac{24}{36}\).

Both numbers can be divided by 2:

12 and 18 can still both be divided by 2:

6 and 9 can both be divided by 3:

So,

You might also notice that 24 and 36 can both be divided by 12 right away:

Both ways are correct.

Helpful thinking

You can think of simplifying as making the numbers smaller while keeping the fraction worth the same amount.

It is like renaming the fraction in an easier way.

Common mistakes to avoid

• Do not divide only the numerator or only the denominator.
• Do not use different numbers for the top and bottom.
• Make sure both divisions are exact whole-number divisions.
• Keep simplifying until the fraction is in simplest form.

Quick check

Is \(\frac{12}{18}\) equivalent to \(\frac{2}{3}\)?

Yes. Divide both 12 and 18 by 6:

Is \(\frac{9}{10}\) already in simplest form?

Yes. 9 and 10 do not have a common factor greater than 1, so it cannot be simplified.

Summary

Equivalent fractions have the same value, even if they look different.

To generate an equivalent fraction by division, divide the numerator and denominator by the same number.

Keep dividing by common factors until the fraction is in simplest form.

When you simplify fractions, you make them easier to read and compare, but the amount stays the same.

Fraction Comparison with Common Denominators

Fractions tell us about parts of a whole. When we compare fractions, we decide which fraction is greater, which is smaller, or if they are equal.

In this lesson, we will learn how to compare fractions that have the same denominator. This is called comparing fractions with common denominators.

When fractions have the same denominator, they are cut into the same size parts. That makes comparing them much easier.

For example, in the fractions \(\frac{3}{8}\) and \(\frac{5}{8}\), both fractions have a denominator of 8. That means both wholes are divided into 8 equal parts. Since the parts are the same size, we only need to look at how many parts each fraction has.

Important idea: If the denominators are the same, compare the numerators.

Remember:

• The denominator is the bottom number. It tells how many equal parts the whole is divided into.
• The numerator is the top number. It tells how many parts we have.

So when the denominators match, the fraction with the larger numerator is the greater fraction.

Here is the rule:

$$\text{If } \frac{a}{b} \text{ and } \frac{c}{b} \text{ have the same denominator, compare } a \text{ and } c.$$

If \(a > c\), then \(\frac{a}{b} > \frac{c}{b}\).

If \(a < c\), then \(\frac{a}{b} < \frac{c}{b}\).

If \(a = c\), then \(\frac{a}{b} = \frac{c}{b}\).

You can think of it like this: if two pizzas are each cut into 6 equal slices, then someone with 5 slices has more pizza than someone with 2 slices.

Worked Example 1

Compare \(\frac{2}{7}\) and \(\frac{5}{7}\).

1. Look at the denominators: both are 7.
2. The pieces are the same size.
3. Compare the numerators: 2 and 5.
4. Since 2 is less than 5, \(\frac{2}{7}\) is less than \(\frac{5}{7}\).

Answer: $$\frac{2}{7} < \frac{5}{7}$$

Worked Example 2

Compare \(\frac{6}{9}\) and \(\frac{4}{9}\).

1. Both denominators are 9.
2. So the parts are the same size.
3. Compare the numerators: 6 and 4.
4. Since 6 is greater than 4, \(\frac{6}{9}\) is greater than \(\frac{4}{9}\).

Answer: $$\frac{6}{9} > \frac{4}{9}$$

Worked Example 3

Compare \(\frac{3}{10}\) and \(\frac{3}{10}\).

1. Both denominators are 10.
2. Compare the numerators: 3 and 3.
3. They are the same.

Answer: $$\frac{3}{10} = \frac{3}{10}$$

Worked Example 4

Jada ate \(\frac{7}{12}\) of a sandwich. Ben ate \(\frac{9}{12}\) of a sandwich. Who ate more?

1. Both fractions have denominator 12.
2. So both sandwiches are divided into the same size parts.
3. Compare the numerators: 7 and 9.
4. Since 9 is greater than 7, Ben ate more.

Answer: $$\frac{9}{12} > \frac{7}{12}$$

How to Compare Fractions with Common Denominators

• Step 1: Check the denominators.
• Step 2: If the denominators are the same, compare the numerators.
• Step 3: Use \(>\), \(<\), or \(=\) to show the comparison.

Helpful reminder about symbols:

• \(>\) means greater than
• \(<\) means less than
• \(=\) means equal to

Think about size and number of parts

If two fractions have the same denominator, the pieces are equal in size. So the only thing that changes is how many pieces there are.

For example:

• \(\frac{1}{5}\) is less than \(\frac{4}{5}\) because 1 fifth is less than 4 fifths.
• \(\frac{8}{11}\) is greater than \(\frac{2}{11}\) because 8 elevenths is more than 2 elevenths.

Common Mistake

Some students look at both numbers in the fraction and get confused. But when the denominators are already the same, you do not need to compare the denominators again. Just compare the numerators.

For example, to compare \(\frac{5}{8}\) and \(\frac{3}{8}\), do not worry that both have 8 on the bottom. That is actually what makes the problem easier. Just compare 5 and 3.

Try thinking with a picture in your mind

Imagine two same-size chocolate bars. Each bar is broken into 8 equal pieces.

• One person has \(\frac{2}{8}\).
• Another person has \(\frac{6}{8}\).

The pieces are the same size, but 6 pieces is more than 2 pieces. So \(\frac{6}{8}\) is greater.

Quick Practice Ideas

• Compare \(\frac{1}{4}\) and \(\frac{3}{4}\)
• Compare \(\frac{7}{8}\) and \(\frac{5}{8}\)
• Compare \(\frac{6}{6}\) and \(\frac{6}{6}\)

The answers would be:

• $$\frac{1}{4} < \frac{3}{4}$$
• $$\frac{7}{8} > \frac{5}{8}$$
• $$\frac{6}{6} = \frac{6}{6}$$

Summary

Fractions with common denominators have equal-size parts. That means we can compare them by looking at the numerators.

If the numerator is bigger, the fraction is bigger. If the numerator is smaller, the fraction is smaller. If the numerators are equal, the fractions are equal.

So always remember: same denominator, compare the numerator.

Fraction Comparison with Common Numerators

Fractions tell us about parts of a whole. In a fraction, the numerator is the top number, and the denominator is the bottom number.

In this lesson, we will learn how to compare fractions that have the same numerator. This means the top numbers are equal, but the bottom numbers are different.

For example, in \(\frac{3}{4}\) and \(\frac{3}{8}\), both fractions have the same numerator: \(3\).

