Arithmetic Operations with Rational Numbers

Integer Addition and the Zero Pair Model

When we add positive and negative integers, it helps to see what is happening. Two useful models are the number line and the zero pair model.

A positive integer is a whole number greater than 0, such as 1, 2, or 7. A negative integer is a whole number less than 0, such as , , or .

In this lesson, you will learn how to add integers by using:

• vector movements on a number line, and
• two-color counters to make zero pairs.

These models help explain why the rules for integer addition work.

1. Review: opposites and additive inverses

Every integer has an opposite. The opposite of 5 is . The opposite of  is 3.

Opposites are also called additive inverses because when you add a number and its opposite, the sum is 0.

For example:

$$5 + (-5) = 0$$

$$-3 + 3 = 0$$

2. What is a zero pair?

In the zero pair model, we use two kinds of counters:

• one color for +1
• another color for 1

One positive counter and one negative counter make a zero pair, because together they are worth 0.

$$+1 + (-1) = 0$$

So any time you see one positive counter matched with one negative counter, you can remove both without changing the total value.

3. Integer addition with counters

To add integers using counters:

1. Show the first integer with counters.
2. Show the second integer with counters.
3. Combine all the counters.
4. Remove all the zero pairs.
5. Count what is left.

If more positive counters are left, the sum is positive. If more negative counters are left, the sum is negative.

4. Integer addition on a number line

A number line also helps us model addition.

• Start at the first number.
• If you add a positive number, move right.
• If you add a negative number, move left.

You can think of each addend as a movement, or vector, on the number line.

For example, to find  + 5:

• start at 
• move 5 units right
• land on 3

So, $$-2 + 5 = 3$$

5. Important patterns in integer addition

There are some patterns you will notice:

• Adding a positive integer moves the value up.
• Adding a negative integer moves the value down.
• Adding opposites gives 0.
• When signs are different, the counters cancel in zero pairs.

You do not have to memorize rules first. The models show the rules naturally.

6. Worked Examples

Example 1: Add two positive integers

Find $$3 + 2$$

Using counters: Show 3 positive counters, then 2 more positive counters. There are 5 positive counters total.

No zero pairs can be made because there are no negative counters.

So, $$3 + 2 = 5$$

Using the number line: Start at 3. Move 2 units right. You land on 5.

Example 2: Add a positive and a negative integer

Find $$4 + (-3)$$

Using counters: Show 4 positive counters and 3 negative counters.

Match 3 positive counters with 3 negative counters to make 3 zero pairs.

After removing the zero pairs, 1 positive counter is left.

So, $$4 + (-3) = 1$$

Using the number line: Start at 4. Adding  means move 3 units left. You land on 1.

Example 3: Add a negative and a positive integer

Find $$-6 + 2$$

Using counters: Show 6 negative counters and 2 positive counters.

Make 2 zero pairs. Remove them.

There are 4 negative counters left.

So, $$-6 + 2 = -4$$

Using the number line: Start at . Move 2 units right because you are adding 2. You land on .

Example 4: Add two negative integers

Find $$-3 + (-4)$$

Using counters: Show 3 negative counters and 4 more negative counters.

Now there are 7 negative counters total. There are no positive counters, so no zero pairs can be removed.

So, $$-3 + (-4) = -7$$

Using the number line: Start at . Then move 4 units left because you are adding . You land on .

7. What if the sum is zero?

Sometimes all the counters cancel out.

For example, find $$-5 + 5$$

Show 5 negative counters and 5 positive counters. Every negative counter matches with a positive counter to make a zero pair.

After removing all 5 zero pairs, nothing is left. The value is 0.

So, $$-5 + 5 = 0$$

8. How to decide the sign of the answer

When both integers have the same sign:

• add the numbers
• keep the common sign

Examples:

$$2 + 6 = 8$$

$$-2 + (-6) = -8$$

When the integers have different signs:

• make zero pairs
• subtract the smaller absolute value from the larger absolute value
• use the sign of the number with the larger absolute value

Remember: absolute value means distance from 0.

Example:

$$-8 + 3$$

The absolute values are 8 and 3. Since 8 is larger, more negative counters remain after making zero pairs.

$$-8 + 3 = -5$$

9. Common mistakes to avoid

• Do not always add the numbers and keep a positive sign. The signs matter.
• Do not forget direction on the number line. Positive means right, negative means left.
• Do not remove counters unless they form a zero pair. A zero pair must be one positive and one negative.
• Do not confuse subtraction with adding a negative. In this lesson, focus on what the second addend tells you: move right for positive, left for negative.

10. Try these on your own

• $$5 + (-2)$$
• $$-7 + 4$$
• $$-3 + (-2)$$
• $$6 + (-6)$$

Think about each one using counters or a number line. Ask yourself:

• Can I make zero pairs?
• Which direction do I move on the number line?
• What counters are left after canceling?

11. Brief Summary

Integer addition can be understood with models, not just rules. On a number line, adding a positive means move right and adding a negative means move left.

In the zero pair model, one positive counter and one negative counter make 0. When adding integers with different signs, zero pairs cancel, and the counters left show the answer.

These models help you see why sums like $$4 + (-3) = 1$$ and $$-6 + 2 = -4$$ are true.

Integer subtraction as additive inverse means that when you subtract an integer, you can rewrite the subtraction as adding the opposite.

This is one of the most important ideas for working with positive and negative numbers. It helps you solve subtraction problems correctly and understand what is happening on a number line.

The key rule is:

$$a-b=a+(-b)$$

This says: subtracting a number is the same as adding its opposite.

For example, subtracting 5 is the same as adding  5. Subtracting  3 is the same as adding 3.

Opposites are numbers the same distance from 0 on the number line, but on different sides.

• The opposite of 4 is  4
• The opposite of  7 is 7
• The opposite of 0 is 0

So when you see subtraction, you can think:

1. Keep the first number the same.
2. Change subtraction to addition.
3. Change the second number to its opposite.

This is sometimes called Keep, Change, Change.

Example pattern:

$$6-(-2)=6+2$$

We kept 6, changed subtraction to addition, and changed  2 to its opposite, 2.

Why does this work? Think about what subtraction means. Subtraction can mean finding the difference or distance between two numbers, and it can also mean removing a value. With integers, using adding the opposite gives a clear and reliable method every time.

On a number line, adding a positive number moves you to the right. Adding a negative number moves you to the left.

