Fraction Operations

Adding and Subtracting Like Fractions

Fractions help us describe parts of a whole. When we add or subtract like fractions, we are working with fractions that have the same denominator.

The denominator tells us how many equal parts the whole is split into. If the denominators are the same, the pieces are the same size. That makes adding and subtracting much easier.

For example, in the fractions \(\frac{2}{8}\) and \(\frac{3}{8}\), both denominators are 8. That means both fractions are made of eighths, so we can combine or compare them directly.

Important idea: When adding or subtracting like fractions, the denominator stays the same. We only add or subtract the numerators.

Here is the basic rule:

This works because the pieces are the same size. If you have 2 eighths and add 3 more eighths, you now have 5 eighths.

Step-by-step for adding like fractions:

• Check that the denominators are the same.
• Add the numerators.
• Keep the denominator the same.
• Simplify if needed.

Step-by-step for subtracting like fractions:

• Check that the denominators are the same.
• Subtract the numerators.
• Keep the denominator the same.
• Simplify if needed.

Let’s look at what each part means:

• The numerator is the top number. It tells how many parts we have.
• The denominator is the bottom number. It tells the size of the parts by showing how many equal parts make one whole.

If the denominator is 6, the whole is split into 6 equal parts. So \(\frac{1}{6}\) means one sixth, \(\frac{4}{6}\) means four sixths, and so on.

Worked Example 1: Simple addition

Add \(\frac{2}{7} + \frac{3}{7}\).

The denominators are both 7, so these are like fractions.

Add the numerators:

Answer: \(\frac{5}{7}\)

Worked Example 2: Simple subtraction

Subtract \(\frac{6}{9} - \frac{2}{9}\).

The denominators are both 9, so subtract the numerators and keep the denominator.

Answer: \(\frac{4}{9}\)

Worked Example 3: Add and simplify

Add \(\frac{3}{10} + \frac{5}{10}\).

The denominators match, so add the numerators:

Now simplify \(\frac{8}{10}\). Both 8 and 10 can be divided by 2:

Answer: \(\frac{4}{5}\)

Worked Example 4: Result greater than 1

Add \(\frac{5}{6} + \frac{4}{6}\).

Add the numerators and keep the denominator:

This fraction is greater than 1 whole. Since \(\frac{6}{6} = 1\), we can think of \(\frac{9}{6}\) as 1 whole and \(\frac{3}{6}\) left over:

Now simplify \(\frac{3}{6}\) to \(\frac{1}{2}\):

Answer: \(1\frac{1}{2}\)

Using a picture idea

Imagine a pizza cut into 8 equal slices. If you eat \(\frac{2}{8}\) of the pizza and your friend eats \(\frac{3}{8}\), together you ate \(\frac{5}{8}\) of the pizza.

Because all the slices are the same size, you can count the slices together. That is why you add the numerators but keep the denominator the same.

Now imagine you had \(\frac{7}{8}\) of a pizza left and then ate \(\frac{2}{8}\) more. You would subtract:

Watch out for these common mistakes:

• Do not add the denominators. For example, \(\frac{1}{5} + \frac{2}{5}\) is not \(\frac{3}{10}\). It is \(\frac{3}{5}\).
• Do not subtract the denominators. For example, \(\frac{4}{7} - \frac{1}{7}\) is not \(\frac{3}{0}\). It is \(\frac{3}{7}\).
• Remember to simplify when possible. For example, \(\frac{2}{6}\) can be simplified to \(\frac{1}{3}\).

Helpful check: Ask yourself, “Are the pieces the same size?” If the denominators are the same, the answer is yes. Then you can add or subtract the numerators.

Practice thinking:

1. \(\frac{1}{4} + \frac{2}{4} = \frac{3}{4}\)
2. \(\frac{5}{8} - \frac{1}{8} = \frac{4}{8} = \frac{1}{2}\)
3. \(\frac{3}{12} + \frac{6}{12} = \frac{9}{12} = \frac{3}{4}\)

Let’s review the big idea:

• Like fractions have the same denominator.
• When denominators are the same, the parts are the same size.
• Add or subtract the numerators.
• Keep the denominator the same.
• Simplify if you can.

Once you understand that the denominator tells the size of the pieces, adding and subtracting like fractions becomes much easier. You are simply combining or comparing equal-sized parts.

Adding and Subtracting Unlike Fractions

Fractions can be added and subtracted, but there is one important rule: the fractions must name parts of the same size.

When fractions have the same denominator, they already have parts of the same size. For example, in \(\frac{1}{8}\) and \(\frac{3}{8}\), both fractions are made of eighths, so they are easy to combine.

But when fractions have unlike denominators, the parts are different sizes. For example, thirds and fourths are not the same size, so you cannot add or subtract them right away.