Big Idea: When two fractions have the same numerator, the fraction with the smaller denominator is larger.

Why does this happen? The denominator tells how many equal parts the whole is cut into. If a whole is cut into more parts, each part is smaller.

So if you take the same number of parts, but the parts are smaller, the fraction is smaller too.

Let’s think about pizzas:

• If one pizza is cut into 4 equal slices, each slice is fairly big.
• If another pizza is cut into 8 equal slices, each slice is smaller.

If you take 3 slices from each pizza, then \(\frac{3}{4}\) is more pizza than \(\frac{3}{8}\), because fourths are bigger pieces than eighths.

We can write this comparison as:

$$\frac{3}{4} > \frac{3}{8}$$

Rule to Remember:

• Same numerator
• Compare the denominators
• Smaller denominator \(\to\) larger fraction
• Larger denominator \(\to\) smaller fraction

This works because the same number of pieces is being counted, but the size of each piece changes.

How to Compare Fractions with Common Numerators

1. Check that the numerators are the same.
2. Look at the denominators.
3. Decide which denominator is smaller.
4. The fraction with the smaller denominator is greater.

Now let’s work through some examples.

Example 1: Compare \(\frac{1}{2}\) and \(\frac{1}{5}\)

The numerators are the same: both are \(1\).

Now compare the denominators: \(2\) and \(5\).

Since \(2\) is smaller than \(5\), halves are bigger than fifths.

So:

$$\frac{1}{2} > \frac{1}{5}$$

Example 2: Compare \(\frac{2}{3}\) and \(\frac{2}{6}\)

The numerators are both \(2\).

The denominators are \(3\) and \(6\).

Since \(3\) is smaller than \(6\), thirds are bigger pieces than sixths.

Taking 2 bigger pieces gives a larger fraction.

So:

$$\frac{2}{3} > \frac{2}{6}$$

Example 3: Compare \(\frac{4}{7}\) and \(\frac{4}{5}\)

The numerators are both \(4\).

The denominators are \(7\) and \(5\).

Since \(5\) is smaller than \(7\), fifths are bigger than sevenths.

So \(\frac{4}{5}\) is larger than \(\frac{4}{7}\).

We write:

$$\frac{4}{7} < \frac{4}{5}$$

Example 4: Compare \(\frac{6}{10}\) and \(\frac{6}{12}\)

The numerators match: both are \(6\).

The denominators are \(10\) and \(12\).

Since \(10\) is smaller than \(12\), tenths are bigger than twelfths.

So:

$$\frac{6}{10} > \frac{6}{12}$$

Helpful Thinking Tip

If the numerator stays the same, imagine taking the same number of pieces from two different wholes cut into different numbers of parts.

If one whole is cut into many tiny parts, each part is small. If the other whole is cut into fewer parts, each part is bigger.

So with the same numerator, you want to ask: Which fraction has bigger pieces?

Try These Ideas in Your Head

• \(\frac{5}{6}\) or \(\frac{5}{9}\)? Since \(6 < 9\), \(\frac{5}{6}\) is larger.
• \(\frac{3}{10}\) or \(\frac{3}{4}\)? Since \(4 < 10\), \(\frac{3}{4}\) is larger.
• \(\frac{7}{8}\) or \(\frac{7}{12}\)? Since \(8 < 12\), \(\frac{7}{8}\) is larger.

Common Mistake to Avoid

Do not think that a bigger denominator means a bigger fraction.

For fractions with the same numerator, a bigger denominator means the whole is split into more parts, so each part is smaller.

That means the fraction is smaller, not bigger.

For example:

$$\frac{2}{8} < \frac{2}{3}$$

Even though \(8\) is bigger than \(3\), eighths are smaller pieces than thirds.

Summary

• The numerator tells how many parts you have.
• The denominator tells how many equal parts make the whole.
• If two fractions have the same numerator, compare the denominators.
• The smaller denominator makes the larger fraction.
• The larger denominator makes the smaller fraction.

Remember: same number of pieces, bigger pieces means bigger fraction.

Fraction Comparison Using Benchmarks

Fractions can be compared in different ways. One helpful way is to use benchmarks.

Benchmarks are special fractions that are easy to understand and easy to picture. The most common benchmarks are:

• 
  strong\(0\)
  /strong
• 
  strong\(\frac{1}{2}\)
  /strong
• 
  strong\(1\)
  /strong

We can compare a fraction to one of these benchmarks to decide if it is smaller or larger than another fraction.

This is useful because you do not always need to make common denominators or draw a detailed model. Sometimes you can tell just by thinking about where each fraction is compared to 0, \(\frac{1}{2}\), or 1.

Main Idea 1: Compare to \(\frac{1}{2}\)

A very useful benchmark is \(\frac{1}{2}\). To decide if a fraction is less than, equal to, or greater than \(\frac{1}{2}\), compare the numerator to half of the denominator.

• If the numerator is less than half of the denominator, the fraction is less than \(\frac{1}{2}\).
• If the numerator is equal to half of the denominator, the fraction is equal to \(\frac{1}{2}\).
• If the numerator is greater than half of the denominator, the fraction is greater than \(\frac{1}{2}\).

For example, in \(\frac{3}{8}\), half of 8 is 4. Since 3 is less than 4, \(\frac{3}{8} < \frac{1}{2}\).

In \(\frac{5}{8}\), half of 8 is 4. Since 5 is greater than 4, \(\frac{5}{8} > \frac{1}{2}\).

Main Idea 2: Compare to 1

The fraction \(1\) means one whole. A fraction is:

• less than 1 when the numerator is less than the denominator
• equal to 1 when the numerator equals the denominator

For example:

• \(\frac{6}{7} < 1\) because 6 is less than 7
• \(\frac{7}{7} = 1\)

When two fractions are both close to 1, the one missing fewer pieces is greater.

For example, compare \(\frac{7}{8}\) and \(\frac{5}{6}\).

Both are less than 1, but:

• \(\frac{7}{8}\) is missing \(\frac{1}{8}\)
• \(\frac{5}{6}\) is missing \(\frac{1}{6}\)

Since \(\frac{1}{8}\) is a smaller missing part than \(\frac{1}{6}\), \(\frac{7}{8}\) is closer to 1. So:

$$\frac{7}{8} > \frac{5}{6}$$

Main Idea 3: Compare to 0

Fractions close to 0 are very small fractions. If two fractions are both close to 0, the one with fewer or smaller pieces may be smaller.

For example, compare \(\frac{1}{6}\) and \(\frac{1}{4}\).

