Operations with Fractions

Adding and Subtracting Fractions with Like Denominators

Fractions help us describe parts of a whole. When we add or subtract fractions, we are combining parts or taking parts away.

In this lesson, we will learn how to add and subtract fractions with like denominators. This means the fractions have the same number on the bottom.

For example, in the fractions \(\frac{2}{7}\) and \(\frac{4}{7}\), the denominator is 7 in both fractions. These are fractions with like denominators.

Important idea: The denominator tells the size of the parts. If the denominators are the same, the parts are the same size. That is why we can combine or compare them easily.

Think about pizza slices. If two pizzas are both cut into 8 equal slices, then each slice is the same size. So adding \(\frac{3}{8}\) and \(\frac{2}{8}\) means combining 3 slices and 2 slices of the same size.

The rule for adding fractions with like denominators:

Keep the denominator the same, and add the numerators.

The rule for subtracting fractions with like denominators:

Keep the denominator the same, and subtract the numerators.

Why do we keep the denominator? Because the denominator tells how many equal parts the whole is divided into. When the parts are the same size, we are only changing how many parts we have, not the size of the parts.

So in \(\frac{5}{9}+\frac{2}{9}\), the ninths stay as ninths. We are just combining 5 ninths and 2 ninths to make 7 ninths.

Here is a step-by-step method you can use every time:

1. Check that the denominators are the same.
2. Add or subtract the numerators.
3. Keep the denominator the same.
4. Simplify the answer if possible.
5. If the answer is greater than 1, write it as a mixed number if needed.

Worked Example 1: Simple Addition

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

Step 1: The denominators are the same, so these are like denominators.

Step 2: Add the numerators: \(1+3=4\).

Step 3: Keep the denominator 6.

Step 4: Simplify if possible. Both 4 and 6 can be divided by 2.

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

Worked Example 2: Simple Subtraction

Subtract \(\frac{7}{10}-\frac{2}{10}\).

Step 1: The denominators are both 10, so the fractions have like denominators.

Step 2: Subtract the numerators: \(7-2=5\).

Step 3: Keep the denominator 10.

Step 4: Simplify. Both 5 and 10 can be divided by 5.

So the final answer is \(\frac{1}{2}\).

Worked Example 3: Addition Greater Than 1

Add \(\frac{5}{8}+\frac{6}{8}\).

Step 1: The denominators are the same.

Step 2: Add the numerators: \(5+6=11\).

Step 3: Keep the denominator 8.

This fraction is greater than 1 because the numerator is larger than the denominator. It is an improper fraction.

We can write it as a mixed number. Since \(8\) goes into \(11\) one whole time with 3 left over:

So the final answer is \(1\frac{3}{8}\).

Worked Example 4: Subtraction with a Larger Fraction

Subtract \(\frac{9}{12}-\frac{4}{12}\).

Step 1: The denominators are both 12.

Step 2: Subtract the numerators: \(9-4=5\).

Step 3: Keep the denominator 12.

Step 4: Check whether it can be simplified. Since 5 and 12 do not have a common factor greater than 1, the fraction is already in simplest form.

So the final answer is \(\frac{5}{12}\).

Common Mistakes to Avoid

• Do not add the denominators. For example, \(\frac{2}{7}+\frac{3}{7}\) is not \(\frac{5}{14}\). It is \(\frac{5}{7}\).
• Do not subtract the denominators. For example, \(\frac{6}{9}-\frac{2}{9}\) is not \(\frac{4}{0}\). It is \(\frac{4}{9}\).
• Always simplify when possible. For example, \(\frac{3}{8}+\frac{1}{8}=\frac{4}{8}=\frac{1}{2}\).
• Watch for answers greater than 1. If needed, change an improper fraction to a mixed number.

Quick Practice Ideas

• \(\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\)
• \(\frac{6}{11}-\frac{4}{11}=\frac{2}{11}\)
• \(\frac{3}{4}+\frac{2}{4}=\frac{5}{4}=1\frac{1}{4}\)
• \(\frac{7}{9}-\frac{3}{9}=\frac{4}{9}\)

Summary

When adding and subtracting fractions with like denominators, the denominator stays the same because the pieces are the same size.

You only add or subtract the numerators.

After finding the answer, always check whether you should simplify it or write it as a mixed number.

If you remember this idea—same denominator, combine the numerators, keep the denominator—you will be able to solve many fraction problems correctly.

Finding Common Denominators for Unlike Fractions

When we add or subtract fractions, the fractions need to talk about the same-sized parts. The denominator tells us what size the parts are.

If two fractions have the same denominator, they are already using the same-sized parts. For example, in \(\frac{3}{8}\) and \(\frac{1}{8}\), both fractions are divided into eighths.

But if fractions have unlike denominators, such as \(\frac{1}{3}\) and \(\frac{1}{4}\), they are divided into different-sized parts. Thirds and fourths are not the same size, so we cannot add or subtract them right away.

To solve this, we find a common denominator. A common denominator is a number that both denominators can divide into evenly.

The best common denominator to use is usually the least common denominator, or LCD. The LCD is the smallest number that both denominators go into evenly. It comes from the Least Common Multiple (LCM) of the denominators.

Why do we use the LCD?

• It makes the numbers smaller and easier to work with.
• It helps us write equivalent fractions more efficiently.
• It is especially helpful when adding and subtracting.

Step-by-step: How to find a common denominator

1. Look at the denominators.
2. Find their least common multiple (LCM).
3. Use that LCM as the common denominator.
4. Rewrite each fraction as an equivalent fraction with the new denominator.

Remember: When you change the denominator, you must also change the numerator by the same factor so the fraction keeps the same value.

For example,

$$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}$$

These are all equivalent fractions. They look different, but they represent the same amount.