To solve these problems, we change the fractions into equivalent fractions with a common denominator. Then we can add or subtract safely.

What is a common denominator?

A common denominator is a number that both denominators can divide into evenly. We often use the least common denominator, which is the smallest number both denominators can go into.

For example:

• For \(2\) and \(3\), a common denominator is \(6\).
• For \(4\) and \(6\), a common denominator is \(12\).
• For \(3\) and \(5\), a common denominator is \(15\).

Steps for adding or subtracting unlike fractions

1. Find a common denominator.
2. Rename each fraction as an equivalent fraction with that denominator.
3. Add or subtract the numerators.
4. Keep the denominator the same.
5. Simplify if needed.

Important reminder: When adding or subtracting fractions, only the numerators change. The denominator stays the same after the fractions have been renamed to the common denominator.

Worked Example 1: Add simple unlike fractions

Solve \(\frac{1}{2}+\frac{1}{3}\).

Step 1: Find a common denominator for \(2\) and \(3\). The least common denominator is \(6\).

Step 2: Rename each fraction in sixths.

Step 3: Add the numerators.

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

Worked Example 2: Subtract unlike fractions

Solve \(\frac{3}{4}-\frac{1}{6}\).

Step 1: Find a common denominator for \(4\) and \(6\). The least common denominator is \(12\).

Step 2: Rename each fraction in twelfths.

Step 3: Subtract the numerators.

Answer: \(\frac{3}{4}-\frac{1}{6}=\frac{7}{12}\)

Worked Example 3: Add fractions and simplify

Solve \(\frac{2}{5}+\frac{1}{10}\).

Step 1: Find a common denominator for \(5\) and \(10\). The least common denominator is \(10\).

Step 2: Rename \(\frac{2}{5}\) in tenths.

The second fraction is already in tenths:

Step 3: Add the numerators.

Step 4: Simplify.

Answer: \(\frac{2}{5}+\frac{1}{10}=\frac{1}{2}\)

Worked Example 4: Subtract a mixed number idea using fractions less than 1

Solve \(\frac{5}{6}-\frac{1}{4}\).

Step 1: Find a common denominator for \(6\) and \(4\). The least common denominator is \(12\).

Step 2: Rename each fraction in twelfths.

Step 3: Subtract the numerators.

Answer: \(\frac{5}{6}-\frac{1}{4}=\frac{7}{12}\)

How to find equivalent fractions

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

Example:

This works because the fraction still has the same value. It is just written using different-sized parts.

How to find a common denominator

One helpful way is to list multiples of each denominator until you find one they share.

• Multiples of \(3\): \(3, 6, 9, 12, 15\)
• Multiples of \(5\): \(5, 10, 15\)

The first shared multiple is \(15\), so \(15\) is the least common denominator.

Common mistakes to avoid

• Do not add the denominators. For example, \(\frac{1}{2}+\frac{1}{3}\) is not \(\frac{2}{5}\).
• Do not subtract the denominators. The denominator is kept once both fractions have the same denominator.
• Rename both fractions correctly. If the denominator changes, the numerator must change too.
• Simplify your final answer when possible.

Quick check

Try thinking through these:

• \(\frac{1}{3}+\frac{1}{6}\) → common denominator: \(6\)
• \(\frac{3}{8}-\frac{1}{4}\) → common denominator: \(8\)
• \(\frac{2}{3}+\frac{1}{5}\) → common denominator: \(15\)

Summary

To add or subtract unlike fractions, first find a common denominator. Then rename the fractions as equivalent fractions with that denominator. After that, add or subtract the numerators, keep the denominator the same, and simplify if needed.

Remember: fractions must be talking about equal-sized parts before you combine them. Once the denominators match, the problem becomes much easier.

Adding and Subtracting Mixed Numbers

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

In this lesson, you will learn how to add and subtract mixed numbers. You will also learn what to do when you need to regroup, which means trading 1 whole for a fraction so the problem can be solved more easily.

Mixed numbers are used in real life all the time. For example, you might measure \(1\frac{1}{2}\) cups of flour or walk \(2\frac{3}{4}\) miles. Knowing how to add and subtract them helps you solve everyday problems.

Step 1: Understand the parts of a mixed number

Every mixed number has:

• a whole number
• a fraction

For example, in \(3\frac{2}{5}\):

• the whole number is \(3\)
• the fraction is \(\frac{2}{5}\)

When adding or subtracting mixed numbers, it helps to work with the whole numbers and the fractions in an organized way.

Step 2: Add mixed numbers with the same denominator

If the fractions have the same denominator, add the whole numbers and add the fractions.