Both fractions have 1 piece. But when a whole is cut into 6 equal parts, each part is smaller than when a whole is cut into 4 equal parts. So:

$$\frac{1}{6} < \frac{1}{4}$$

How to Use Benchmarks to Compare Fractions

1. Look at both fractions.
2. Ask: Is each fraction less than, equal to, or greater than \(\frac{1}{2}\)?
3. Ask: Is each fraction close to 0 or close to 1?
4. Use that information to decide which fraction is greater.

Sometimes one fraction is below \(\frac{1}{2}\) and the other is above \(\frac{1}{2}\). Then it is easy to compare them.

For example, any fraction greater than \(\frac{1}{2}\) is larger than any fraction less than \(\frac{1}{2}\).

Worked Example 1

Compare \(\frac{3}{8}\) and \(\frac{5}{8}\).

Use the benchmark \(\frac{1}{2}\).

• Half of 8 is 4.
• In \(\frac{3}{8}\), 3 is less than 4, so \(\frac{3}{8} < \frac{1}{2}\).
• In \(\frac{5}{8}\), 5 is greater than 4, so \(\frac{5}{8} > \frac{1}{2}\).

One fraction is less than \(\frac{1}{2}\), and the other is greater than \(\frac{1}{2}\). So:

$$\frac{3}{8} < \frac{5}{8}$$

Worked Example 2

Compare \(\frac{2}{5}\) and \(\frac{1}{2}\).

Half of 5 is 2 and a half. Since 2 is less than 2 and a half, \(\frac{2}{5}\) is less than \(\frac{1}{2}\).

So:

$$\frac{2}{5} < \frac{1}{2}$$

Worked Example 3

Compare \(\frac{4}{7}\) and \(\frac{5}{9}\).

Use the benchmark \(\frac{1}{2}\).

• Half of 7 is 3 and a half. Since 4 is greater than 3 and a half, \(\frac{4}{7} > \frac{1}{2}\).
• Half of 9 is 4 and a half. Since 5 is greater than 4 and a half, \(\frac{5}{9} > \frac{1}{2}\).

Both fractions are greater than \(\frac{1}{2}\), so we need to think a little more.

Let us see which one is closer to \(\frac{1}{2}\):

• \(\frac{4}{7}\) is just a little more than \(\frac{1}{2}\)
• \(\frac{5}{9}\) is also just a little more than \(\frac{1}{2}\)

Now think about their size compared to 1. Both are not very close to 1, so we compare how far above \(\frac{1}{2}\) they are. \(\frac{4}{7}\) is a little more above half than \(\frac{5}{9}\).

So:

$$\frac{4}{7} > \frac{5}{9}$$

Worked Example 4

Compare \(\frac{7}{8}\) and \(\frac{6}{7}\).

Use the benchmark \(1\).

• \(\frac{7}{8}\) is missing \(\frac{1}{8}\) to make 1.
• \(\frac{6}{7}\) is missing \(\frac{1}{7}\) to make 1.

Since \(\frac{1}{8}\) is smaller than \(\frac{1}{7}\), \(\frac{7}{8}\) is closer to 1.

So:

$$\frac{7}{8} > \frac{6}{7}$$

Helpful Tips

• If one fraction is less than \(\frac{1}{2}\) and the other is greater than \(\frac{1}{2}\), the one greater than \(\frac{1}{2}\) is larger.
• If both fractions are close to 1, compare what part is missing.
• If both fractions are unit fractions, like \(\frac{1}{3}\) and \(\frac{1}{5}\), the fraction with the larger denominator is smaller.
• Benchmarks help you think about size without always doing a lot of work.

Be Careful

Do not just look at which numerator is bigger or which denominator is bigger. You must think about the whole fraction.

For example, \(\frac{3}{5}\) and \(\frac{4}{9}\): even though 4 is bigger than 3, \(\frac{4}{9}\) is not bigger than \(\frac{3}{5}\). Use a benchmark:

• \(\frac{3}{5} > \frac{1}{2}\)
• \(\frac{4}{9} < \frac{1}{2}\)

So:

$$\frac{3}{5} > \frac{4}{9}$$

Summary

Benchmarks are easy fractions like \(0\), \(\frac{1}{2}\), and \(1\). We can compare other fractions by thinking about whether they are less than, equal to, or greater than these benchmarks.

Using benchmarks helps you understand the size of fractions quickly. It is a smart way to compare fractions and decide which one is greater or smaller.

Fractions Greater Than One

Sometimes a fraction does not name just part of one whole. Sometimes it names more than one whole. These are called fractions greater than one.

You already know that a fraction has two parts:

• The denominator tells how many equal parts make 1 whole.
• The numerator tells how many of those parts we have.

For example, in \(\frac{3}{4}\), the 4 means the whole is split into 4 equal parts, and the 3 means we have 3 of those parts.

But what if we have more parts than fit in one whole? Then the fraction can be greater than 1.

For example, in \(\frac{5}{4}\), the denominator is 4, so 1 whole is made of 4 fourths:

If we have 5 fourths, that is 1 whole and 1 more fourth. So \(\frac{5}{4}\) is greater than 1.

A fraction is greater than 1 when the numerator is larger than the denominator. That means we have enough equal parts to make at least 1 whole, and then maybe more.

Here are some examples of fractions greater than 1:

• \(\frac{3}{2}\)
• \(\frac{5}{4}\)
• \(\frac{7}{3}\)
• \(\frac{9}{8}\)

Notice that in each fraction, the numerator is bigger than the denominator.

It also helps to remember these special cases:

• If the numerator is smaller than the denominator, the fraction is less than 1.
• If the numerator is equal to the denominator, the fraction is exactly 1.
• If the numerator is greater than the denominator, the fraction is greater than 1.

For example:

• \(\frac{2}{5} < 1\)
• \(\frac{5}{5} = 1\)
• \(\frac{6}{5} > 1\)

Thinking with unit fractions

A unit fraction has 1 on top, like \(\frac{1}{4}\) or \(\frac{1}{3}\). Fractions are built from unit fractions.

For example:

Since 4 fourths make 1 whole, we can group them:

This shows clearly that \(\frac{5}{4}\) is greater than 1.

How to see fractions greater than one with pictures

Imagine circles, rectangles, or bars split into equal parts.

• If a shape is divided into 4 equal parts, then each part is \(\frac{1}{4}\).
• 4 fourths fill 1 whole shape.
• If you have 5 fourths, you fill 1 whole shape and then 1 more fourth in another shape.