•  Adding 3 means move 3 units right.
•  Adding  3 means move 3 units left.

Because subtraction becomes adding the opposite, a subtraction problem tells you which direction to move after changing it to addition.

For example:

$$5-2=5+(-2)$$

Start at 5. Add  2, so move 2 units left. You land on 3.

And:

$$5-(-2)=5+2$$

Start at 5. Add 2, so move 2 units right. You land on 7.

This helps explain an important idea:

• Subtracting a positive moves left.
• Subtracting a negative moves right.

Now lets work through some examples from easier to harder.

Worked Example 1

Solve:

$$8-3$$

Use adding the opposite:

$$8-3=8+(-3)$$

Now add:

$$8+(-3)=5$$

Answer: 5

Number line thinking: start at 8 and move 3 units left.

Worked Example 2

Solve:

$$4-(-6)$$

Change subtraction to addition and change the second number to its opposite:

$$4-(-6)=4+6$$

Now add:

$$4+6=10$$

Answer: 10

Number line thinking: start at 4. Since subtracting  6 becomes adding 6, move 6 units right.

Worked Example 3

Solve:

$$-3-5$$

Rewrite using the opposite:

$$-3-5=-3+(-5)$$

Now add two negative numbers. Move 5 units left from  3:

$$-3+(-5)=-8$$

Answer:  8

Worked Example 4

Solve:

$$-7-(-4)$$

Use adding the opposite:

$$-7-(-4)=-7+4$$

Now add 4 to  7:

$$-7+4=-3$$

Answer:  3

Number line thinking: start at  7 and move 4 units right.

Subtraction as distance on a number line is another useful way to understand integer subtraction.

For example, in the problem

$$3-(-2)$$

you can also think about the distance from  2 to 3 on the number line. From  2 to 3 is 5 units, so the answer is 5.

This matches the additive inverse method:

$$3-(-2)=3+2=5$$

Here is another one:

$$-1-4=-1+(-4)=-5$$

Starting at  1 and moving 4 units left lands at  5.

Common mistakes to avoid

• Do not keep the subtraction sign. After rewriting, subtraction changes to addition.
• Do not forget to change the second number to its opposite. In $$6-(-2)$$, the opposite of  2 is 2.
• Watch double negatives carefully. A problem like $$-4-(-3)$$ becomes $$-4+3$$.
• Start at the first number. The first number tells where you begin on the number line.

Helpful reminders

•  Subtract positive  add negative
•  Subtract negative  add positive
•  Adding a negative moves left
•  Adding a positive moves right

Lets compare a few problems:

• $$6-2=6+(-2)=4$$
• $$6-(-2)=6+2=8$$
• $$-6-2=-6+(-2)=-8$$
• $$-6-(-2)=-6+2=-4$$

Notice how changing the sign of the second number changes the direction and the result.

How to solve any integer subtraction problem

1. Look at the subtraction problem.
2. Keep the first integer.
3. Change subtraction to addition.
4. Change the second integer to its opposite.
5. Add.

For example:

$$9-(-7)$$

becomes

$$9+7=16$$

And

$$-2-8$$

becomes

$$-2+(-8)=-10$$

Summary

Integer subtraction can always be rewritten as adding the opposite. This means:

$$a-b=a+(-b)$$

Using this rule makes subtraction with positive and negative numbers much easier. It also matches what happens on a number line: adding positive numbers moves right, and adding negative numbers moves left.

When you remember Keep, Change, Change, you have a reliable way to solve integer subtraction problems correctly.

Multiplicative Sign Patterns help us figure out the sign of a product when multiplying positive and negative integers. This lesson will show why the sign rules work, not just what to memorize.

When we multiply integers, there are two parts to think about:

• the sign of the answer: positive or negative
• the value of the answer: the product of the number parts

For example, in \((-3)\times 4\), the value part is \(3\times 4=12\). The sign part tells us whether the answer is \(12\) or \(-12\).

Let’s start with patterns you may already know.

If we multiply a positive number by positive numbers, we get positive answers:

Notice what happens as the second factor goes down by 1 each time. The products go down by 3 each time:

• \(3\times 4=12\)
• \(3\times 3=9\)
• \(3\times 2=6\)
• \(3\times 1=3\)
• \(3\times 0=0\)

If we keep the pattern going, we subtract 3 again:

• \(3\times (-1)=-3\)
• \(3\times (-2)=-6\)
• \(3\times (-3)=-9\)

This pattern shows that a positive times a negative is negative.

We can also think about multiplication as repeated addition when the second number is positive.

For example:

This means negative times positive is negative.

So far we have learned:

• positive \(\times\) positive = positive
• positive \(\times\) negative = negative
• negative \(\times\) positive = negative

Now let’s discover the last rule: what happens with negative \(\times\) negative?

Look at this pattern:

• \((-2)\times 3=-6\)
• \((-2)\times 2=-4\)
• \((-2)\times 1=-2\)
• \((-2)\times 0=0\)

Again, the second factor goes down by 1 each time. The products must keep going up by 2 each time:

• \((-2)\times (-1)=2\)
• \((-2)\times (-2)=4\)
• \((-2)\times (-3)=6\)

This pattern suggests that negative times negative is positive.

We can also use the distributive property to check this idea. The distributive property says:

Let’s use it with \((-3)\times (2+(-2))\). Since \(2+(-2)=0\), we know:

Now distribute:

Substitute what we know:

To make the total equal 0, the missing product must be \(6\):

So the rule is confirmed: a negative times a negative is positive.

Here are the multiplication sign rules all together:

• Same signs \(\rightarrow\) positive answer
• Different signs \(\rightarrow\) negative answer

You can also write them like this:

• \((+)\times (+)=(+)\)
• \((+)\times (-)=(-)\)
• \((- )\times (+)=(-)\)
• \((- )\times (-)=(+)\)

After deciding the sign, multiply the number parts as usual.

Worked Example 1

Find \(5\times (-7)\).

The signs are different, so the answer is negative.

Multiply the number parts: \(5\times 7=35\).

So,

Worked Example 2

Find \((-6)\times 4\).

The signs are different, so the answer is negative.

Multiply the number parts: \(6\times 4=24\).

So,

Worked Example 3

Find \((-8)\times (-3)\).

The signs are the same, so the answer is positive.

Multiply the number parts: \(8\times 3=24\).

So,

Worked Example 4

Find \((-2)\times (-5)\times 3\).