Example idea:

Add the whole numbers:

Add the fractions:

Put them together:

This works because the pieces are the same size. Fifths can be added to fifths.

Step 3: What if the fraction part adds to 1 whole or more?

Sometimes the fractions add up to an improper fraction, which means the numerator is greater than or equal to the denominator.

When that happens, change the fraction part into a mixed number or whole number, then add it to the whole-number part.

For example:

Add the whole numbers:

Add the fractions:

Now rewrite \(\frac{5}{4}\) as:

Add that extra whole:

So:

Step 4: Subtract mixed numbers with the same denominator

If the top mixed number has a larger fraction part than the bottom mixed number, subtract the whole numbers and subtract the fractions.

For example:

Subtract the whole numbers:

Subtract the fractions:

Put them together:

You can simplify \(\frac{3}{6}\) to \(\frac{1}{2}\), so the final answer is:

Step 5: Subtract mixed numbers when you need to regroup

Sometimes the fraction on top is smaller than the fraction on the bottom. Then you cannot subtract the fractions right away.

In that case, regroup by taking 1 whole from the whole number and turning it into a fraction with the same denominator.

For example:

At first, \(\frac{1}{5}-\frac{3}{5}\) does not work because \(1<3\).

So regroup \(4\frac{1}{5}\). Take 1 whole from the 4. That leaves 3 wholes. The 1 whole becomes \(\frac{5}{5}\) because the denominator is 5.

Add that to the fraction part:

Now subtract:

Subtract the whole numbers:

Subtract the fractions:

Answer:

How regrouping works

Here is the important idea:

The fraction you use depends on the denominator in the problem.

So if you need to regroup:

• take 1 away from the whole number
• change that 1 whole into a fraction with the same denominator
• add it to the fraction part

For example:

• \(7\frac{2}{8}=6\frac{10}{8}\)
• \(3\frac{1}{4}=2\frac{5}{4}\)
• \(6\frac{3}{7}=5\frac{10}{7}\)

Worked Example 1: Easy addition

Solve:

Add the whole numbers:

Add the fractions:

Answer:

Worked Example 2: Addition that makes an extra whole

Solve:

Add the whole numbers:

Add the fractions:

Rewrite \(\frac{9}{6}\) as a mixed number:

Now combine with 3:

Simplify \(\frac{3}{6}\) to \(\frac{1}{2}\):

Answer:

Worked Example 3: Subtraction without regrouping

Solve:

Subtract the whole numbers:

Subtract the fractions:

Answer:

Worked Example 4: Subtraction with regrouping

Solve:

You cannot do \(\frac{2}{4}-\frac{3}{4}\), so regroup.

Take 1 whole from 7. That leaves 6. Change the 1 whole into \(\frac{4}{4}\).

Add it to \(\frac{2}{4}\):

Now subtract:

Subtract the whole numbers:

Subtract the fractions:

Answer:

Tips to remember

• Add or subtract the whole numbers and fraction parts carefully.
• The fractions in this lesson have the same denominator, so keep the denominator the same.
• If the fraction part in addition is greater than or equal to 1 whole, rename it.
• If the fraction part in subtraction is too small, regroup.
• Simplify your fraction if you can.

Common mistakes to avoid

• Do not add the denominators when the denominators are the same.
• Do not forget to regroup in subtraction when the top fraction is smaller.
• Do not leave an improper fraction in your final mixed-number answer unless your teacher says it is okay.
• Do not forget to simplify, like changing \(\frac{2}{4}\) to \(\frac{1}{2}\).

Quick check

1. \(3\frac{2}{7}+1\frac{4}{7}\)
2. \(5\frac{3}{8}-2\frac{1}{8}\)
3. \(2\frac{5}{6}+4\frac{3}{6}\)
4. \(6\frac{1}{3}-2\frac{2}{3}\)

You can solve these by asking:

• Are the denominators the same?
• Am I adding or subtracting?
• Do I need to regroup?
• Can I simplify my answer?

Summary

To add mixed numbers, add the whole numbers and add the fractions. If the fraction part makes 1 whole or more, rename it and add the extra whole.

To subtract mixed numbers, subtract the whole numbers and fractions if the top fraction is large enough. If not, regroup 1 whole into a fraction, then subtract.

With practice, adding and subtracting mixed numbers becomes easier. Work step by step, keep your fractions organized, and check whether your answer should be simplified.

Multiplying Fractions by Whole Numbers

Sometimes we need to add the same fraction again and again. For example, if you eat \\(\frac{1}{4}\\) of a pizza each day for 3 days, how much pizza did you eat in all?

This is where multiplying fractions by whole numbers helps. It is a faster way to show repeated addition.

For example:

$$3 \times \frac{1}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4}$$

So, multiplying a fraction by a whole number means adding that fraction the given number of times.