So \(\frac{5}{4}\) means one full whole and part of another whole.

How to place fractions greater than one on a number line

A number line helps show where a fraction belongs.

Let us place \(\frac{5}{4}\) on a number line. From 0 to 1, split the space into 4 equal parts. Then keep going from 1 to 2 using the same-size parts.

Because \(\frac{4}{4}=1\), the next fourth after 1 is \(\frac{5}{4}\). So \(\frac{5}{4}\) is just a little more than 1.

On a number line, fractions greater than 1 are found to the right of 1.

Worked Example 1: Is the fraction greater than 1?

Look at \(\frac{7}{6}\).

1. Compare the numerator and denominator.
2. The numerator is 7.
3. The denominator is 6.
4. Since \(7 > 6\), the fraction is greater than 1.

Answer: \(\frac{7}{6} > 1\)

Worked Example 2: Find the whole and the extra part

Look at \(\frac{6}{5}\).

We know that 5 fifths make 1 whole:

Now break \(\frac{6}{5}\) into 5 fifths and 1 more fifth:

Answer: \(\frac{6}{5}\) is 1 whole and \(\frac{1}{5}\) more.

Worked Example 3: Build a fraction from unit fractions

Suppose you have 8 pieces, and each piece is \(\frac{1}{3}\).

That means the total amount is:

Since 3 thirds make 1 whole, group the thirds:

Answer: \(\frac{8}{3}\) is greater than 1. It is 2 wholes and \(\frac{2}{3}\) more.

Worked Example 4: Compare fractions to 1 on a number line

Which fraction is farther to the right: \(\frac{4}{3}\) or \(\frac{5}{3}\)?

Both fractions use thirds, so compare how many thirds there are.

• \(\frac{4}{3}=1+\frac{1}{3}\)
• \(\frac{5}{3}=1+\frac{2}{3}\)

Since \(1+\frac{2}{3}\) is more than \(1+\frac{1}{3}\), \(\frac{5}{3}\) is farther to the right.

Answer: \(\frac{5}{3}\)

Helpful steps for understanding fractions greater than one

• First, look at the denominator. It tells what kind of parts you have.
• Next, ask how many of those parts make 1 whole.
• Then, compare the numerator to the denominator.
• If the numerator is bigger, the fraction is greater than 1.
• You can also break the fraction into 1 whole plus extra parts.

Common mistakes to avoid

• Do not think every fraction is less than 1. Some fractions are more than 1.
• Do not look only at the numerator. You must compare it to the denominator.
• Remember that \(\frac{4}{4}=1\), \(\frac{3}{3}=1\), and \(\frac{8}{8}=1\). When the top and bottom are equal, the fraction is exactly 1.

Let’s review with quick checks

• Is \(\frac{2}{7}\) greater than 1? No, because \(2 < 7\).
• Is \(\frac{9}{9}\) greater than 1? No, it equals 1.
• Is \(\frac{10}{9}\) greater than 1? Yes, because \(10 > 9\).
• Is \(\frac{12}{4}\) greater than 1? Yes, because \(12 > 4\).

Summary

Fractions greater than one name amounts larger than a whole. They happen when the numerator is greater than the denominator.

You can understand them by thinking about equal parts, grouping enough parts to make 1 whole, drawing models, or placing them on a number line. When you see a fraction like \(\frac{5}{4}\), think: 4 fourths make 1 whole, so 5 fourths means 1 whole and 1 extra fourth.

Mixed numbers and improper fractions are two ways to show the same amount.

A mixed number has a whole number and a fraction, like \(2\frac{1}{3}\).

An improper fraction has a numerator greater than the denominator, like \(\frac{7}{3}\).

In this lesson, you will learn how to switch between these two forms and understand what they mean.

First, remember what a fraction means. The denominator tells how many equal parts make 1 whole. The numerator tells how many of those parts we have.

For example, in \(\frac{3}{4}\), the denominator is 4, so 1 whole is split into 4 equal parts. The numerator is 3, so we have 3 of those 4 parts.

A mixed number shows wholes and extra parts.

• In \(1\frac{2}{5}\), the 1 means 1 whole.
• The \(\frac{2}{5}\) means 2 more fifths.
• So \(1\frac{2}{5}\) means 1 whole and 2 out of 5 more equal parts.

An improper fraction shows the total number of equal parts.

For example, \(\frac{7}{5}\) means 7 fifths. Since 5 fifths make 1 whole, \(\frac{7}{5}\) is more than 1 whole.

You can think of it like this:

So \(\frac{7}{5}\) and \(1\frac{2}{5}\) are equivalent. They name the same amount.

How to change a mixed number into an improper fraction

1. Multiply the whole number by the denominator.
2. Add the numerator.
3. Keep the same denominator.

This works because each whole is made of that many fractional parts.

For example, in \(3\frac{2}{4}\):

• There are 3 wholes.
• Each whole has 4 fourths.
• So 3 wholes = \(3 \times 4 = 12\) fourths.
• Then add 2 more fourths.
• That makes \(14\) fourths.

How to change an improper fraction into a mixed number

1. Ask: How many full groups of the denominator fit into the numerator?
2. The number of full groups is the whole number.
3. The leftover parts become the numerator of the fraction.
4. Keep the same denominator.

For example, in \(\frac{11}{4}\):

• 4 fourths make 1 whole.
• 11 fourths contains 2 full groups of 4 fourths.
• That uses 8 fourths.
• There are 3 fourths left.

Worked Example 1

Change \(1\frac{3}{4}\) into an improper fraction.

Step 1: Multiply the whole number by the denominator.

Step 2: Add the numerator.

Step 3: Keep the denominator 4.

Worked Example 2

Change \(2\frac{2}{3}\) into an improper fraction.

2 wholes means 2 groups of 3 thirds.

Add the extra 2 thirds.

Keep the denominator 3.

Worked Example 3

Change \(\frac{9}{2}\) into a mixed number.

2 halves make 1 whole. Now see how many groups of 2 are in 9.

• 8 makes 4 groups of 2.
• 1 is left over.

So there are 4 wholes and 1 half left.

Worked Example 4

Change \(\frac{14}{5}\) into a mixed number.

5 fifths make 1 whole. Find how many groups of 5 are in 14.

• 10 is 2 groups of 5.
• 4 is left over.

So there are 2 wholes and 4 fifths left.