Multiply step by step.

First, \((-2)\times (-5)\). The signs are the same, so the result is positive:

Now multiply by 3:

So,

A helpful strategy is:

1. Look at the signs.
2. Decide whether the answer will be positive or negative.
3. Multiply the number parts.

Be careful about a common mistake: some students think any problem with a negative number has a negative answer. That is not true. If both factors are negative, the product is positive.

For example:

Another important idea is that multiplying by 0 always gives 0, no matter what the sign is:

Summary

Multiplicative sign patterns show us how signs behave in multiplication. By using number patterns, repeated addition, and the distributive property, we can understand the rules instead of just memorizing them.

• same signs \(\rightarrow\) positive product
• different signs \(\rightarrow\) negative product

Once you know the sign rule, multiply the number parts to get the final answer.

Division Involving Zero and Signed Quotients

When we divide rational numbers, we need to pay close attention to signs and to the special role of zero.

In this lesson, you will learn how to divide positive and negative numbers, what happens when zero is involved, and why dividing by zero is undefined.

1. What is a quotient?

The answer to a division problem is called the quotient.

For example, in \(12 \div 3 = 4\), the quotient is \(4\).

When rational numbers are positive or negative, the quotient can also be positive or negative.

2. Rules for signed quotients

To divide signed numbers, first decide whether the answer will be positive or negative. Then divide the absolute values.

• Same signs → positive quotient
• Different signs → negative quotient

These sign rules are the same for integers, fractions, and decimals.

Here is the pattern:

3. How to divide signed numbers

1. Look at the signs.
2. Use the sign rules to decide whether the quotient is positive or negative.
3. Divide the numbers as usual.

Example: \((-20) \div 5\)

• The signs are different, so the answer is negative.
• \(20 \div 5 = 4\)
• So, \((-20) \div 5 = -4\)

4. Division when zero is the dividend

The dividend is the number being divided. In \(a \div b\), the dividend is \(a\).

If the dividend is zero, and the divisor is not zero, the quotient is always zero.

Why? Because we ask, “How many groups of \(5\) are in \(0\)?” There are no groups, so the answer is \(0\).

Another way to think about it is with multiplication. Since

it makes sense that

This works for any nonzero divisor:

5. Division when zero is the divisor

The divisor is the number you divide by. In \(a \div b\), the divisor is \(b\).

If the divisor is zero, the expression is undefined.

That means it does not have a value in ordinary arithmetic.

For example:

Why can’t we divide by zero?

Remember that division can be checked with multiplication.

If \(6 \div 0\) had an answer, then there would have to be some number \(n\) such that

But multiplying any number by zero always gives zero, never \(6\).

So there is no number that works, which means \(6 \div 0\) is undefined.

Now think about \(0 \div 0\). If it had just one answer, then we would need a number \(n\) such that

But this is true for every number. For example, \(0 \cdot 1 = 0\), \(0 \cdot 5 = 0\), and \(0 \cdot (-3) = 0\).

Since there is not one single answer, \(0 \div 0\) is also undefined.

6. Important zero facts to remember

• \(0 \div a = 0\) for any nonzero number \(a\)
• \(a \div 0\) is undefined for any number \(a\)
• \(0 \div 0\) is undefined

7. Worked examples

Example 1: Divide signed integers

Find \((-18) \div (-3)\).

Step 1: The signs are the same, so the quotient is positive.

Step 2: Divide the absolute values: \(18 \div 3 = 6\).

Answer:

Example 2: Divide numbers with different signs

Find \(24 \div (-6)\).

Step 1: The signs are different, so the quotient is negative.

Step 2: Divide the absolute values: \(24 \div 6 = 4\).

Answer:

Example 3: Zero as the dividend

Find \(0 \div (-7)\).

Zero divided by any nonzero number is zero.

Answer:

Example 4: Zero as the divisor

Find \((-15) \div 0\).

Division by zero is undefined.

Answer:

8. Common mistakes to avoid

• Mistake: Thinking \(a \div 0 = 0\)
  That is not true. Division by zero is undefined.
• Mistake: Forgetting the sign rule
  Always check whether the signs are the same or different.
• Mistake: Thinking \(0 \div a\) is undefined
  If \(a \ne 0\), then the answer is \(0\).

9. Quick check

Try these on your own:

1. \((-32) \div 8\)
2. \((-45) \div (-9)\)
3. \(0 \div 12\)
4. \(7 \div 0\)

Answers:

1. \(-4\)
2. \(5\)
3. \(0\)
4. undefined

10. Summary

When dividing signed numbers, use the signs to decide whether the quotient is positive or negative. Same signs give a positive quotient, and different signs give a negative quotient.

Zero divided by any nonzero number is zero. But any number divided by zero is undefined, because no number can make the multiplication check work.

Fraction Addition and Subtraction Algorithms

Fractions are numbers that name parts of a whole. When we add or subtract fractions, we are combining or comparing parts.

The most important rule is this: you can only add or subtract fractions easily when they refer to the same-sized parts. In math, that means the fractions must have the same denominator.

For example, eighths can be added to eighths, and fifths can be subtracted from fifths. But you cannot directly combine thirds and fourths until you rewrite them as equal-sized parts.

Key Idea: The denominator tells the size of the parts. The numerator tells how many parts you have.

So when adding or subtracting fractions:

• Keep the denominator the same only after the fractions have a common denominator.
• Add or subtract the numerators.
• Simplify if possible.

1. Adding and subtracting fractions with the same denominator

If two fractions already have the same denominator, the job is simple. Add or subtract the numerators and keep the denominator.

In general:

Example 1: Same denominator

Find:

The denominators are already the same, so add the numerators:

Answer: \(\frac{7}{10}\)

Now try subtraction:

Simplify \(\frac{4}{12}\) by dividing top and bottom by 4:

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

2. Fractions with different denominators

If the denominators are different, first find a common denominator. A very efficient choice is the least common multiple (LCM) of the denominators.

The LCM is the smallest number that both denominators divide into evenly.

For example:

• Multiples of 3: 3, 6, 9, 12, 15, 18, ...
• Multiples of 4: 4, 8, 12, 16, 20, ...

The least common multiple of 3 and 4 is 12, so 12 is the least common denominator.

Algorithm for adding or subtracting fractions with different denominators

1. Find the LCM of the denominators.
2. Rewrite each fraction as an equivalent fraction with that denominator.
3. Add or subtract the numerators.
4. Keep the common denominator.
5. Simplify, and change to a mixed number if needed.