Important idea: When you multiply a fraction by a whole number, the denominator stays the same. The denominator tells the size of the parts. The numerator gets multiplied by the whole number because you are counting more parts.

Here is the rule:

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

In words:

• Multiply the whole number by the numerator.
• Keep the denominator the same.
• Simplify if needed.
• If the answer is greater than 1, you can write it as a mixed number.

Why does this make sense?

If you have \\(\frac{2}{5}\\) and you take it 4 times, then you have 4 groups of 2 fifths.

That means:

$$4 \times \frac{2}{5} = \frac{2}{5} + \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{8}{5}$$

You now have 8 fifths. The pieces are still fifths, so the denominator stays 5.

You can also think with a model.

Imagine a rectangle split into 4 equal parts. One part is \\(\frac{1}{4}\\). If you shade \\(\frac{1}{4}\\) three times, then 3 out of 4 parts are shaded. That gives \\(\frac{3}{4}\\).

So:

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

This shows that multiplying by a whole number can be seen as making more equal groups of the same fraction.

Steps for multiplying a fraction by a whole number

1. Write the multiplication problem.
2. Multiply the whole number and the numerator.
3. Keep the denominator the same.
4. Simplify the fraction if possible.
5. If needed, change an improper fraction to a mixed number.

Worked Example 1

Find \\(2 \times \frac{3}{8}\\).

Multiply the whole number by the numerator:

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

Simplify \\(\frac{6}{8}\\):

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

So the answer is \\(\frac{3}{4}\\).

You can also check with repeated addition:

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

Worked Example 2

Find \\(5 \times \frac{1}{6}\\).

Multiply the numerator by 5:

$$5 \times \frac{1}{6} = \frac{5 \times 1}{6} = \frac{5}{6}$$

This fraction is already simplified.

So the answer is \\(\frac{5}{6}\\).

Worked Example 3

Find \\(4 \times \frac{2}{3}\\).

Multiply the numerator by 4:

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

Now change \\(\frac{8}{3}\\) to a mixed number.

Since \\(3\\) goes into \\(8\\) two whole times with \\(2\\) left over:

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

So the answer is \\(2\frac{2}{3}\\).

Check with repeated addition:

$$\frac{2}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3} = \frac{8}{3} = 2\frac{2}{3}$$

Worked Example 4

A ribbon piece is \\(\frac{3}{10}\\) meter long. Mia uses 3 pieces. How much ribbon does she use in all?

This means:

$$3 \times \frac{3}{10} = \frac{9}{10}$$

So Mia uses \\(\frac{9}{10}\\) meter of ribbon.

When the answer is greater than 1

Sometimes multiplying a fraction by a whole number gives more than one whole. That is okay.

Example:

$$3 \times \frac{3}{4} = \frac{9}{4} = 2\frac{1}{4}$$

This means three groups of \\(\frac{3}{4}\\) make \\(2\frac{1}{4}\\).

Common mistakes to avoid

• Do not multiply the denominator by the whole number. Only the numerator is multiplied.
• Do not forget to simplify. For example, \\(\frac{4}{6} = \frac{2}{3}\\).
• Do not forget mixed numbers. If your fraction is improper, change it if needed.

Look at this mistake:

$$3 \times \frac{2}{5} = \frac{6}{15}$$

This is not correct because the denominator should stay 5.

The correct answer is:

$$3 \times \frac{2}{5} = \frac{6}{5} = 1\frac{1}{5}$$

Helpful tip

If you are not sure, rewrite the multiplication as addition.

For example:

$$4 \times \frac{1}{7} = \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{4}{7}$$

This can help you see why the denominator stays the same.

Summary

• Multiplying a fraction by a whole number means repeated addition.
• Multiply the whole number and the numerator.
• Keep the denominator the same.
• Simplify your answer if possible.
• If the answer is greater than 1, write it as a mixed number if needed.

Remember:

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

With practice, you will see that multiplying fractions by whole numbers is just counting fractional parts in equal groups.

Multiplying Fractions by Fractions

Sometimes we need to find a part of a part. That is what happens when we multiply one fraction by another fraction.

For example, what does \(\frac{1}{2} \times \frac{3}{4}\) mean? It means one-half of three-fourths. We are finding part of an amount that is already a fraction.

In this lesson, you will learn two ways to understand this:

• with an area model, and
• with a simple multiply across rule.

1. What does multiplying fractions mean?

When whole numbers are multiplied, we can think of repeated groups. But with fractions, multiplication often means finding a part of another part.

So if we have:

$$\frac{2}{3} \times \frac{3}{5}$$

we are finding two-thirds of three-fifths.

2. Using an area model

An area model helps us see why fraction multiplication works.