A quick way to think about it

• Mixed number to improper fraction: whole number parts + extra fraction parts
• Improper fraction to mixed number: make as many wholes as you can, then write the leftover part

Watch out for these common mistakes:

• Do not add the denominator when changing a mixed number to an improper fraction. Only multiply the whole number by the denominator, then add the numerator.
• Do not change the denominator. The size of the parts stays the same.
• When changing to a mixed number, the fraction part must be less than 1 whole. That means the numerator should be smaller than the denominator.

Try thinking about wholes and parts.

Suppose you have \(\frac{13}{6}\). Since 6 sixths make 1 whole, 12 sixths make 2 wholes, and 1 sixth is left.

Suppose you have \(4\frac{1}{2}\). Four wholes means 8 halves, and 1 more half makes 9 halves.

Summary

Mixed numbers and improper fractions can show the same quantity in different ways. To change a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the denominator. To change an improper fraction to a mixed number, make as many wholes as possible and write the leftover fraction.

Decomposing Fractions into Sums means breaking one fraction into smaller fractions that add up to the same amount.

For example, the fraction \(\frac{3}{5}\) can be broken into:

It can also be broken into other sums, like:

All of these are correct because the pieces still make the same whole fraction.

Big idea: When fractions have the same denominator, you can add the numerators.

The denominator tells how many equal parts the whole is split into. The numerator tells how many of those parts we have.

• The denominator stays the same when we decompose into fractions with like denominators.
• The numerators must add up to the original numerator.

So if you want to decompose \(\frac{5}{6}\), you need fractions with denominator 6, and their numerators must add to 5.

Some possible ways are:

Each sum equals \(\frac{5}{6}\).

Unit fractions are fractions with 1 on top, like \(\frac{1}{4}\) or \(\frac{1}{9}\).

Every fraction can be written as a sum of unit fractions. This helps us understand what the fraction really means.

For example:

This shows that \(\frac{4}{7}\) means 4 copies of \(\frac{1}{7}\).

We can also decompose fractions in different ways, not just into unit fractions.

Here are some important steps to follow:

1. Look at the denominator.
2. Keep that denominator the same in each part.
3. Break the numerator into smaller numbers that add to the original numerator.
4. Write each part as a fraction.

Let’s practice with worked examples.

Example 1: Decompose \(\frac{2}{5}\)

The denominator is 5, so each fraction in the sum must have denominator 5.

The numerator is 2, so we need numbers that add to 2.

One way is 1 and 1.

This is a sum of unit fractions.

Example 2: Decompose \(\frac{4}{6}\) in two different ways

The denominator stays 6.

The numerator 4 can be broken into 1 and 3.

The numerator 4 can also be broken into 2 and 2.

Both are correct because \(1+3=4\) and \(2+2=4\).

Example 3: Decompose \(\frac{5}{8}\) as a sum of unit fractions

We need 5 copies of \(\frac{1}{8}\).

This works because adding five eighths-sized pieces gives \(\frac{5}{8}\).

Example 4: Find two ways to decompose \(\frac{7}{9}\)

The denominator is 9, so all parts must be ninths.

One way is to split 7 into 3 and 4.

Another way is to split 7 into 1, 2, and 4.

Both sums equal \(\frac{7}{9}\).

Helpful picture idea: Imagine a rectangle split into 8 equal parts. If 5 parts are shaded, that is \(\frac{5}{8}\). You can think of the shaded parts as:

• 5 single parts: \(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\)
• 2 parts and 3 parts: \(\frac{2}{8}+\frac{3}{8}\)

The total shaded amount does not change. Only the way we describe it changes.

Watch out for these mistakes:

• Do not change the denominator when decomposing into sums with like denominators.
• Make sure the numerators add correctly. For example, \(\frac{3}{7}\neq\frac{1}{7}+\frac{1}{7}\), because \(1+1=2\), not 3.
• Check your total. Ask, “Do these fractions together make the original fraction?”

Let’s look at one more quick check:

This is correct because the denominator stays 10 and \(2+4=6\).

Here is another correct decomposition:

That is correct because \(1+1+1+3=6\).

Summary: To decompose a fraction into a sum, keep the denominator the same and break the numerator into parts that add to the original numerator. A fraction can be decomposed in many different ways. Writing fractions as sums helps us understand them better.

Adding Fractions with Like Denominators

Fractions help us show parts of a whole. When we add fractions, we are putting parts together.

In this lesson, you will learn how to add fractions that have the same denominator. These are called like denominators.

For example, in the fraction \(\frac{3}{8}\), the top number is the numerator. It tells how many parts we have. The bottom number is the denominator. It tells how many equal parts the whole is split into.

When fractions have the same denominator, the pieces are the same size. That means we can add the numerators and keep the denominator the same.

Main Idea:

To add fractions with like denominators:

• Add the numerators.
• Keep the denominator the same.

We can write the rule like this:

This works because the denominator tells the size of the parts, and the size of the parts does not change when we add them together.

Think of it like slices of a pizza. If each slice is \(\frac{1}{6}\) of the pizza, then adding 2 slices and 3 more slices gives 5 slices, and each slice is still \(\frac{1}{6}\) of the pizza.

Step-by-Step Method

1. Check that the denominators are the same.
2. Add the numerators.
3. Write the sum over the same denominator.
4. If needed, decide whether the answer is less than 1 whole, equal to 1 whole, or more than 1 whole.

Worked Example 1

Add \(\frac{1}{5}+\frac{2}{5}\).

The denominators are both 5, so the pieces are the same size.

Add the numerators: \(1+2=3\).

Keep the denominator 5.

So, the answer is \(\frac{3}{5}\).

Worked Example 2

Add \(\frac{3}{8}+\frac{4}{8}\).

The denominators are both 8, so we can add the numerators.

\(3+4=7\)

Keep the denominator 8.

So, the answer is \(\frac{7}{8}\).

Worked Example 3

Add \(\frac{4}{10}+\frac{5}{10}\).

The denominators are the same, so add the numerators: \(4+5=9\).

Keep the denominator 10.

So, the answer is \(\frac{9}{10}\).

Worked Example 4

Add \(\frac{3}{6}+\frac{4}{6}\).

The denominators are both 6, so add the numerators.

\(3+4=7\)

Keep the denominator 6.

This answer is more than 1 whole because 7 sixths is more than 6 sixths.

We know that:

So \(\frac{7}{6}\) means 1 whole and \(\frac{1}{6}\).

Important Things to Remember

• The denominator tells the size of the pieces.
• If the denominators are the same, the pieces are the same size.
• Only the numerators get added.
• The denominator stays the same.

A Quick Visual Idea

Imagine a bar split into 4 equal parts.