How to find the LCM

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

Example: Find the LCM of 3 and 4.

• Multiples of 3: 3, 6, 9, 12, 15, ...
• Multiples of 4: 4, 8, 12, 16, ...

The first common multiple is 12, so the LCD is 12.

Worked Example 1: Simple common denominator

Find a common denominator for \(\frac{1}{3}\) and \(\frac{1}{4}\).

Step 1: Find the LCM of 3 and 4.

• Multiples of 3: 3, 6, 9, 12
• Multiples of 4: 4, 8, 12

The LCM is 12.

Step 2: Rewrite each fraction with denominator 12.

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

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

So the fractions with a common denominator are \(\frac{4}{12}\) and \(\frac{3}{12}\).

Worked Example 2: Using the common denominator to add

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

Step 1: Find the LCM of 5 and 2.

• Multiples of 5: 5, 10, 15, ...
• Multiples of 2: 2, 4, 6, 8, 10, ...

The LCM is 10.

Step 2: Rewrite both fractions with denominator 10.

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

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

Step 3: Add the numerators.

$$\frac{4}{10}+\frac{5}{10}=\frac{9}{10}$$

So,

$$\frac{2}{5}+\frac{1}{2}=\frac{9}{10}$$

Worked Example 3: Using the common denominator to subtract

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

Step 1: Find the LCM of 4 and 6.

• Multiples of 4: 4, 8, 12, 16, ...
• Multiples of 6: 6, 12, 18, ...

The LCM is 12.

Step 2: Rewrite both fractions with denominator 12.

$$\frac{3}{4}=\frac{3\times 3}{4\times 3}=\frac{9}{12}$$

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

Step 3: Subtract the numerators.

$$\frac{9}{12}-\frac{2}{12}=\frac{7}{12}$$

So,

$$\frac{3}{4}-\frac{1}{6}=\frac{7}{12}$$

Worked Example 4: A case where one denominator is already a multiple of the other

Find a common denominator for \(\frac{5}{6}\) and \(\frac{1}{3}\), then add them.

Step 1: Find the LCM of 6 and 3.

• Multiples of 6: 6, 12, 18, ...
• Multiples of 3: 3, 6, 9, 12, ...

The LCM is 6.

This means one fraction already has the LCD.

Step 2: Rewrite only the fraction that needs changing.

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

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

Step 3: Add the numerators.

$$\frac{5}{6}+\frac{2}{6}=\frac{7}{6}$$

So,

$$\frac{5}{6}+\frac{1}{3}=\frac{7}{6}$$

Important ideas to remember

• You can only add or subtract fractions easily when the denominators are the same.
• The common denominator should be a common multiple of both denominators.
• The least common denominator is usually the easiest one to use.
• To make an equivalent fraction, multiply the numerator and denominator by the same number.
• Do not add or subtract the denominators.

Common mistakes

• Mistake: Adding denominators.
  Example: \(\frac{1}{3}+\frac{1}{4}\neq\frac{2}{7}\)
• Mistake: Changing the denominator but not the numerator.
  If \(\frac{1}{4}\) becomes twelfths, it must become \(\frac{3}{12}\), not \(\frac{1}{12}\).
• Mistake: Choosing a common denominator that works, but forgetting to rewrite both fractions correctly.

Quick check

Try these on your own:

1. Find a common denominator for \(\frac{1}{2}\) and \(\frac{1}{5}\).
2. Rewrite \(\frac{3}{8}\) and \(\frac{1}{6}\) with the LCD.
3. Add \(\frac{1}{4}+\frac{2}{3}\).
4. Subtract \(\frac{5}{6}-\frac{1}{4}\).

Brief summary

Finding a common denominator helps us change unlike fractions into equivalent fractions with the same denominator. We usually use the least common denominator, which is the least common multiple of the denominators. Once the denominators match, we can add or subtract the numerators and keep the denominator the same.

Adding and Subtracting Fractions with Unlike Denominators

Fractions can be added and subtracted, but there is one very important rule: the denominators must match first.

If the denominators are already the same, the job is easy. But when the denominators are different, we must rewrite the fractions so they have a common denominator.

In this lesson, you will learn how to add and subtract fractions with unlike denominators and how to check if your answer makes sense.

1. What do the numerator and denominator mean?

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

• The numerator is the top number: 3
• The denominator is the bottom number: 4

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

This is why we cannot add fractions with different denominators right away. For example, \(\frac{1}{2}\) and \(\frac{1}{3}\) are cut into different-sized pieces. We must first change them into fractions with pieces of the same size.

2. What is a common denominator?

A common denominator is a number that both denominators can divide into evenly.

For example, for \(\frac{1}{2}\) and \(\frac{1}{3}\), the denominators are 2 and 3. A common denominator is 6, because:

• 2 goes into 6
• 3 goes into 6

The least common denominator is the smallest common denominator. Using the least common denominator usually makes the work easier.

3. How to add or subtract fractions with unlike denominators

1. Find a common denominator.
2. Rewrite each fraction as an equivalent fraction with that denominator.
3. Add or subtract the numerators.
4. Keep the denominator the same.
5. Simplify if possible.
6. Check if the answer is reasonable.

4. Finding equivalent fractions

Equivalent fractions name the same amount.

For example:

$$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}$$

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

Example:

$$\frac{2}{3}=\frac{2\times 4}{3\times 4}=\frac{8}{12}$$

5. Worked Example 1: Simple addition

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

Step 1: Find a common denominator.

The denominators are 2 and 3. The least common denominator is 6.

Step 2: Rewrite each fraction.

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

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

Step 3: Add the numerators.

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

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

Reasonableness check: \(\frac{1}{2}\) is about 0.5 and \(\frac{1}{3}\) is about 0.33, so the sum should be about 0.83. Since \(\frac{5}{6}\approx 0.83\), the answer makes sense.