Example 2: Add fractions with different denominators

Find:

Step 1: Find the LCM of 3 and 4.

The LCM is 12.

Step 2: Rewrite each fraction with denominator 12.

To change \(\frac{2}{3}\) into twelfths, multiply top and bottom by 4:

To change \(\frac{1}{4}\) into twelfths, multiply top and bottom by 3:

Step 3: Add the numerators.

Answer: \(\frac{11}{12}\)

Example 3: Subtract fractions with different denominators

Find:

Step 1: Find the LCM of 6 and 8.

Multiples of 6: 6, 12, 18, 24, ...

Multiples of 8: 8, 16, 24, ...

So the LCM is 24.

Step 2: Rewrite each fraction with denominator 24.

because \(6\times 4=24\), so multiply the numerator by 4 too.

because \(8\times 3=24\), so multiply the numerator by 3 too.

Step 3: Subtract the numerators.

Answer: \(\frac{17}{24}\)

3. Why equivalent fractions matter

Equivalent fractions name the same amount in different ways. For example:

When we rewrite fractions to have a common denominator, we are not changing their value. We are only changing the size of the pieces so both fractions use the same-sized parts.

This is why the algorithm works.

4. Adding and subtracting mixed numbers

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

There are two common ways to work with mixed numbers:

• Rewrite them as improper fractions.
• Or add/subtract the whole numbers and fractions separately when possible.

Using improper fractions is a reliable method, especially when denominators are different.

Example 4: Add mixed numbers

Find:

Step 1: Rewrite as improper fractions.

Step 2: Find the LCM of 5 and 3.

The LCM is 15.

Step 3: Rewrite each fraction with denominator 15.

Step 4: Add.

Step 5: Write as a mixed number.

Answer: \(3\frac{11}{15}\)

5. Subtracting mixed numbers carefully

When subtracting mixed numbers, rewriting as improper fractions helps avoid mistakes.

For example, in a problem like \(3\frac{1}{4}-1\frac{2}{3}\), the fraction parts have different denominators, so improper fractions make the work clearer.

Let’s solve it.

Example 5: Subtract mixed numbers

Step 1: Rewrite as improper fractions.

Step 2: Find the LCM of 4 and 3.

The LCM is 12.

Step 3: Rewrite each fraction with denominator 12.

Step 4: Subtract.

Step 5: Write as a mixed number.

Answer: \(1\frac{7}{12}\)

6. Common mistakes to avoid

• Do not add the denominators. For example, \(\frac{1}{4}+\frac{1}{4}\neq\frac{2}{8}\). The correct answer is \(\frac{2}{4}=\frac{1}{2}\).
• Do not subtract the denominators. Only the numerators change after you have a common denominator.
• Do not forget to use equivalent fractions. If denominators are different, rewrite first.
• Always simplify your final answer if possible.
• Check mixed numbers by converting back if needed.

7. Quick checklist for every problem

1. Are the denominators the same?
2. If not, what is the LCM?
3. Did I rewrite each fraction correctly?
4. Did I add or subtract only the numerators?
5. Did I simplify the answer?
6. If needed, did I write the answer as a mixed number?

Summary

To add or subtract fractions, the fractions must have a common denominator. The best choice is often the least common multiple of the denominators.

Once the fractions are rewritten as equivalent fractions with the same denominator, add or subtract the numerators, keep the denominator, and simplify. The same idea works for mixed numbers, especially when you first rewrite them as improper fractions.

Fraction Multiplication and Scaling

Fractions can be multiplied just like whole numbers, but they also have an important meaning: scaling.

When you multiply by a fraction, you are often finding part of a part or changing the size of something. For example, finding \\(\frac{1}{2}\\) of \\(\frac{3}{4}\\) means taking half of three-fourths.

In this lesson, you will learn how to multiply fractions and mixed numbers, and how to understand multiplication as a way to shrink or stretch a quantity.

1. What does fraction multiplication mean?

There are two big ways to think about multiplying fractions:

• Part of a part: For example, \\(\frac{2}{3} \times \frac{3}{5}\\) means two-thirds of three-fifths.
• Scaling: For example, multiplying by \\(\frac{1}{2}\\) makes a number half as large, while multiplying by \\(\frac{3}{2}\\) makes it larger.

If you multiply by a fraction less than 1, the answer gets smaller. If you multiply by a fraction greater than 1, the answer gets larger.

• If \\(0 < \text{fraction} < 1\\), the product is smaller.
• If the fraction equals 1, the number stays the same.
• If the fraction is greater than 1, the product is larger.

2. How to multiply fractions

To multiply two fractions, multiply the numerators and multiply the denominators.

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

Steps:

1. Multiply the top numbers (numerators).
2. Multiply the bottom numbers (denominators).
3. Simplify the answer if possible.

Example pattern:

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

3. Why this works: area model

One way to understand fraction multiplication is with an area model. Imagine a rectangle.

If you shade \\(\frac{3}{4}\\) of the rectangle in one direction and then shade \\(\frac{2}{3}\\) of it in the other direction, the overlapping part shows \\(\frac{2}{3} \times \frac{3}{4}\\).

The overlap covers:

$$ \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2} $$

So \\(\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}\\).

This shows that multiplying fractions finds the area of the overlapping part, or the part of a part.

4. Multiplying whole numbers and fractions

A whole number can be written as a fraction with denominator 1.

For example, \\(3 = \frac{3}{1}\\).

So if you multiply a whole number by a fraction, use the same rule.

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

This also makes sense as scaling: 3 groups of \\(\frac{2}{5}\\) equals \\(\frac{6}{5}\\).

5. Multiplying mixed numbers

A mixed number must be changed into an improper fraction before multiplying.

For example:

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

Then multiply as usual.

Steps for mixed numbers:

1. Convert each mixed number to an improper fraction.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify.
5. Change back to a mixed number if needed.

6. Simplifying before multiplying

Sometimes you can simplify before multiplying. This is called cross-canceling.

If a numerator and a denominator have a common factor, divide them before multiplying. This keeps the numbers smaller and easier to work with.

Example:

$$ \frac{2}{3} \times \frac{9}{10} $$

The 9 and 3 can be simplified:

$$ \frac{2}{\cancel{3}} \times \frac{\cancel{9}^{3}}{10} = \frac{2}{1} \times \frac{3}{10} = \frac{6}{10} = \frac{3}{5} $$

7. Multiplication as scaling

Scaling means changing the size of a number by a factor.