Imagine a rectangle that stands for 1 whole.

• First, divide it into equal parts one way.
• Then shade one fraction.
• Next, divide it the other way.
• Shade the second fraction in a different direction.
• The overlapping part shows the product.

Let us look at \(\frac{1}{2} \times \frac{3}{4}\).

• Divide the rectangle into 4 equal columns and shade 3 of them. This shows \(\frac{3}{4}\).
• Now divide the same rectangle into 2 equal rows and shade 1 row in the other direction. This shows \(\frac{1}{2}\).
• The overlapping shaded part is the answer.

The whole rectangle now has \(2 \times 4 = 8\) equal small parts.

The overlap covers 3 of those 8 parts, so:

$$\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$$

This is why multiplying fractions gives a smaller piece when both fractions are less than 1.

3. The rule: multiply numerators and multiply denominators

There is a quick rule for multiplying fractions:

• Multiply the numerators (top numbers).
• Multiply the denominators (bottom numbers).

In general:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

This works because the area model creates a new grid:

• the denominator tells how many equal parts the whole is split into,
• and the numerator tells how many of those parts are shaded.

4. Worked Examples

Example 1: Multiply \(\frac{1}{2} \times \frac{3}{4}\)

Multiply the numerators:

$$1 \times 3 = 3$$

Multiply the denominators:

$$2 \times 4 = 8$$

So:

$$\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$$

Answer: \(\frac{3}{8}\)

Example 2: Multiply \(\frac{2}{3} \times \frac{4}{5}\)

Multiply the numerators:

$$2 \times 4 = 8$$

Multiply the denominators:

$$3 \times 5 = 15$$

So:

$$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$$

Answer: \(\frac{8}{15}\)

Example 3: Multiply \(\frac{3}{4} \times \frac{2}{3}\)

Multiply the numerators:

$$3 \times 2 = 6$$

Multiply the denominators:

$$4 \times 3 = 12$$

So first we get:

$$\frac{3}{4} \times \frac{2}{3} = \frac{6}{12}$$

Now simplify. Both 6 and 12 can be divided by 6:

$$\frac{6}{12} = \frac{1}{2}$$

Answer: \(\frac{1}{2}\)

Example 4: Multiply \(\frac{5}{6} \times \frac{3}{10}\)

Multiply the numerators:

$$5 \times 3 = 15$$

Multiply the denominators:

$$6 \times 10 = 60$$

So first we get:

$$\frac{5}{6} \times \frac{3}{10} = \frac{15}{60}$$

Simplify by dividing both numbers by 15:

$$\frac{15}{60} = \frac{1}{4}$$

Answer: \(\frac{1}{4}\)

5. How to simplify your answer

After multiplying, always check whether the fraction can be written in a simpler form.

To simplify, divide the numerator and denominator by the same number.

For example:

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

because both 4 and 8 can be divided by 4.

6. A helpful way to think about the answer

When you multiply two fractions that are both less than 1, the answer is usually smaller than either fraction.

For example:

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

One-third of a half is a very small part, so \(\frac{1}{6}\) makes sense.

This is a great way to check if your answer is reasonable.

7. Steps to multiply fractions

1. Multiply the numerators.
2. Multiply the denominators.
3. Write the product as a fraction.
4. Simplify if possible.

Example of the steps:

$$\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}$$

8. Common mistakes to avoid

• Do not add the numerators and denominators. For multiplication, you multiply them.
• Do not forget to simplify if the answer can be reduced.
• Make sure the answer makes sense. If you are finding a part of a part, the answer should often be smaller.

For example, this is not correct:

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

That used addition thinking, not multiplication.

The correct multiplication is:

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

9. Quick practice thinking

Try these in your head or on paper:

• \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\)
• \(\frac{2}{5} \times \frac{1}{4} = \frac{2}{20} = \frac{1}{10}\)
• \(\frac{3}{8} \times \frac{2}{3} = \frac{6}{24} = \frac{1}{4}\)

Summary

Multiplying fractions means finding a part of a part. You can show it with an area model by shading two fractions and looking at the overlap.

To multiply fractions, multiply the top numbers and multiply the bottom numbers:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Then simplify if needed. With practice, fraction multiplication becomes quick and easy.

Multiplication as Scaling means that multiplication does not always make a number bigger in the same way. Sometimes multiplication stretches a number, and sometimes it shrinks a number.

You may already know multiplication as equal groups, like 4 groups of 3. That idea is still true. But in 5th grade, we also learn another important meaning of multiplication: scaling.

When we scale something, we change its size by a factor. A factor is the number we multiply by. If the factor is greater than 1, the amount gets bigger. If the factor is less than 1, the amount gets smaller.