6. Worked Example 2: Subtraction

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

Step 1: Find a common denominator.

The denominators are 4 and 6. The least common denominator is 12.

Step 2: Rewrite each fraction.

$$\frac{3}{4}=\frac{9}{12}$$

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

Step 3: Subtract the numerators.

$$\frac{9}{12}-\frac{2}{12}=\frac{7}{12}$$

Answer: \(\frac{7}{12}\)

Reasonableness check: \(\frac{3}{4}\) is 0.75 and \(\frac{1}{6}\) is a little less than 0.2. The difference should be a little more than 0.5. Since \(\frac{7}{12}\approx 0.58\), the answer is reasonable.

7. Worked Example 3: When one denominator is a multiple of the other

Find \(\frac{5}{8}+\frac{1}{4}\).

Step 1: Find a common denominator.

The denominators are 8 and 4. Since 8 is a multiple of 4, we can use 8 as the common denominator.

Step 2: Rewrite each fraction.

$$\frac{5}{8}=\frac{5}{8}$$

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

Step 3: Add the numerators.

$$\frac{5}{8}+\frac{2}{8}=\frac{7}{8}$$

Answer: \(\frac{7}{8}\)

Reasonableness check: \(\frac{5}{8}\) is a little more than one-half, and \(\frac{1}{4}\) is 0.25. Together they should be less than 1. \(\frac{7}{8}\) is less than 1, so the answer makes sense.

8. Worked Example 4: Answer needs simplifying

Find \(\frac{2}{5}+\frac{4}{15}\).

Step 1: Find a common denominator.

The denominators are 5 and 15. Since 15 is a multiple of 5, use 15.

Step 2: Rewrite each fraction.

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

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

Step 3: Add the numerators.

$$\frac{6}{15}+\frac{4}{15}=\frac{10}{15}$$

Step 4: Simplify.

Both 10 and 15 can be divided by 5.

$$\frac{10}{15}=\frac{2}{3}$$

Answer: \(\frac{2}{3}\)

Reasonableness check: \(\frac{2}{5}=0.4\) and \(\frac{4}{15}\approx 0.27\). The total is about 0.67, which matches \(\frac{2}{3}\approx 0.67\).

9. Common mistakes to avoid

• Do not add the denominators. For example, \(\frac{1}{2}+\frac{1}{3}\neq\frac{2}{5}\).
• Do not subtract the denominators. The denominator stays the same after you rewrite both fractions with a common denominator.
• Do not forget to simplify. An answer like \(\frac{10}{15}\) should be simplified to \(\frac{2}{3}\).
• Make sure the fractions are equivalent when rewritten. Multiply the numerator and denominator by the same number.

10. How to check if your answer is reasonable

After solving, ask yourself:

• Is the sum greater than each addend when adding?
• Is the difference smaller than the number you started with when subtracting?
• Is the answer close to what you would expect from benchmark fractions like \(\frac{1}{2}\), \(\frac{1}{4}\), or 1?

Example: For \(\frac{1}{2}+\frac{1}{3}\), the answer should be more than \(\frac{1}{2}\) but less than 1. So \(\frac{5}{6}\) is reasonable.

11. Quick step-by-step guide

Use this every time:

1. Look at the denominators.
2. Find the least common denominator.
3. Rename the fractions.
4. Add or subtract the numerators.
5. Keep the denominator.
6. Simplify.
7. Check if the answer makes sense.

Summary

To add or subtract fractions with unlike denominators, you must first change them to equivalent fractions with a common denominator. Then you add or subtract only the numerators and keep the denominator the same. Finally, simplify your answer and check that it is reasonable.

Remember: same denominator first, then solve.

Adding and Subtracting Mixed Numbers with Regrouping

Mixed numbers are numbers that have a whole number and a fraction together, like \(3\frac{1}{4}\) or \(7\frac{5}{6}\).

In this lesson, you will learn how to add and subtract mixed numbers, especially when you need to regroup. Regrouping means taking 1 whole and rewriting it as a fraction so the problem can be solved more easily.

This is very important in subtraction. Sometimes the fraction on top is smaller than the fraction on the bottom, so you cannot subtract the fractional parts right away. When that happens, you regroup.

1. Review: What is a mixed number?

A mixed number has two parts:

• a whole number
• a fraction

Example: In \(4\frac{2}{3}\), the whole number is \(4\) and the fraction is \(\frac{2}{3}\).

2. Adding mixed numbers

To add mixed numbers, follow these steps:

1. Add the whole numbers.
2. Add the fractions.
3. If the fractional sum is greater than or equal to 1, regroup it into a whole number and a fraction.
4. Simplify if needed.

Worked Example 1: Add without much regrouping

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

Add the whole numbers:

$$2 + 3 = 5$$

Add the fractions:

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

Put them together:

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

Worked Example 2: Add and regroup

Solve: \(4\frac{3}{8} + 2\frac{7}{8}\)

Add the whole numbers:

$$4 + 2 = 6$$

Add the fractions:

$$\frac{3}{8} + \frac{7}{8} = \frac{10}{8}$$

The fraction \(\frac{10}{8}\) is greater than 1. Rewrite it:

$$\frac{10}{8} = \frac{8}{8} + \frac{2}{8} = 1\frac{2}{8}$$

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

$$6 + 1\frac{2}{8} = 7\frac{2}{8}$$

Simplify \(\frac{2}{8}\):

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

Final answer:

$$4\frac{3}{8} + 2\frac{7}{8} = 7\frac{1}{4}$$

3. Subtracting mixed numbers

To subtract mixed numbers, you often subtract the whole numbers and the fractions separately.