Here is how multiplication changes a number:

• Multiplying by \\(\frac{1}{2}\\) makes it half as big.
• Multiplying by \\(\frac{3}{4}\\) makes it three-fourths as big.
• Multiplying by \\(\frac{5}{4}\\) makes it bigger, because \\(\frac{5}{4} > 1\\).

Examples:

• \\(8 \times \frac{1}{2} = 4\\), so the number is smaller.
• \\(8 \times \frac{3}{2} = 12\\), so the number is larger.

This helps you check if your answer makes sense.

If you multiply 6 by \\(\frac{2}{3}\\), your answer should be less than 6, because \\(\frac{2}{3}\\) is less than 1.

Worked Example 1: Multiply two simple fractions

Find \\(\frac{3}{5} \times \frac{2}{7}\\).

Step 1: Multiply the numerators.

$$ 3 \times 2 = 6 $$

Step 2: Multiply the denominators.

$$ 5 \times 7 = 35 $$

Step 3: Write the product.

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

Answer: \\(\frac{6}{35}\\)

Worked Example 2: Multiply a whole number and a fraction

Find \\(4 \times \frac{3}{8}\\).

Write 4 as a fraction:

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

Now multiply:

$$ \frac{4}{1} \times \frac{3}{8} = \frac{12}{8} $$

Simplify:

$$ \frac{12}{8} = \frac{3}{2} = 1\frac{1}{2} $$

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

Notice that multiplying by \\(\frac{3}{8}\\) makes 4 smaller, because \\(\frac{3}{8} < 1\\).

Worked Example 3: Multiply mixed numbers

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

Step 1: Convert to improper fractions.

$$ 1\frac{1}{2} = \frac{3}{2}, \qquad 2\frac{1}{3} = \frac{7}{3} $$

Step 2: Multiply.

$$ \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} $$

Step 3: Simplify.

$$ \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2} $$

Answer: \\(3\frac{1}{2}\\)

You can also notice that one factor, \\(2\frac{1}{3}\\), is greater than 1, so the answer should be greater than \\(1\frac{1}{2}\\).

Worked Example 4: Scaling in a word problem

A ribbon is 12 inches long. You use \\(\frac{3}{4}\\) of it. How many inches of ribbon do you use?

This means find \\(\frac{3}{4}\\) of 12:

$$ 12 \times \frac{3}{4} = \frac{12}{1} \times \frac{3}{4} = \frac{36}{4} = 9 $$

Answer: 9 inches

This is scaling because \\(\frac{3}{4}\\) makes 12 smaller.

8. Common mistakes to avoid

• Adding instead of multiplying: \\(\frac{1}{2} \times \frac{1}{3}\\) is not \\(\frac{2}{5}\\). Multiply tops and bottoms.
• Forgetting to simplify: Always reduce your final answer if possible.
• Not converting mixed numbers: Change mixed numbers to improper fractions before multiplying.
• Ignoring size: If you multiply by a fraction less than 1, the answer should be smaller than the starting number.

9. Quick check ideas

Ask yourself:

• Did I multiply numerator by numerator and denominator by denominator?
• Did I simplify my answer?
• If I multiplied by a fraction less than 1, is my answer smaller?
• If I multiplied by a number greater than 1, is my answer larger?

Summary

To multiply fractions, multiply the numerators and multiply the denominators. Mixed numbers must be changed to improper fractions first.

Fraction multiplication can mean part of a part, and it can also mean scaling. Multiplying by a fraction less than 1 makes a quantity smaller, while multiplying by a fraction greater than 1 makes it larger.

Understanding both the rule and the meaning helps you solve problems correctly and check whether your answer makes sense.

Fraction Division and Reciprocals

Fractions can be added, subtracted, multiplied, and divided. In this lesson, we will focus on dividing fractions and understanding reciprocals.

This is an important skill because division with fractions shows up in many math problems and real-life situations, like sharing food, measuring ingredients, or figuring out how many groups can be made.

By the end of this lesson, you should be able to:

• Explain what a reciprocal is.
• Divide fractions by multiplying by the reciprocal.
• Divide positive and negative rational numbers written as fractions.
• Understand why the rule works using simple fraction reasoning.

1. What is a reciprocal?

A reciprocal of a number is what you get when you flip the fraction.

If a fraction is \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\), as long as the number is not 0.

Examples:

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

A number times its reciprocal equals 1.

For example:

This is why reciprocals are also called multiplicative inverses.

Important: 0 does not have a reciprocal, because no number multiplied by 0 can equal 1.

2. What does fraction division mean?

Division asks, "How many groups?" or "How much is in each group?"

For example, consider:

This asks, “How many \(\frac{1}{4}\) pieces fit into \(\frac{1}{2}\)?”

If you picture a whole split into 4 equal parts, then \(\frac{1}{2}\) is the same as \(\frac{2}{4}\). There are 2 one-fourths in \(\frac{2}{4}\).

So:

This makes sense because two fourths fit into one half.

3. The rule for dividing fractions

To divide by a fraction, multiply by its reciprocal.

In symbols:

This is often called "Keep, Change, Flip":

• Keep the first fraction the same.
• Change division to multiplication.
• Flip the second fraction to its reciprocal.

Example of the rule:

Then multiply straight across:

4. Why does this rule work?

Let’s think about a simpler division problem with whole numbers:

This works because \(3 \times 2 = 6\).

Division can be thought of as finding a missing number:

This means:

So division and multiplication are connected.

Now look at:

Dividing by \(\frac{4}{5}\) is the same as multiplying by the number that turns \(\frac{4}{5}\) into 1. That number is its reciprocal, \(\frac{5}{4}\), because:

So:

Another way to see this is with a complex fraction, which is a fraction inside a fraction:

We want to make the bottom equal to 1. Multiply the top and bottom by \(\frac{5}{4}\):

The denominator becomes 1, so we get:

That is why dividing by a fraction is the same as multiplying by its reciprocal.

5. Steps for dividing fractions

1. Write the problem clearly.
2. Keep the first fraction.
3. Change division to multiplication.
4. Flip the second fraction.
5. Multiply the numerators and denominators.
6. Simplify the answer.