For example, if you multiply 8 by 2, you are making 8 twice as large. If you multiply 8 by \(\frac{1}{2}\), you are making 8 half as large.

So multiplication can do three different things:

• Multiply by a number greater than 1: the value increases.
• Multiply by 1: the value stays the same.
• Multiply by a fraction less than 1: the value decreases.

Let’s look at these ideas more closely.

1. Multiplying by a number greater than 1 makes the quantity larger.

If the factor is greater than 1, the original number is scaled up. This means it gets bigger.

Examples of numbers greater than 1 are \(2\), \(3\), and \(\frac{5}{4}\).

If we start with 6 and multiply by 2, we get:

$$6 \times 2 = 12$$

The 6 became larger. It was scaled up by a factor of 2.

If we multiply 6 by \(\frac{5}{4}\), we also get a larger number because \(\frac{5}{4}\) is greater than 1.

$$6 \times \frac{5}{4} = \frac{30}{4} = 7\frac{1}{2}$$

Since \(7\frac{1}{2} > 6\), the number increased.

2. Multiplying by 1 keeps the quantity the same.

The number 1 does not change the size of a number when you multiply.

$$9 \times 1 = 9$$

This means scaling by 1 keeps the original amount exactly the same size.

3. Multiplying by a fraction less than 1 makes the quantity smaller.

This is the big idea in this lesson. A fraction less than 1 represents part of a whole. When you multiply by it, you take only part of the original amount.

Examples of fractions less than 1 are \(\frac{1}{2}\), \(\frac{3}{4}\), and \(\frac{2}{5}\).

If we start with 8 and multiply by \(\frac{1}{2}\), we get:

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

The number got smaller. It was scaled down to half its size.

If we multiply 8 by \(\frac{3}{4}\), we get:

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

Again, the result is smaller than 8 because \(\frac{3}{4}\) is less than 1.

Why does multiplying by a fraction less than 1 make a number smaller?

Think about the meaning of \(8 \times \frac{3}{4}\). It means three-fourths of 8. Three-fourths is only part of a whole, so the result must be less than 8.

You can picture 8 objects being split into 4 equal parts. Each part is worth 2. Then take 3 of those parts:

$$8 \div 4 = 2$$

$$2 \times 3 = 6$$

So:

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

Using bar models or strips can help us see scaling.

Imagine a bar that shows a length of 12. If we want \(\frac{1}{3}\) of 12, we divide the bar into 3 equal parts. Each part is 4. One part is \(\frac{1}{3}\) of the whole, so:

$$12 \times \frac{1}{3} = 4$$

The bar got smaller because we took only one of the three equal parts.

Now imagine multiplying 12 by \(\frac{4}{3}\). Since \(\frac{4}{3}\) is greater than 1, the bar becomes larger than 12.

$$12 \times \frac{4}{3} = 16$$

This shows that fractions can either make a number bigger or smaller. It depends on whether the fraction is greater than 1 or less than 1.

A helpful comparison rule is:

• If the factor is greater than 1, the product is greater than the starting number.
• If the factor is equal to 1, the product is equal to the starting number.
• If the factor is less than 1, the product is less than the starting number.

This is very useful when checking whether an answer makes sense.

For example, if a student says:

$$10 \times \frac{1}{5} = 50$$

we know right away this cannot be correct. Since \(\frac{1}{5}\) is less than 1, the answer should be less than 10, not greater than 10.

Worked Example 1

Find \(7 \times 3\). Explain what happens to 7.

Step 1: Notice that 3 is greater than 1.

Step 2: This means 7 will be scaled up.

Step 3: Multiply.

$$7 \times 3 = 21$$

Answer: The number 7 became larger. Multiplying by 3 made it 3 times as large.

Worked Example 2

Find \(10 \times \frac{1}{2}\). Explain what happens to 10.

Step 1: Notice that \(\frac{1}{2}\) is less than 1.

Step 2: This means 10 will be scaled down.

Step 3: Multiply.

$$10 \times \frac{1}{2} = 5$$

Answer: The number 10 became smaller. Multiplying by \(\frac{1}{2}\) made it half as large.

Worked Example 3

Find \(16 \times \frac{3}{4}\). Explain why the answer is smaller than 16.

Step 1: \(\frac{3}{4}\) is less than 1, so the result should be less than 16.

Step 2: Divide 16 into 4 equal parts.

$$16 \div 4 = 4$$

Step 3: Take 3 of those parts.

$$4 \times 3 = 12$$

So:

$$16 \times \frac{3}{4} = 12$$

Answer: The answer is smaller than 16 because \(\frac{3}{4}\) means only part of 16.

Worked Example 4

Find \(12 \times \frac{5}{3}\). Explain why the answer is greater than 12.