But there is a problem sometimes: the top fraction may be smaller than the bottom fraction.

For example, in \(5\frac{1}{4} - 2\frac{3}{4}\), you cannot do \(\frac{1}{4} - \frac{3}{4}\) because the top fraction is smaller.

That is when you regroup.

4. How regrouping works

Suppose you have \(5\frac{1}{4}\).

You can take 1 whole from the 5 and turn it into fourths:

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

So:

$$5\frac{1}{4} = 4 + \left(\frac{4}{4} + \frac{1}{4}\right) = 4\frac{5}{4}$$

This does not change the value. It only rewrites the number in a way that helps you subtract.

Steps for subtracting mixed numbers with regrouping

1. Check the fractions.
2. If the top fraction is smaller, subtract 1 from the top whole number.
3. Change that 1 whole into a fraction with the same denominator.
4. Add it to the top fraction.
5. Subtract the fractions.
6. Subtract the whole numbers.
7. Simplify if needed.

Worked Example 3: Subtract with regrouping

Solve: \(5\frac{1}{4} - 2\frac{3}{4}\)

Look at the fractions:

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

You cannot subtract because \(\frac{1}{4}\) is less than \(\frac{3}{4}\).

Regroup \(5\frac{1}{4}\):

Take 1 whole from 5, leaving 4.

Change that 1 whole into fourths:

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

Add it to the fraction:

$$\frac{4}{4} + \frac{1}{4} = \frac{5}{4}$$

So:

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

Now subtract:

Whole numbers:

$$4 - 2 = 2$$

Fractions:

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

Put them together:

$$2\frac{2}{4}$$

Simplify the fraction:

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

Final answer:

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

Worked Example 4: A harder subtraction example

Solve: \(7\frac{2}{3} - 4\frac{5}{3}\)

First, check the fractions:

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

You cannot subtract because \(\frac{2}{3}\) is less than \(\frac{5}{3}\).

Regroup \(7\frac{2}{3}\):

Take 1 whole from 7, leaving 6.

Change 1 whole into thirds:

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

Add it to \(\frac{2}{3}\):

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

So:

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

Now subtract:

Whole numbers:

$$6 - 4 = 2$$

Fractions:

$$\frac{5}{3} - \frac{5}{3} = 0$$

Final answer:

$$7\frac{2}{3} - 4\frac{5}{3} = 2$$

5. Important reminder about denominators

When adding or subtracting fractions in mixed numbers, the fractions must have the same denominator.

In this lesson, the examples already had matching denominators. If the denominators are different, you first rewrite the fractions so they have a common denominator.

Example:

To work with \(\frac{1}{2}\) and \(\frac{3}{4}\), you can rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\).

6. Common mistakes to avoid

• Do not forget to regroup when the top fraction is smaller.
• Do not change only the whole number. You must also turn the borrowed 1 into a fraction.
• Make sure the denominator stays the same when regrouping. For example, 1 whole becomes \(\frac{4}{4}\) if the denominator is 4.
• Always simplify your final fraction if possible.

7. Quick strategy check

When you see a subtraction problem with mixed numbers, ask yourself:

• Are the denominators the same?
• Is the top fraction big enough to subtract the bottom fraction?
• If not, can I regroup 1 whole?

Summary

To add mixed numbers, add the whole numbers and fractions separately. If the fraction part is 1 or more, regroup it into a whole number and a fraction.

To subtract mixed numbers with regrouping, check whether the top fraction is smaller than the bottom fraction. If it is, borrow 1 whole from the whole-number part, rewrite that 1 as a fraction with the same denominator, and then subtract.

Remember: regrouping does not change the value of the number. It only changes how the number is written so the subtraction is possible.

With practice, you will get faster at seeing when regrouping is needed and how to do it correctly.

Multiplying Fractions by Whole Numbers

In this lesson, you will learn how to multiply a fraction by a whole number.

This is an important skill because it helps you find repeated fractional amounts. For example, if you have 3 groups of rac{1}{4}, or if you want to find 5 times rac{2}{3}, you are multiplying a fraction by a whole number.

There are two main ways to think about this:

• Repeated addition: adding the same fraction again and again
• Scaling: making a fraction larger by multiplying it by a whole number

Lets learn both ideas.

1. Multiplying a fraction by a whole number as repeated addition

If you multiply a fraction by a whole number, you can think of it as adding that fraction over and over.

For example,

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

\(\frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\)

When adding fractions with the same denominator, you keep the denominator and add the numerators:

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

So,

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

2. A quick rule for multiplying a fraction by a whole number

To multiply a whole number by a fraction, write the whole number as a fraction over 1:

\(4 = \frac{4}{1}\)

Then multiply numerators and multiply denominators:

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

This gives the same answer as repeated addition.

3. What happens when the answer is greater than 1?

Sometimes the product is an improper fraction, which means the numerator is greater than the denominator.

For example, \(\frac{4}{3}\) is more than 1 whole.

You can change it to a mixed number:

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

So,

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

4. Multiplying to find part of a whole quantity

Multiplying fractions by whole numbers also helps you find fractional parts of a set or amount.

For example, if one ribbon piece is \(\frac{2}{5}\) meter long, then 3 pieces have length:

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

This means 3 pieces together are \(1\frac{1}{5}\) meters long.

5. Understanding multiplication as scaling

When you multiply a fraction by a whole number, you are making it larger by that number of times.

For example,

\(2 \times \frac{3}{4}\) means 2 times as much as \(\frac{3}{4}\).

That is why:

$$ 2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2} = 1\frac{1}{2} $$

You started with \(\frac{3}{4}\), and doubling it gave you \(1\frac{1}{2}\).

Steps for multiplying a fraction by a whole number

1. Write the whole number as a fraction over 1.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify if possible.
5. Change to a mixed number if needed.