6. Worked Examples

Example 1: Basic fraction division

Find \(\frac{3}{5} \div \frac{1}{2}\).

Step 1: Keep the first fraction, change to multiplication, and flip the second fraction.

Step 2: Multiply.

Step 3: Write as a mixed number if needed.

Answer: \(\frac{6}{5}\) or \(1\frac{1}{5}\)

Example 2: Divide a whole number by a fraction

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

Step 1: Write 4 as a fraction.

Step 2: Change division to multiplication and flip the second fraction.

Step 3: Multiply.

Answer: \(6\)

This makes sense: if each group is \(\frac{2}{3}\), then 6 groups make 4 wholes.

Example 3: Divide fractions with negatives

Find \(-\frac{3}{8} \div \frac{1}{4}\).

Step 1: Keep, change, flip.

Step 2: Multiply.

Step 3: Simplify.

Answer: \(-\frac{3}{2}\) or \(-1\frac{1}{2}\)

Example 4: Negative divided by negative

Find \(-\frac{5}{6} \div -\frac{10}{9}\).

Step 1: Keep, change, flip.

Step 2: Notice that a negative times a negative is positive.

Step 3: Multiply.

Step 4: Simplify.

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

7. Tips for simplifying

Sometimes you can simplify before multiplying. This makes the numbers smaller and the work easier.

For example:

Before multiplying, simplify across:

• 2 and 4 both divide by 2
• 15 and 9 both divide by 3

So the problem becomes:

This is the same answer, but with easier multiplication.

8. Common mistakes to avoid

• Do not flip the first fraction. Only flip the second fraction.
• Do not forget to change division to multiplication.
• Do not forget signs. A negative divided by a positive is negative. A negative divided by a negative is positive.
• Always simplify at the end if needed.
• Never try to take the reciprocal of 0. Zero has no reciprocal.

9. Quick check for understanding

• What is the reciprocal of \(\frac{7}{9}\)? \(\frac{9}{7}\)
• What is the reciprocal of \(-4\)? \(-\frac{1}{4}\)
• What is \(\frac{1}{3} \div \frac{2}{5}\)?

Use the rule:

10. Summary

A reciprocal is the flipped form of a fraction. When you divide by a fraction, you multiply by its reciprocal.

The rule is:

This works because multiplying a fraction by its reciprocal gives 1. Once the division problem is turned into multiplication, multiply straight across and simplify.

Remember: Keep, Change, Flip — and watch your signs when working with negative rational numbers.

Complex Fractions are fractions where the numerator, the denominator, or both already contain fractions.

For example, \(\frac{\frac{1}{2}}{\frac{3}{4}}\) is a complex fraction because it is a fraction divided by another fraction.

This lesson will show you how to simplify complex fractions by using what you already know about dividing fractions. Once you understand the pattern, complex fractions become much easier.

Main idea: A fraction bar means division. So a complex fraction is really a division problem.

For example,

To divide fractions, keep the first fraction, change division to multiplication, and flip the second fraction.

Then multiply straight across and simplify if needed.

Steps for simplifying a complex fraction:

1. Rewrite the complex fraction as a division problem.

2. Change division to multiplication by the reciprocal of the second fraction.

3. Multiply the numerators and multiply the denominators.

4. Simplify the answer.

Important reminder: The reciprocal of a fraction is the fraction turned upside down.

• The reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).

• The reciprocal of \(\frac{7}{2}\) is \(\frac{2}{7}\).

If there are whole numbers mixed in, rewrite them as fractions first.

• \(2 = \frac{2}{1}\)

• \(3 = \frac{3}{1}\)

Let’s look at some examples, starting easy and building up.

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

Step 1: Rewrite as division.

Step 2: Multiply by the reciprocal of \(\frac{1}{2}\).

Step 3: Multiply.

Step 4: Simplify.

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

Example 2: Simplify \(\frac{\frac{5}{6}}{\frac{10}{9}}\)

Rewrite as division:

Change to multiplication by flipping the second fraction:

Multiply:

Simplify:

So, \(\frac{\frac{5}{6}}{\frac{10}{9}} = \frac{3}{4}\).

Example 3: Simplify \(\frac{\frac{-2}{3}}{\frac{4}{5}}\)

This example includes a negative fraction. The steps are the same.

Rewrite as division:

Multiply by the reciprocal:

Multiply:

Simplify:

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

Example 4: Simplify \(\frac{\frac{3}{8}}{2}\)

The denominator is a whole number, so rewrite it as a fraction.

Now rewrite the complex fraction:

Change division to multiplication:

Multiply:

So, the simplified answer is \(\frac{3}{16}\).

A useful shortcut: If the top and bottom are each just one fraction, you can think of the fraction bar as division right away.

For example,

Watch out for these common mistakes:

• Forgetting to flip the second fraction. In division, only the second fraction gets flipped.

• Flipping the first fraction by mistake. Keep the first fraction the same.

• Not simplifying at the end. Always check whether the fraction can be reduced.

• Ignoring negative signs. A negative divided by a positive is negative.

Quick check: Try to think through these on your own.

• \(\frac{\frac{1}{3}}{\frac{2}{5}}\)

• \(\frac{\frac{-4}{7}}{\frac{2}{3}}\)

• \(\frac{\frac{5}{9}}{3}\)

If you solve them, remember to rewrite as division, multiply by the reciprocal, and simplify.

Summary: A complex fraction is just a fraction that contains fractions. The fraction bar means division, so simplify by rewriting it as a division problem. Then use the fraction division rule: keep the first fraction, change to multiplication, flip the second fraction, multiply, and simplify.

Signed Decimal Operations means working with decimals that can be positive or negative.

In this lesson, you will learn how to add, subtract, multiply, and divide signed decimals. You will also learn how to keep track of place value and how to decide whether your answer should be positive or negative.

These skills are important because decimals and negative numbers appear in real life all the time, such as in money, temperature, elevation, and measurements.

First, remember what a signed number is:

• A positive number is greater than 0, like \(3.5\).
• A negative number is less than 0, like \(-3.5\).

You can think of positive and negative numbers as showing direction on a number line. Positive numbers go to the right of 0, and negative numbers go to the left of 0.

Rule 1: Adding signed decimals

When adding signed decimals, use the same ideas as adding signed integers. The decimal part does not change the sign rules.

• If the signs are the same, add the numbers and keep the sign.
• If the signs are different, subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value.

Absolute value means how far a number is from 0. For example, \(|-4.2| = 4.2\).