Step 1: Compare \(\frac{5}{3}\) to 1. Since \(\frac{5}{3} = 1\frac{2}{3}\), it is greater than 1.

Step 2: This means 12 will be scaled up.

Step 3: Multiply.

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

Answer: The answer is greater than 12 because multiplying by a number greater than 1 makes the quantity larger.

Tips for solving multiplication as scaling problems

1. Look at the factor you are multiplying by.
2. Ask: Is it greater than 1, equal to 1, or less than 1?
3. Predict whether the answer should be bigger, the same, or smaller.
4. Then solve the problem.
5. Check if your answer makes sense.

Common mistakes to avoid

• Thinking multiplication always makes numbers bigger.
• Forgetting that fractions less than 1 make a quantity smaller.
• Not checking whether the answer makes sense.
• Confusing \(\frac{5}{4}\) and \(\frac{4}{5}\). One is greater than 1, but the other is less than 1.

Quick check

Without solving exactly, decide whether each product is greater than, equal to, or less than the starting number.

• \(9 \times \frac{2}{3}\) → less than 9
• \(9 \times 1\) → equal to 9
• \(9 \times \frac{7}{6}\) → greater than 9

This kind of thinking helps you understand multiplication better, not just compute answers.

Summary

Multiplication as scaling means multiplication changes the size of a quantity.

• If you multiply by a number greater than 1, the quantity gets larger.
• If you multiply by 1, the quantity stays the same.
• If you multiply by a fraction less than 1, the quantity gets smaller.

When you solve a multiplication problem, always think about the factor first. Then you can predict whether the answer should be bigger or smaller, and that helps you check your work.

Dividing Unit Fractions by Whole Numbers

Today we will learn how to divide a unit fraction by a whole number.

A unit fraction is a fraction with a 1 on top, like \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{8}\).

When we divide a unit fraction by a whole number, we are asking: If we split this small piece into equal smaller pieces, how big is each piece?

For example, if you have \(\frac{1}{2}\) of a sandwich and share it equally with 2 people, each person gets half of the half. That amount is smaller than \(\frac{1}{2}\).

Big idea: Dividing by a whole number makes the piece smaller because you are splitting it into more equal parts.

How to think about it with pictures

Start with one whole. Then divide the whole into equal parts to show the unit fraction. After that, split that unit fraction into more equal parts.

• First, find the unit fraction, such as \(\frac{1}{3}\).
• Then divide that one part into the number of equal groups named by the whole number.
• The new pieces are smaller fractions of the whole.

Here is the rule we will use:

This means when you divide a unit fraction by a whole number, you multiply the denominator by that whole number.

Why does that work? The denominator tells how many equal parts make the whole. If one of those parts gets split into even more equal pieces, the whole is now made of more total equal parts.

For example, if you start with \(\frac{1}{4}\), the whole is divided into 4 equal parts. If each fourth is split into 3 equal parts, then the whole is divided into \(4 \times 3 = 12\) equal parts. So:

Worked Example 1

Find \(\frac{1}{2} \div 2\).

Step 1: Start with \(\frac{1}{2}\). This means one whole is split into 2 equal parts.

Step 2: Divide that half into 2 equal smaller pieces.

Step 3: Now the whole has 4 equal parts, so each small piece is \(\frac{1}{4}\).

Meaning: Half of something shared equally between 2 people gives each person \(\frac{1}{4}\).

Worked Example 2

Find \(\frac{1}{3} \div 2\).

We take one third and split it into 2 equal parts.

The whole was in 3 equal parts. Splitting each third into 2 parts makes \(3 \times 2 = 6\) equal parts in the whole.

Check: Is \(\frac{1}{6}\) smaller than \(\frac{1}{3}\)? Yes. That makes sense because dividing should make the piece smaller here.

Worked Example 3

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

Use the rule: multiply the denominator by 4.

So each piece is \(\frac{1}{20}\).

Picture idea: Imagine a strip divided into 5 equal parts. Shade 1 part for \(\frac{1}{5}\). Then cut that shaded part into 4 equal pieces. Each tiny piece is \(\frac{1}{20}\) of the whole strip.

Worked Example 4

Find \(\frac{1}{8} \div 3\).

Multiply the denominator 8 by 3.

So splitting \(\frac{1}{8}\) into 3 equal parts gives \(\frac{1}{24}\) for each part.

What to remember

• A unit fraction has 1 as the numerator.
• Dividing a unit fraction by a whole number makes a smaller fraction.
• To divide \(\frac{1}{b}\) by a whole number \(n\), multiply the denominator by \(n\).

Common mistake to avoid

Do not multiply the numerator by the whole number. The numerator stays 1 because you are still talking about one small piece of the whole.