In general,

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

This works because only the numerator changes when you have repeated groups of the same fraction.

Worked Example 1

Find \(3 \times \frac{1}{4}\).

Method 1: Repeated addition

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

Method 2: Multiply

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

Answer: \(\frac{3}{4}\)

Worked Example 2

Find \(5 \times \frac{2}{7}\).

Write 5 as \(\frac{5}{1}\):

$$ 5 \times \frac{2}{7} = \frac{5}{1} \times \frac{2}{7} $$

Multiply numerators and denominators:

$$ \frac{5 \times 2}{1 \times 7} = \frac{10}{7} $$

Change to a mixed number:

$$ \frac{10}{7} = 1\frac{3}{7} $$

Answer: \(1\frac{3}{7}\)

Worked Example 3

A jogger runs \(\frac{3}{5}\) mile each lap. How far does the jogger run in 4 laps?

This means:

$$ 4 \times \frac{3}{5} $$

Multiply:

$$ 4 \times \frac{3}{5} = \frac{4}{1} \times \frac{3}{5} = \frac{12}{5} $$

Change to a mixed number:

$$ \frac{12}{5} = 2\frac{2}{5} $$

Answer: The jogger runs \(2\frac{2}{5}\) miles.

Worked Example 4

A recipe uses \(\frac{3}{8}\) cup of sugar for one batch. How much sugar is needed for 6 batches?

We multiply:

$$ 6 \times \frac{3}{8} = \frac{6}{1} \times \frac{3}{8} = \frac{18}{8} $$

Simplify:

$$ \frac{18}{8} = \frac{9}{4} $$

Change to a mixed number:

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

Answer: The recipe needs \(2\frac{1}{4}\) cups of sugar.

Common mistakes to avoid

• Do not multiply the denominator by the whole number only. The denominator stays the same in problems like \(3 \times \frac{2}{5}\), so the answer is \(\frac{6}{5}\), not \(\frac{2}{15}\).
• Do not forget to simplify. For example, \(\frac{18}{8}\) should simplify to \(\frac{9}{4}\).
• Do not forget mixed numbers when needed. If your answer is improper, it is often helpful to write it as a mixed number too.

Helpful check

You can check if your answer makes sense by thinking about size.

If you multiply a fraction by a whole number greater than 1, the answer should usually be larger than the original fraction.

For example, \(5 \times \frac{1}{6}\) should be bigger than \(\frac{1}{6}\). And it is:

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

Summary

Multiplying a fraction by a whole number means adding the fraction repeatedly or finding that many groups of the fraction.

To solve, multiply the whole number by the numerator and keep the denominator the same:

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

Then simplify your answer, and change it to a mixed number if needed.

Multiplying Fractions by Fractions via Area Models

When we multiply whole numbers, we can think of multiplication as groups of something. For example, \(3 \times 4\) means 3 groups of 4.

When we multiply fractions, the idea is a little different. A fraction times a fraction often means “a part of a part.” This is why area models are so helpful. They let us see what the product means.

In this lesson, you will learn how to:

• use an area model to multiply two fractions,
• understand why the answer is the overlapping part, and
• connect the picture to the standard multiplication rule.

1. What is an area model?

An area model is a rectangle used to show fractions. We divide the rectangle into equal parts in one direction, then divide it again in the other direction. The overlap shows the product.

Think of one fraction as shading the rectangle across, and the other fraction as shading it up and down. The part that has both shadings is the answer.

For example, \(\frac{1}{2} \times \frac{1}{3}\) means:

• take \(\frac{1}{2}\) of the rectangle,
• then take \(\frac{1}{3}\) of that same whole,
• and look at the part where the two shadings overlap.

2. How to build an area model

1. Draw one rectangle to represent 1 whole.
2. Divide it into equal parts using the denominator of the first fraction.
3. Shade the number of parts shown by the numerator of the first fraction.
4. Now divide the same rectangle in the other direction using the denominator of the second fraction.
5. Shade the number of parts shown by the numerator of the second fraction in the new direction.
6. Count the overlapping parts.
7. Write the overlap as a fraction of the whole rectangle.

Important idea: The total number of small pieces in the rectangle is found by multiplying the denominators. The number of overlapping pieces is found by multiplying the numerators.

That is why:

This rule matches what we see in the area model.

3. Why does “part of a part” make the answer smaller?

If you take a fraction of something, you are taking only part of it. If you then take a fraction of that part, you usually get an even smaller piece.

For example, \(\frac{1}{2} \times \frac{1}{3}\) is not bigger than \(\frac{1}{2}\) or \(\frac{1}{3}\). It is a smaller piece: one out of six equal parts.

So when you multiply two fractions less than 1, the product is usually less than either factor.

4. Worked Example 1

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

Step 1: Draw a rectangle for 1 whole.

Step 2: Divide it into 2 equal columns and shade 1 column. This shows \(\frac{1}{2}\).

Step 3: Divide the same rectangle into 3 equal rows and shade 1 row in the other direction. This shows \(\frac{1}{3}\).

Step 4: Count the overlap.

The rectangle now has \(2 \times 3 = 6\) equal small parts. Only 1 small part is in the overlap.

So:

Check with the rule:

The picture and the rule match.

5. Worked Example 2

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

Use the area model:

• Divide the rectangle into 3 equal columns and shade 2 of them.
• Then divide the same rectangle into 4 equal rows and shade 3 of them in the other direction.

Now the rectangle has:

equal small parts.

The overlap covers:

small parts.

So the product is:

This fraction can be simplified:

So the final answer is:

Check with the rule:

6. Worked Example 3

Find \(\frac{4}{5} \times \frac{2}{3}\).