When adding decimals, it is very important to line up the decimal points.

Example:

$$-2.4 + (-1.35)$$

Both numbers are negative, so add their absolute values:

$$2.40 + 1.35 = 3.75$$

Since both numbers were negative, the answer is:

$$-3.75$$

Rule 2: Subtracting signed decimals

Subtracting a signed decimal can be turned into an addition problem.

Use this rule:

$$a - b = a + (-b)$$

In words, subtracting a number is the same as adding its opposite.

Examples of opposites:

• The opposite of \(2.7\) is \(-2.7\).
• The opposite of \(-2.7\) is \(2.7\).

So if you see:

$$4.8 - (-1.2)$$

Change it to:

$$4.8 + 1.2$$

Then add:

$$6.0$$

Rule 3: Multiplying signed decimals

To multiply signed decimals, follow two steps:

1. Multiply the numbers as if they were whole numbers.
2. Place the decimal point correctly.
3. Use the sign rules to decide whether the answer is positive or negative.

Sign rules for multiplication:

• Positive \(\times\) Positive = Positive
• Negative \(\times\) Negative = Positive
• Positive \(\times\) Negative = Negative
• Negative \(\times\) Positive = Negative

Decimal rule for multiplication:

Count the total number of decimal places in both factors. The product must have that many decimal places.

For example:

$$1.2 \times 0.3$$

Ignore decimals first:

$$12 \times 3 = 36$$

There are 2 decimal places total: 1 in \(1.2\) and 1 in \(0.3\). So place the decimal in the product:

$$0.36$$

Rule 4: Dividing signed decimals

To divide signed decimals, first use the sign rules, then make the divisor a whole number.

Sign rules for division:

• Positive \(\div\) Positive = Positive
• Negative \(\div\) Negative = Positive
• Positive \(\div\) Negative = Negative
• Negative \(\div\) Positive = Negative

Decimal rule for division:

Move the decimal point in the divisor to the right until the divisor becomes a whole number. Move the decimal point in the dividend the same number of places.

For example:

$$-4.8 \div 0.6$$

The divisor is \(0.6\), so move the decimal 1 place right:

$$0.6 \to 6$$

Move the decimal in \(-4.8\) one place right too:

$$-4.8 \to -48$$

Now divide:

$$-48 \div 6 = -8$$

The answer is:

$$-8$$

Important place value ideas

• When adding or subtracting, line up the decimal points.
• When multiplying, multiply first, then count decimal places.
• When dividing, shift decimals to make the divisor a whole number.

Worked Example 1: Adding signed decimals

Find:

$$3.7 + (-5.2)$$

The signs are different, so subtract the smaller absolute value from the larger absolute value:

$$5.2 - 3.7 = 1.5$$

Since \(-5.2\) has the larger absolute value, the answer is negative:

$$-1.5$$

Worked Example 2: Subtracting signed decimals

Find:

$$-6.4 - 2.75$$

Rewrite as addition:

$$-6.4 + (-2.75)$$

Line up decimals:

$$-6.40 + (-2.75)$$

Add the absolute values:

$$6.40 + 2.75 = 9.15$$

Both numbers are negative, so the answer is:

$$-9.15$$

Worked Example 3: Multiplying signed decimals

Find:

$$-1.5 \times 2.4$$

First decide the sign:

Negative \(\times\) Positive = Negative

Now multiply ignoring decimals:

$$15 \times 24 = 360$$

Count decimal places: \(1.5\) has 1 decimal place and \(2.4\) has 1 decimal place, for a total of 2.

Place the decimal in the product:

$$3.60 = 3.6$$

Apply the sign:

$$-3.6$$

Worked Example 4: Dividing signed decimals

Find:

$$7.56 \div (-0.9)$$

First decide the sign:

Positive \(\div\) Negative = Negative

Make the divisor a whole number by moving the decimal 1 place right:

$$-0.9 \to -9$$

Move the decimal in the dividend 1 place right too:

$$7.56 \to 75.6$$

Now divide:

$$75.6 \div 9 = 8.4$$

Apply the sign:

$$-8.4$$

Common mistakes to avoid

• Do not forget the sign rules. A correct decimal answer with the wrong sign is still incorrect.
• Do not line up digits by the left side when adding or subtracting. Always line up decimal points.
• Do not count decimal places when dividing. That rule is for multiplication, not division.
• Be careful when subtracting negatives. Subtracting a negative becomes adding a positive.

Helpful check

After solving, ask yourself:

• Does the sign make sense?
• Is the decimal point in a reasonable place?
• Did I use the correct operation rule?

Summary

Signed decimal operations use the same sign rules as signed integers. For addition and subtraction, focus on signs and line up decimal points. For multiplication, multiply first and then place the decimal based on total decimal places. For division, make the divisor a whole number and then divide, using sign rules to decide whether the final answer is positive or negative.

Order of Operations with Nested Grouping helps us evaluate expressions correctly when there are many operations and different kinds of grouping symbols.

When an expression has nested grouping, it means one group is inside another group, such as brackets around parentheses, or a fraction bar around a grouped expression. We must work carefully from the innermost group outward.

This is especially important when working with rational numbers, which include positive and negative fractions, decimals, and integers. A small mistake with signs or grouping can change the entire answer.

To evaluate expressions correctly, we follow the order of operations.

Order of Operations Rules

• Step 1: Simplify expressions inside grouping symbols first.
  Grouping symbols include parentheses \\( ( ) \\), brackets \\( [ ] \\), braces \\( \{ \} \\), and fraction bars.
• Step 2: If there are groups inside groups, do the innermost group first.
• Step 3: Evaluate exponents.
• Step 4: Multiply and divide from left to right.
• Step 5: Add and subtract from left to right.

A helpful way to think about it is:

1. Go inside the smallest grouping.
2. Work outward one layer at a time.
3. At each layer, use exponents, multiplication/division, then addition/subtraction.

Important note about fraction bars: A fraction bar acts like a grouping symbol. In the expression

you must simplify the numerator \\(4+2\\) and the denominator \\(3-1\\) before dividing.

Important note about negative numbers: Be careful when a negative sign is next to grouping symbols.

For example, in

you first evaluate the parentheses: \\(3-5=-2\\). Then apply the negative sign outside:

The sign outside the group affects the value of the whole group.