If you shade \(\frac{1}{4}\) and then shade \(\frac{2}{4}\) more, you have shaded 3 of the 4 equal parts.

You are counting how many equal parts are shaded. The size of each part stays \(\frac{1}{4}\).

Common Mistake

Some students try to add both the top and bottom numbers. That is not correct for adding fractions with like denominators.

For example:

Instead, keep the denominator the same and only add the numerators:

Let’s Review with a Real-Life Example

Mia drank \(\frac{2}{8}\) of a bottle of juice in the morning and \(\frac{3}{8}\) of the bottle in the afternoon. How much did she drink in all?

The denominators are both 8, so add the numerators.

Mia drank \(\frac{5}{8}\) of the bottle in all.

Summary

When you add fractions with like denominators, you are adding parts that are the same size. Add the numerators and keep the denominator the same.

Always check that the denominators match first. If they do, adding is simple: count how many equal parts you have altogether.

Subtracting Fractions with Like Denominators

Fractions tell us about parts of a whole. When we subtract fractions, we are finding out how much is left after some parts are taken away.

In this lesson, we will learn how to subtract fractions that have like denominators. Like denominators means the fractions have the same bottom number.

For example, in \(\frac{5}{8}\) and \(\frac{2}{8}\), both fractions have 8 on the bottom. That means they are divided into the same size parts, so we can subtract them easily.

What the denominator and numerator mean

• The denominator is the bottom number. It tells how many equal parts the whole is split into.
• The numerator is the top number. It tells how many parts we have.

When fractions have the same denominator, the pieces are the same size. So, when we subtract, we only subtract the number of pieces we have. The size of the pieces stays the same.

That means:

We subtract the numerators and keep the denominator the same.

Why this works

Imagine a pizza cut into 6 equal slices. If you have \(\frac{5}{6}\) of the pizza and eat \(\frac{2}{6}\), you started with 5 sixths and took away 2 sixths. That leaves 3 sixths.

The slices are still sixths, so the denominator stays 6.

Using a number line

A number line can help you see fraction subtraction. Suppose you start at \(\frac{5}{8}\). If you subtract \(\frac{2}{8}\), you move left 2 eighths. You land on \(\frac{3}{8}\).

This shows that subtraction means moving backward by the amount being taken away.

Steps for subtracting fractions with like denominators

1. Check that the denominators are the same.
2. Subtract the numerators.
3. Keep the denominator the same.
4. If possible, write the answer in simplest form.

Worked Example 1

Find \(\frac{4}{7} - \frac{1}{7}\).

Step 1: The denominators are the same, so we can subtract.

Step 2: Subtract the numerators: \(4 - 1 = 3\).

Step 3: Keep the denominator 7.

Answer: \(\frac{3}{7}\)

Worked Example 2

Find \(\frac{6}{9} - \frac{4}{9}\).

The denominators are both 9, so subtract the top numbers.

Answer: \(\frac{2}{9}\)

Worked Example 3

Find \(\frac{7}{10} - \frac{3}{10}\).

Subtract the numerators and keep the denominator the same.

This answer can be simplified because 4 tenths is the same as 2 fifths.

Answer: \(\frac{4}{10}\), or in simplest form, \(\frac{2}{5}\)

Worked Example 4

A ribbon is \(\frac{8}{12}\) of a meter long. Maria cuts off \(\frac{5}{12}\) of a meter. How much ribbon is left?

We subtract the fractions because some ribbon is taken away.

So \(\frac{3}{12}\) of a meter is left.

This can also be simplified to \(\frac{1}{4}\).

Answer: \(\frac{3}{12}\), or \(\frac{1}{4}\)

Let’s connect this to a picture in your mind

Think of a bar split into 8 equal parts. If 6 parts are shaded, that is \(\frac{6}{8}\). If 2 shaded parts are taken away, 4 shaded parts remain. So:

You are removing some of the same-sized parts, so the denominator does not change.

Common mistake to avoid

Do not subtract the denominators.

For example, this is not correct:

Instead, keep the denominator the same:

The denominator tells the size of the parts. The size of the parts does not change when you take some away.

Helpful tips

• If the bottom numbers match, subtract the top numbers.
• Keep the denominator the same.
• Use a picture or number line if you want to see the parts being taken away.
• Check whether your answer can be simplified.

Quick practice to think about

• \(\frac{5}{6} - \frac{1}{6} = \frac{4}{6}\)
• \(\frac{9}{11} - \frac{3}{11} = \frac{6}{11}\)
• \(\frac{7}{8} - \frac{7}{8} = 0\)

In the last problem, all the parts were taken away, so nothing is left.

Summary

To subtract fractions with like denominators, make sure the denominators are the same. Then subtract the numerators and keep the denominator the same.

You can think about fraction subtraction as taking away equal-size parts from a whole. A number line or picture can help you see what is left.

Adding Mixed Numbers with Like Denominators

A mixed number has a whole number and a fraction together, like \(2\frac{3}{8}\) or \(5\frac{1}{4}\).

In this lesson, you will learn how to add mixed numbers when the denominators are the same. The denominator is the bottom number in a fraction.

When mixed numbers have like denominators, it means the fractions have the same bottom number. That makes adding easier, because you can add the fraction parts without changing the denominator.

For example, in \(3\frac{2}{5} + 1\frac{1}{5}\), both fractions have denominator 5. So these are mixed numbers with like denominators.

Big idea: Add the whole numbers. Then add the fractions. If the fraction part is large enough to make a whole, regroup it.

Step-by-step method

1. Add the whole numbers.
2. Add the fractions.
3. Keep the same denominator.
4. If the fraction is an improper fraction (the top number is greater than or equal to the bottom number), change it into a whole number and a fraction.
5. Add any new whole number to the whole-number sum.
6. Simplify if needed.

Here is the pattern:

Then check if \(\frac{b+d}{n}\) can make one or more whole numbers.

Important reminder: When the denominators are the same, you add the numerators, but the denominator stays the same.

You do not add the denominators.

Worked Example 1: No regrouping needed

Add \(2\frac{1}{6} + 3\frac{4}{6}\).

First, add the whole numbers:

Next, add the fractions:

Put them together:

Answer: \(2\frac{1}{6} + 3\frac{4}{6} = 5\frac{5}{6}\)

In this example, the fraction part stayed less than 1 whole, so we did not need to regroup.

Worked Example 2: Regrouping to make 1 whole

Add \(4\frac{3}{8} + 2\frac{6}{8}\).