Area model steps:

• Divide the rectangle into 5 equal columns and shade 4 columns.
• Divide the same rectangle into 3 equal rows and shade 2 rows in the other direction.

The whole rectangle is divided into:

equal parts.

The overlap is:

parts.

So:

This fraction is already simplified, so the final answer is:

Check with the rule:

7. Connecting the picture to the standard algorithm

After using area models a few times, you may notice a pattern:

• the numerators tell how many parts are shaded,
• the denominators tell how many equal parts the whole is split into.

That is why multiplying fractions works like this:

This is called the standard algorithm for multiplying fractions.

Steps for the standard algorithm:

1. Multiply the numerators.
2. Multiply the denominators.
3. Simplify the fraction if possible.

Example:

8. Common mistakes to avoid

• Do not add the denominators. In multiplication, you multiply denominators.
• Do not add the numerators. You multiply numerators too.
• Remember to simplify if the answer can be reduced.
• Use one whole rectangle for both fractions in the area model. Both fractions must refer to the same whole.

Example of a mistake:

Someone says:

That is incorrect because they added the numerators and denominators. Instead, multiply:

9. How to know if your answer makes sense

Ask yourself:

• Is my answer less than 1 when I multiply two fractions less than 1?
• Does my answer seem smaller than the fractions I started with?
• Did I multiply both numerators and denominators?
• Did I simplify my answer?

For instance, \(\frac{1}{2} \times \frac{1}{3}\) should be smaller than both \(\frac{1}{2}\) and \(\frac{1}{3}\). Since \(\frac{1}{6}\) is smaller, it makes sense.

10. Quick practice thinking

If you wanted to find \(\frac{2}{5} \times \frac{1}{4}\), you could think:

• Split the rectangle into 5 columns and shade 2.
• Split the same rectangle into 4 rows and shade 1.
• The whole has \(5 \times 4 = 20\) parts.
• The overlap has \(2 \times 1 = 2\) parts.

So:

Summary

Multiplying fractions means finding a part of a part.

An area model helps by showing:

• one fraction shaded in one direction,
• the other fraction shaded in the other direction,
• and the overlap as the product.

To multiply fractions:

Then simplify if needed.

Remember:

• Multiply numerators.
• Multiply denominators.
• The answer shows the overlapping part of the whole.
• Area models help you understand why the rule works.

With practice, you will be able to use both the picture and the rule to solve fraction multiplication problems with confidence.

Scaling and Resizing via Fraction Multiplication

When we multiply by a fraction, we are often resizing a number or an amount.

Sometimes multiplication makes something bigger, and sometimes it makes something smaller. The key idea is this: the size of the fraction tells us what will happen to the product.

In this lesson, you will learn how to predict whether the answer to a multiplication problem will be greater than, less than, or equal to the starting number.

1. What does scaling mean?

Scaling means changing the size of something by multiplying.

For example, if a picture is enlarged, it is scaled up. If a recipe is cut in half, it is scaled down.

Fractions are very useful for scaling because they can represent parts of a whole.

• Multiplying by a number greater than 1 makes the amount bigger.
• Multiplying by 1 keeps the amount the same.
• Multiplying by a fraction less than 1 makes the amount smaller.

So before solving, you can often predict what the answer will be like.

2. Compare the fraction to 1

The most important step is to ask: Is the fraction less than 1, equal to 1, or greater than 1?

Here is how to tell:

• If the numerator is smaller than the denominator, the fraction is less than 1. Example: \(\frac{3}{4}\).
• If the numerator and denominator are the same, the fraction is equal to 1. Example: \(\frac{5}{5}=1\).
• If the numerator is larger than the denominator, the fraction is greater than 1. Example: \(\frac{6}{5}\).

This helps us predict the product.

3. Predicting the size of the product

Suppose you start with a number, such as 8.

• \(8 \times \frac{1}{2}\) will be less than 8 because \(\frac{1}{2}\) is less than 1.
• \(8 \times 1 = 8\), so the product is equal to 8.
• \(8 \times \frac{3}{2}\) will be greater than 8 because \(\frac{3}{2}\) is greater than 1.

This idea works for whole numbers, fractions, and mixed numbers.

4. Why does this happen?

Multiplication can mean “take groups of” or “take a part of.”

When you multiply by a fraction less than 1, you are taking only part of the original amount. That makes the result smaller.

When you multiply by 1, you keep the whole amount, so it stays the same.

When you multiply by a number greater than 1, you are taking more than one full copy of the amount. That makes the result bigger.

5. Visual thinking

Imagine a ribbon that is 12 inches long.

• If you take \(\frac{1}{2}\) of it, the new length is shorter.
• If you take \(1\) of it, the length stays the same.
• If you stretch it by a factor of \(\frac{3}{2}\), the new length is longer.

So multiplication can act like a shrink tool or an enlarge tool.

6. Worked Examples

Example 1: Predict before solving

Will \(9 \times \frac{2}{3}\) be greater than 9, less than 9, or equal to 9?

Step 1: Compare \(\frac{2}{3}\) to 1.

Since \(\frac{2}{3}\) is less than 1, the product will be less than 9.

Step 2: Solve to check.

And 6 is less than 9, just as we predicted.

Example 2: Multiplying by 1

Will \(\frac{7}{8} \times 1\) be greater than, less than, or equal to \(\frac{7}{8}\)?

Since the factor is 1, the product will be equal to \(\frac{7}{8}\).

Multiplying by 1 does not change the number.

Example 3: A fraction greater than 1

Will \(5 \times \frac{6}{5}\) be greater than 5, less than 5, or equal to 5?

Step 1: Compare \(\frac{6}{5}\) to 1.

Since 6 is greater than 5, \(\frac{6}{5}\) is greater than 1.

So the product will be greater than 5.