For example:

Incorrect: \(\frac{1}{4} \div 2 = \frac{2}{4}\)

Correct:

Why? Because the fourth is being split into 2 smaller equal parts, making eighths.

Try thinking with words

• \(\frac{1}{2} \div 3\) means “one half split into 3 equal parts.”
• \(\frac{1}{6} \div 2\) means “one sixth split into 2 equal parts.”
• \(\frac{1}{10} \div 5\) means “one tenth split into 5 equal parts.”

Using the rule:

Summary

Dividing a unit fraction by a whole number means splitting that fraction into smaller equal pieces.

If you divide \(\frac{1}{b}\) by \(n\), the answer is \(\frac{1}{b \times n}\).

Always ask yourself: Did my answer get smaller? If yes, that is a good sign that your work makes sense.

Dividing Whole Numbers by Unit Fractions

Today we will learn how to divide a whole number by a unit fraction.

A whole number is a counting number like 1, 2, 3, 4, or 10.

A unit fraction is a fraction with 1 on top. Examples are \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), and \(\frac{1}{8}\).

When we divide a whole number by a unit fraction, we are really asking:

How many of those small fractional pieces fit into the whole number?

For example, \(4 \div \frac{1}{2}\) means:

How many halves are in 4 wholes?

This idea is easier to understand if we picture each whole being split into equal parts.

Main Idea

If each whole is split into halves, then 1 whole has 2 halves.

If each whole is split into thirds, then 1 whole has 3 thirds.

If each whole is split into fourths, then 1 whole has 4 fourths.

So when dividing by a unit fraction, the answer gets larger because we are counting many small pieces inside the whole number.

For example:

There are 4 fourths in 1 whole.

Using a Visual Model

Imagine 3 rectangles, and each rectangle stands for 1 whole.

If each rectangle is cut into 2 equal parts, then each whole has 2 halves.

So 3 wholes have:

That means:

We can use the same thinking for any unit fraction.

• Divide by \(\frac{1}{2}\): count halves
• Divide by \(\frac{1}{3}\): count thirds
• Divide by \(\frac{1}{4}\): count fourths
• Divide by \(\frac{1}{5}\): count fifths

A Helpful Pattern

Look at these examples:

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

The denominator tells how many pieces are in each whole.

So dividing by \(\frac{1}{n}\) is like multiplying by \(n\).

That means:

This works because each whole contains \(n\) pieces of size \(\frac{1}{n}\).

Worked Example 1

Solve:

Ask: How many halves are in 2 wholes?

Each whole has 2 halves.

So 2 wholes have:

Answer:

Worked Example 2

Solve:

Ask: How many fifths are in 5 wholes?

Each whole has 5 fifths.

So 5 wholes have:

Answer:

Worked Example 3

Solve:

Ask: How many thirds are in 4 wholes?

Each whole has 3 thirds.

So 4 wholes have:

Answer:

Worked Example 4

Solve:

Ask: How many fourths are in 7 wholes?

Each whole has 4 fourths.

So 7 wholes have:

Answer:

Using Reciprocal Multiplication

There is also a quick way to solve these problems.

To divide by a fraction, we can multiply by its reciprocal.

The reciprocal of \(\frac{1}{4}\) is \(4\). The reciprocal of \(\frac{1}{3}\) is \(3\). The reciprocal of \(\frac{1}{2}\) is \(2\).

So:

This matches the picture idea. If each whole has that many equal parts, then we multiply to count all the parts.

Steps to Solve

1. Look at the unit fraction.
2. Use the denominator to see how many pieces are in each whole.
3. Multiply the whole number by that denominator.
4. Write the answer.

Example:

For \(9 \div \frac{1}{3}\):

1. The denominator is 3.
2. Each whole has 3 thirds.
3. \(9 \times 3 = 27\)

So:

Common Mistake to Avoid

Do not think the answer should get smaller just because this is division.

When you divide by a unit fraction, you are counting tiny parts inside the whole number. Since the parts are small, there are many of them.

That is why:

not \(1\).

There are 9 thirds in 3 wholes.

Try the Thinking

• \(1 \div \frac{1}{2}\): How many halves are in 1? Answer: 2
• \(2 \div \frac{1}{4}\): How many fourths are in 2? Answer: 8
• \(6 \div \frac{1}{3}\): How many thirds are in 6? Answer: 18

Summary

Dividing a whole number by a unit fraction means finding how many fractional pieces fit into the whole number.

You can think with a picture, count the equal parts, or use multiplication.

For a unit fraction like \(\frac{1}{n}\), each whole has \(n\) pieces, so:

If you remember to ask, How many of these small pieces are in the whole number? you will be able to solve these problems correctly.