First, add the whole numbers:

Next, add the fractions:

The fraction \(\frac{9}{8}\) is greater than 1 whole. Since \(\frac{8}{8} = 1\), we can break \(\frac{9}{8}\) into:

Now add that extra whole number to the whole-number sum:

Answer: \(4\frac{3}{8} + 2\frac{6}{8} = 7\frac{1}{8}\)

This is called regrouping. The fractions made a new whole.

Worked Example 3: Fraction sum equals exactly 1 whole

Add \(1\frac{2}{5} + 3\frac{3}{5}\).

Add the whole numbers:

Add the fractions:

And \(\frac{5}{5} = 1\).

So add that 1 whole to the whole-number sum:

Answer: \(1\frac{2}{5} + 3\frac{3}{5} = 5\)

Sometimes the answer is a whole number with no fraction part at all.

Worked Example 4: Simplify after adding

Add \(3\frac{1}{4} + 2\frac{1}{4}\).

Add the whole numbers:

Add the fractions:

Put them together:

The fraction \(\frac{2}{4}\) can be simplified to \(\frac{1}{2}\).

Answer: \(3\frac{1}{4} + 2\frac{1}{4} = 5\frac{1}{2}\)

What regrouping means

Think of the fraction part as pieces of a whole. If the pieces add up to enough parts to make a whole, trade those parts for 1 whole.

For example, with eighths:

• \(8/8 = 1\) whole
• \(9/8 = 1\frac{1}{8}\)
• \(13/8 = 1\frac{5}{8}\)

So when adding mixed numbers, always check the fraction sum to see if it can make a whole.

Tips to remember

• Add whole numbers and fractions separately.
• If denominators are the same, keep that denominator.
• Add only the top numbers in the fractions.
• If the fraction is \(\geq 1\), regroup.
• Simplify the fraction if you can.

Common mistakes to avoid

• Mistake: Adding the denominators.
  Correct: In \(\frac{2}{6} + \frac{3}{6}\), the answer is \(\frac{5}{6}\), not \(\frac{5}{12}\).
• Mistake: Forgetting to add the extra whole after regrouping.
  Correct: \(6 + \frac{9}{8} = 6 + 1\frac{1}{8} = 7\frac{1}{8}\).
• Mistake: Forgetting to simplify.
  Correct: \(\frac{2}{4}\) should be written as \(\frac{1}{2}\).

Let’s review the steps one more time

1. Look at the denominators. Make sure they are the same.
2. Add the whole numbers.
3. Add the fractions.
4. If the fraction makes a whole or more, regroup.
5. Write the answer as a mixed number or whole number.
6. Simplify if possible.

Summary

To add mixed numbers with like denominators, add the whole numbers and add the fractions. Keep the same denominator for the fraction part.

If the fraction sum is equal to or greater than 1 whole, regroup by changing it into a whole number and a fraction. Then add that new whole to your answer.

With practice, you will get faster at seeing when the fractions make a whole. Remember: whole numbers together, fractions together, then regroup if needed.

Subtracting Mixed Numbers with Like Denominators

Sometimes we need to subtract one mixed number from another mixed number.

A mixed number has a whole number and a fraction, like \(5\frac{2}{6}\) or \(3\frac{4}{6}\).

In this lesson, we will learn how to subtract mixed numbers when the fractions have the same denominator. We will also learn what to do when the top fraction is too small and we need to borrow or decompose one whole.

Step 1: Check the denominators.

If the denominators are the same, we can work with the fractions more easily. The denominator tells how many equal parts make one whole.

For example, in \(\frac{5}{8}\), the 8 means one whole is split into 8 equal parts.

Step 2: Subtract the whole numbers and fractions separately.

If the fraction on top is large enough, just subtract the whole numbers and subtract the fractions.

Example: $$6\frac{5}{7}-2\frac{3}{7}$$

The denominators are both 7, so they are like denominators.

Subtract the whole numbers: \(6-2=4\)

Subtract the fractions: \(\frac{5}{7}-\frac{3}{7}=\frac{2}{7}\)

So the answer is:

What if the top fraction is smaller?

Look at this problem:

We cannot do \(\frac{1}{5}-\frac{3}{5}\) because there are not enough fifths.

So we decompose one whole from the 4. This means we take 1 away from the whole number and turn it into fifths.

Since one whole equals \(\frac{5}{5}\), we can rewrite:

Why? Because we took 1 whole from 4, leaving 3 wholes, and added \(\frac{5}{5}\) to \(\frac{1}{5}\):

Now subtract:

Whole numbers: \(3-2=1\)

Fractions: \(\frac{6}{5}-\frac{3}{5}=\frac{3}{5}\)

So:

Important idea: When you borrow 1 whole, turn that whole into a fraction with the same denominator.

• If the denominator is 4, then 1 whole is \(\frac{4}{4}\).
• If the denominator is 6, then 1 whole is \(\frac{6}{6}\).
• If the denominator is 9, then 1 whole is \(\frac{9}{9}\).

Steps for subtracting mixed numbers with like denominators

1. Check that the denominators are the same.
2. Compare the fractions.
3. If the top fraction is greater than or equal to the bottom fraction, subtract normally.
4. If the top fraction is smaller, decompose 1 whole from the first mixed number.
5. Rewrite that 1 whole as a fraction with the same denominator.
6. Subtract the whole numbers and the fractions.

Worked Example 1

Solve: $$7\frac{4}{9}-3\frac{2}{9}$$

The denominators are both 9.

The top fraction \(\frac{4}{9}\) is bigger than \(\frac{2}{9}\), so no borrowing is needed.

Subtract the whole numbers: \(7-3=4\)

Subtract the fractions: \(\frac{4}{9}-\frac{2}{9}=\frac{2}{9}\)

Answer:

Worked Example 2

Solve: $$5\frac{2}{6}-1\frac{5}{6}$$

The denominators are both 6.

The top fraction \(\frac{2}{6}\) is smaller than \(\frac{5}{6}\), so we need to decompose 1 whole.

Rewrite \(5\frac{2}{6}\) as \(4\frac{8}{6}\).

That works because:

Now subtract:

Whole numbers: \(4-1=3\)

Fractions: \(\frac{8}{6}-\frac{5}{6}=\frac{3}{6}\)

Answer:

Worked Example 3

Solve: $$8\frac{3}{4}-6\frac{3}{4}$$

The denominators are both 4.

The fractions are equal, so subtract them:

Whole numbers: \(8-6=2\)

Fractions: \(\frac{3}{4}-\frac{3}{4}=\frac{0}{4}=0\)

Answer:

Worked Example 4

Solve: $$6\frac{1}{8}-2\frac{7}{8}$$

The denominators are both 8.