Step 2: Solve.

And 6 is greater than 5.

Example 4: Mixed number scaling

A recipe needs 4 cups of juice. You make only \(\frac{3}{4}\) of the recipe. How much juice do you need?

Since \(\frac{3}{4}\) is less than 1, the answer should be less than 4 cups.

Now multiply:

You need 3 cups of juice.

This makes sense because making less than the whole recipe should use less than 4 cups.

7. A quick rule to remember

• If you multiply by a fraction less than 1, the product is less than the starting number.
• If you multiply by 1, the product is equal to the starting number.
• If you multiply by a number greater than 1, the product is greater than the starting number.

8. Common mistakes to avoid

• Mistake: Thinking multiplication always makes numbers bigger.
  This is not true when multiplying by fractions less than 1.
• Mistake: Forgetting to compare the fraction to 1.
  Always check whether the fraction is less than, equal to, or greater than 1.
• Mistake: Guessing without thinking about the meaning.
  Ask yourself: “Am I taking part of the amount, the whole amount, or more than the whole amount?”

9. How to think through any problem

1. Look at the number you are multiplying by.
2. Decide whether it is less than 1, equal to 1, or greater than 1.
3. Predict whether the product will be smaller, the same, or bigger.
4. Then solve the problem to check your prediction.

10. Brief Summary

Multiplying by a fraction can resize an amount. A fraction less than 1 makes the amount smaller, 1 keeps it the same, and a number greater than 1 makes it larger.

By comparing the multiplier to 1 before solving, you can predict whether the product will be greater than, less than, or equal to the starting number. This is a powerful way to understand fraction multiplication, not just calculate it.

Dividing Unit Fractions and Whole Numbers

In this lesson, you will learn how to divide unit fractions and whole numbers.

A unit fraction is a fraction with a numerator of 1, such as \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), or \(\frac{1}{8}\).

When we divide with unit fractions, there are two important ideas:

• How many equal parts are in a fraction?
• How many fractions fit into a whole number?

We will use pictures in our minds, equal parts, and simple number reasoning to understand both ideas.

1. Dividing a Unit Fraction by a Whole Number

When you divide a unit fraction by a whole number, you are splitting that fraction into smaller equal parts.

For example, imagine \(\frac{1}{2}\) of a sandwich. If you split that half into 3 equal pieces, each piece is smaller than \(\frac{1}{2}\).

So, \(\frac{1}{2} \div 3\) means: What is one-third of one-half?

You can think of it like this:

Why? Because if a whole is cut into 2 equal parts, and then that half is cut into 3 equal parts, the whole is now cut into 6 equal parts.

Here is the pattern:

This means:

• Keep the 1 in the numerator.
• Multiply the denominator by the whole number.

2. Dividing a Whole Number by a Unit Fraction

When you divide a whole number by a unit fraction, you are asking: How many of those fractional pieces fit into the whole number?

For example:

This asks: How many fourths are in 3 wholes?

Each whole has 4 fourths. So 3 wholes have:

So:

You can use this pattern:

This means when dividing a whole number by a unit fraction, count how many of those unit fractions are in the whole number.

Visual Thinking

Visual models can help a lot.

If you have 1 whole and divide it into 5 equal parts, each part is \(\frac{1}{5}\). So there are 5 copies of \(\frac{1}{5}\) in 1 whole.

If you have 2 wholes, there are 10 copies of \(\frac{1}{5}\).

This is why:

Now think about splitting a unit fraction.

If you take \(\frac{1}{3}\) and divide it into 2 equal pieces, each piece is half of \(\frac{1}{3}\), which is \(\frac{1}{6}\).

So:

Steps for Dividing a Unit Fraction by a Whole Number

1. Start with the unit fraction.
2. Split it into the number of equal parts named by the whole number.
3. Multiply the denominator by that whole number.

Example pattern:

Steps for Dividing a Whole Number by a Unit Fraction

1. Ask how many of the unit fractions fit into 1 whole.
2. Then multiply by the number of wholes.

Example pattern:

There are 3 thirds in 1 whole, so there are 12 thirds in 4 wholes.

Worked Example 1

Find \(\frac{1}{4} \div 2\).

This means split \(\frac{1}{4}\) into 2 equal parts.

If one whole is split into 4 equal parts, and then that one part is split into 2 more equal parts, the whole is split into 8 equal parts.

So:

Worked Example 2

Find \(\frac{1}{3} \div 4\).

We are splitting one third into 4 equal parts.

Multiply the denominator by 4:

So each piece is \(\frac{1}{12}\).

Worked Example 3

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

This asks: how many thirds are in 2 wholes?

Each whole has 3 thirds. So 2 wholes have:

Therefore:

Worked Example 4

Find \(5 \div \frac{1}{2}\).

This asks: how many halves are in 5 wholes?

Each whole has 2 halves. So 5 wholes have:

So:

Word Problem Example

A ribbon is \(\frac{1}{2}\) yard long. It is cut into 5 equal pieces. How long is each piece?

This means:

Multiply the denominator by 5:

Each piece is \(\frac{1}{10}\) yard long.

Another Word Problem Example

There are 4 pizzas. Each serving is \(\frac{1}{8}\) of a pizza. How many servings are there?

This means:

Each pizza has 8 eighths, so 4 pizzas have:

So there are 32 servings.

Important Ideas to Remember

• Dividing a unit fraction by a whole number makes the pieces smaller.
• Dividing a whole number by a unit fraction tells how many fractional pieces fit inside.
• Unit fractions always have a numerator of 1.
• Use pictures and equal parts to make sense of the division.