Worked Example 1: Parentheses inside brackets

Evaluate:

Step 1: Simplify the innermost grouping, \\(3+2\\).

Step 2: Simplify the bracket.

Step 3: Divide.

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

This example shows that we do not subtract 8 and 3 first. The parentheses must be completed before the brackets are finished.

Worked Example 2: Negative rational numbers with nested grouping

Evaluate:

Step 1: Work inside the parentheses first.

Step 2: Simplify inside the brackets.

Step 3: Apply the negative sign outside.

Answer: \\(0\\)

Notice that the negative sign outside the bracket affects the whole bracket. But since the bracket equals 0, the result is still 0.

Worked Example 3: Nested grouping with fractions

Evaluate:

Step 1: Simplify the innermost parentheses.

Since \\(\frac{2}{4}=\frac{1}{2}\\), rewrite:

Step 2: Simplify the bracket.

Why? Because \\(\frac{1}{2}+\frac{1}{2}=1\\), so the expression becomes

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

This example shows that the entire numerator is grouped by the fraction bar, so we must finish the numerator before dividing by 2.

Worked Example 4: Exponents and nested grouping

Evaluate:

Step 1: Simplify the innermost parentheses.

Step 2: Evaluate the exponent.

Step 3: Simplify the bracket.

Step 4: Divide.

Answer: \\(2\\)

This example is a good reminder that exponents come after grouping is simplified. First find \\(3-5\\), then square the result.

Common Mistakes to Avoid

• Ignoring the innermost grouping: Always start with the smallest group first.
• Forgetting that a fraction bar is grouping: Simplify the numerator and denominator before dividing.
• Mixing up signs: Negative numbers can change the result, especially when a negative sign is outside parentheses or brackets.
• Doing addition before multiplication: After grouping and exponents, multiply and divide before add and subtract.
• Forgetting left to right: For multiplication and division, or addition and subtraction, work from left to right.

Helpful Strategy

1. Rewrite the expression neatly.
2. Circle or identify the innermost grouping.
3. Simplify one step at a time.
4. Keep track of negative signs carefully.
5. Check whether the result makes sense.

For example, if a bracket becomes 0, then multiplying or adding with that group may become much simpler. Looking for these simplifications can help you avoid mistakes.

Summary

When evaluating expressions with nested grouping, always begin with the innermost grouping symbols and work outward. At each level, follow the order of operations: grouping, exponents, multiplication and division, then addition and subtraction. Remember that fraction bars also act as grouping symbols, and be extra careful with negative numbers.

Evaluating Complex Fractions

A complex fraction is a fraction that has a fraction in the numerator, the denominator, or both.

For example, these are complex fractions:

• \(\frac{\frac{3}{4}}{2}\)
• \(\frac{5}{\frac{1}{3}}\)
• \(\frac{\frac{2}{3}}{\frac{5}{6}}\)

The most important idea is this: the main fraction bar means division.

So when you see a complex fraction, you can read it as one number divided by another number.

For example,

Once you rewrite it as division, you can use the rule for dividing fractions:

In words: keep the first fraction, change division to multiplication, and flip the second fraction.

This is sometimes called multiply by the reciprocal. The reciprocal of a fraction is the fraction turned upside down.

• The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
• The reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\).
• The reciprocal of \(3\) is \(\frac{1}{3}\), because \(3 = \frac{3}{1}\).

Steps for evaluating a complex fraction

1. Identify the numerator and denominator of the large fraction.
2. Rewrite the complex fraction as a division problem.
3. Change division to multiplication by using the reciprocal of the second number.
4. Multiply the fractions.
5. Simplify your answer.

Let’s work through some examples.

Example 1: Fraction in the numerator only

Evaluate:

Step 1: Rewrite as division.

Remember that \(2 = \frac{2}{1}\).

Step 2: Multiply by the reciprocal of \(\frac{2}{1}\).

Step 3: Multiply.

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

This makes sense because dividing \(\frac{3}{4}\) into 2 equal parts gives a smaller number.

Example 2: Fraction in the denominator only

Evaluate:

Step 1: Rewrite as division.

Write \(5\) as a fraction:

Step 2: Multiply by the reciprocal of \(\frac{1}{3}\).

Step 3: Multiply.

Answer: \(15\)

This also makes sense. The question asks, “How many one-thirds are in 5?” There are 15 one-thirds in 5.

Example 3: Fractions in both numerator and denominator

Evaluate:

Step 1: Rewrite as division.

Step 2: Multiply by the reciprocal of the second fraction.

Step 3: Multiply.

Step 4: Simplify.

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

Example 4: Simplify carefully

Evaluate:

Step 1: Rewrite as division.

Step 2: Multiply by the reciprocal of the second fraction.

Step 3: Multiply.

Step 4: Simplify.

Answer: \(\frac{3}{16}\)

Helpful tips

• Always treat the big fraction bar as division.
• Turn whole numbers into fractions by putting them over 1.
• Flip only the second fraction when changing division to multiplication.
• Simplify at the end.

Common mistakes to avoid

• Mistake: Flipping the first fraction.
  Instead, keep the first fraction the same and flip the second one.
• Mistake: Forgetting to write a whole number as a fraction.
  For example, write \(2\) as \(\frac{2}{1}\).
• Mistake: Multiplying straight across before changing division to multiplication.
  First rewrite the complex fraction as division.
• Mistake: Forgetting to simplify the final answer.

Quick check

Try these on your own:

1. \(\frac{\frac{1}{2}}{4}\)
2. \(\frac{3}{\frac{3}{5}}\)
3. \(\frac{\frac{4}{9}}{\frac{2}{3}}\)

Answers:

1. $$\frac{\frac{1}{2}}{4} = \frac{1}{2} \div 4 = \frac{1}{2} \div \frac{4}{1} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$$
2. $$\frac{3}{\frac{3}{5}} = 3 \div \frac{3}{5} = \frac{3}{1} \times \frac{5}{3} = 5$$
3. $$\frac{\frac{4}{9}}{\frac{2}{3}} = \frac{4}{9} \div \frac{2}{3} = \frac{4}{9} \times \frac{3}{2} = \frac{12}{18} = \frac{2}{3}$$

Summary

A complex fraction is a fraction with a fraction inside it.

To evaluate a complex fraction, read the main fraction bar as division. Then keep the first fraction, change division to multiplication, flip the second fraction, multiply, and simplify.

If you follow those steps carefully, you can solve complex fractions with confidence.