The top fraction \(\frac{1}{8}\) is smaller than \(\frac{7}{8}\), so decompose 1 whole from 6.

Rewrite:

Now subtract:

Whole numbers: \(5-2=3\)

Fractions: \(\frac{9}{8}-\frac{7}{8}=\frac{2}{8}\)

So:

Helpful reminder

When you decompose, you are not changing the amount. You are only rewriting it in a way that makes subtraction easier.

For example:

These are equal amounts. They just look different.

Common mistakes to avoid

• Do not subtract the denominators. Keep the denominator the same.
• Do not forget to lower the whole number by 1 when you borrow.
• Do not forget to add the denominator as fractional parts when you decompose one whole.

For example, with \(5\frac{2}{6}\), if you borrow 1 whole, the 5 becomes 4, and the fraction becomes \(\frac{8}{6}\), not \(\frac{3}{6}\).

Let’s review the big idea

Subtract mixed numbers with like denominators by looking at the fraction parts first.

If the first fraction is big enough, subtract normally.

If it is too small, decompose one whole into fractional parts with the same denominator, then subtract.

Summary

Mixed numbers have a whole number and a fraction. When subtracting mixed numbers with the same denominator, subtract the whole numbers and fractions separately if possible.

If the top fraction is smaller than the bottom fraction, decompose 1 whole from the first mixed number. Change that whole into a fraction with the same denominator, add it to the fraction part, and then subtract.

Lesson: Multiplying a Fraction by a Whole Number Conceptually

Sometimes in math, we need to add the same fraction again and again. Instead of writing a long addition sentence, we can use multiplication.

When we multiply a fraction by a whole number, we are finding repeated groups of the same fraction. This lesson will help you understand what that means using pictures in your mind, repeated addition, and simple number patterns.

Important idea: Multiplying a fraction by a whole number means adding that fraction over and over.

For example, \(3 \times \frac{1}{4}\) means:

This is 3 groups of one-fourth.

Before we begin, remember:

• A unit fraction has 1 on top, like \(\frac{1}{2}\), \(\frac{1}{5}\), or \(\frac{1}{8}\).
• A fraction tells how many equal parts we have.
• The denominator is the bottom number. It tells how many equal parts make one whole.
• The numerator is the top number. It tells how many parts we have.

Main Idea 1: Multiply a unit fraction by a whole number

Let’s look at \(4 \times \frac{1}{3}\).

This means 4 groups of one-third:

When we add unit fractions with the same denominator, the denominator stays the same, and we add the numerators:

So:

This answer is more than 1 whole, because 3 thirds make 1 whole, and then there is 1 more third left.

You can think of \(\frac{4}{3}\) as:

Main Idea 2: Multiply a non-unit fraction by a whole number

Now let’s look at a fraction that does not have 1 on top, like \(\frac{2}{5}\).

If we have \(3 \times \frac{2}{5}\), that means 3 groups of two-fifths:

Add the numerators because the denominator stays the same:

So:

This is also more than 1 whole. Since \(\frac{5}{5}=1\), we can see that:

Main Idea 3: Think of multiplication as equal groups

A whole number tells how many groups there are. The fraction tells the size of each group.

• In \(2 \times \frac{3}{4}\), there are 2 groups.
• Each group is \(\frac{3}{4}\).
• So we add \(\frac{3}{4}\) two times.

This gives:

And \(\frac{6}{4}\) is the same as \(1\frac{2}{4}\), or \(1\frac{1}{2}\).

Main Idea 4: A number line can help

A number line is a great way to show repeated addition of fractions.

For example, for \(3 \times \frac{1}{2}\), start at 0 and make 3 jumps of size \(\frac{1}{2}\):

• First jump: \(0 \to \frac{1}{2}\)
• Second jump: \(\frac{1}{2} \to 1\)
• Third jump: \(1 \to 1\frac{1}{2}\)

So:

The number line shows us that multiplying a fraction by a whole number is just making equal jumps again and again.

Worked Example 1

Find \(2 \times \frac{1}{6}\).

Step 1: Write it as repeated addition.

Step 2: Add the fractions.

Answer:

This is also the same as \(\frac{1}{3}\), but the important idea here is seeing it as 2 groups of one-sixth.

Worked Example 2

Find \(5 \times \frac{1}{4}\).

Step 1: Write repeated addition.

Step 2: Add the numerators.

Step 3: Think about wholes.

Since \(\frac{4}{4}=1\), then:

Answer:

Worked Example 3

Find \(3 \times \frac{2}{3}\).

Step 1: Write repeated addition.

Step 2: Add the numerators.

Step 3: Rename the fraction.

Answer:

This makes sense because 3 groups of two-thirds make exactly 2 wholes.

Worked Example 4

A ribbon is \(\frac{3}{8}\) yard long. Mia uses 4 pieces of ribbon. How much ribbon does she use in all?

Step 1: Find the repeated addition.

Step 2: Add the numerators.

Step 3: Think about wholes.

Since \(\frac{8}{8}=1\), then:

And \(\frac{4}{8}\) is the same as \(\frac{1}{2}\), so:

Answer: Mia uses \(1\frac{1}{2}\) yards of ribbon in all.

Helpful Pattern to Notice

When multiplying a fraction by a whole number using repeated addition:

• The denominator stays the same.
• The numerator is added again and again.

For example:

Common Mistakes to Avoid

• Do not add the denominators when the fractions are the same size. In repeated addition, the denominator stays the same.
• Do not forget the meaning of the problem. \(3 \times \frac{1}{5}\) means 3 groups of one-fifth, not \(\frac{3}{15}\).
• Check if your answer is more than 1 whole. If the numerator is greater than the denominator, the answer is more than 1.

Try Thinking These Through

1. \(2 \times \frac{3}{5}\) means \(\frac{3}{5}+\frac{3}{5}\).
2. \(4 \times \frac{1}{2}\) means four jumps of one-half on a number line.
3. \(3 \times \frac{4}{6}\) means 3 equal groups of four-sixths.

Each time, ask yourself: How many groups are there? What fraction is in each group?

Summary

Multiplying a fraction by a whole number means repeated addition. The whole number tells how many groups there are, and the fraction tells the size of each group.

To solve, write the fraction again and again, then add. Keep the denominator the same and add the numerators. Sometimes the answer is less than 1, sometimes it is exactly 1 or more than 1.

If you can think of multiplication as equal groups or equal jumps on a number line, multiplying fractions by whole numbers will make much more sense.