Quick Check

• \(\frac{1}{5} \div 2 = \frac{1}{10}\)
• \(\frac{1}{6} \div 3 = \frac{1}{18}\)
• \(3 \div \frac{1}{2} = 6\)
• \(6 \div \frac{1}{3} = 18\)

Summary

To divide a unit fraction by a whole number, split the fraction into equal parts. The denominator gets multiplied by the whole number.

To divide a whole number by a unit fraction, find how many of those fractional pieces fit into the whole number. Multiply the number of wholes by the denominator of the unit fraction.

Always think about the meaning of the problem: Are you splitting a fraction into smaller equal parts, or are you counting how many fractional pieces fit in a whole number?

Dividing Fractions by Fractions

In this lesson, you will learn how to divide one fraction by another fraction.

At first, fraction division can look tricky. But once you understand what division means and learn one reliable method, it becomes much easier.

We will use two ways to understand fraction division:

• A thinking method using equal-size parts and common denominators
• A shortcut method called multiply by the reciprocal

Both methods give the same answer.

What does division mean?

Division asks, "How many groups?" or "How many times does one amount fit into another?"

For whole numbers, if you divide \(12 \div 3\), you are asking, “How many groups of 3 are in 12?” The answer is 4.

For fractions, the idea is the same. If you divide \(\frac{3}{4} \div \frac{1}{8}\), you are asking, “How many \(\frac{1}{8}\) pieces fit into \(\frac{3}{4}\)?”

Method 1: Use common denominators to understand the division

This method helps you see what is happening.

Suppose we want to find:

To compare these fractions easily, write them with the same denominator.

We know that:

So now the problem becomes:

Now ask: how many \(\frac{1}{8}\) pieces are in \(\frac{6}{8}\)?

There are 6 of them.

So:

This makes sense because six eighths contains 6 pieces of size one eighth.

Another common denominator example

Find:

Write \(\frac{2}{3}\) with denominator 6:

Now divide:

There are 4 pieces of size \(\frac{1}{6}\) in \(\frac{4}{6}\).

So:

This method is especially helpful when one denominator can be changed to match the other easily.

Method 2: Multiply by the reciprocal

This is the main method you will use to divide fractions.

Reciprocal means flip the fraction.

• The reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).
• The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
• The reciprocal of \(\frac{7}{1}\) is \(\frac{1}{7}\).

To divide by a fraction, keep the first fraction the same, change division to multiplication, and flip the second fraction.

This is often called:

Keep, Change, Flip

1. Keep the first fraction
2. Change division to multiplication
3. Flip the second fraction

In symbols:

Then multiply as usual.

Why does this work?

You do not need a complicated proof in 6th Grade, but here is the idea: dividing by a fraction is the same as finding how many of those fractional parts fit inside another amount. Flipping the second fraction helps us count those parts correctly.

The common denominator method and the reciprocal method always match.

Steps for dividing fractions

1. Write the problem.
2. Keep the first fraction.
3. Change \(\div\) to \(\times\).
4. Flip the second fraction.
5. Multiply the numerators.
6. Multiply the denominators.
7. Simplify if needed.

Worked Example 1

Find:

Use keep, change, flip:

Multiply:

Answer: \(2\)

Check with thinking: how many fourths are in one half? Since \(\frac{1}{2} = \frac{2}{4}\), there are 2 fourths in one half.

Worked Example 2

Find:

Use keep, change, flip:

Multiply:

Answer: \(\frac{9}{10}\)

This answer is less than 1, which makes sense because \(\frac{2}{3}\) is larger than \(\frac{3}{5}\), so fewer than one full group fits.

Worked Example 3

Find:

Use keep, change, flip:

Now multiply:

Answer: \(6\)

You can also see this with common denominators. Since \(\frac{4}{7} = \frac{12}{21}\), the problem becomes:

So the two methods agree.

Worked Example 4

Find:

Use keep, change, flip:

Multiply:

Simplify:

Answer: \(\frac{10}{9}\) or \(1\frac{1}{9}\)

This answer is greater than 1. That makes sense because we are asking how many \(\frac{3}{4}\) groups fit into \(\frac{5}{6}\). Since \(\frac{5}{6}\) is a little bigger than \(\frac{3}{4}\), the answer should be a little more than 1.

How to simplify while you work

Sometimes you can make the multiplication easier by simplifying before multiplying.

Example:

Since 21 and 7 share a factor of 7:

Now multiply:

This is a quicker way, but if simplifying first feels confusing, it is okay to multiply first and simplify at the end.

What if there are mixed numbers?

Before dividing, change mixed numbers into improper fractions.

For example, if you had \(1\frac{1}{2} \div \frac{3}{4}\), first change \(1\frac{1}{2}\) to \(\frac{3}{2}\), then divide:

Common mistakes to avoid

• Do not flip the first fraction. Only the second fraction gets flipped.
• Do not multiply before changing division to multiplication.
• Do not forget to simplify.
• Be careful with mixed numbers. Change them to improper fractions first.

Quick sense check

After solving, ask yourself if the answer makes sense.

• If you divide by a fraction smaller than 1, the answer often gets bigger.
• If the two fractions are equal, the answer is 1.
• If the divisor is larger than the first fraction, the answer is less than 1.

Example:

• \(\frac{1}{2} \div \frac{1}{4} = 2\), which is bigger than \(\frac{1}{2}\)
• \(\frac{3}{5} \div \frac{3}{5} = 1\)
• \(\frac{3}{5} \div \frac{4}{5} = \frac{3}{4}\), which is less than 1

Summary

Dividing fractions means finding how many of one fractional amount fit into another.

You can understand fraction division by rewriting fractions with a common denominator, especially when the pieces are easy to compare.

The main method is:

1. Keep the first fraction
2. Change division to multiplication
3. Flip the second fraction

Then multiply and simplify.

With practice, dividing fractions becomes just as manageable as multiplying them.