Fractions and Mixed Numbers | CSE Reviewer

What Is a Fraction?

A fraction is a way to show a part of something. It has two parts: the numerator on top and the denominator on the bottom.

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

• The numerator is 3 (top number)
• The denominator is 4 (bottom number)

The denominator tells how many equal parts make one whole. The numerator tells how many of those parts you have.

Tip: Read fractions as "parts out of whole." \(\frac{3}{4}\) means "3 parts out of 4 equal parts."

Example: If you eat 2 slices of a pizza with 8 slices:

Step 1: You ate 2 slices → numerator = 2
Step 2: Total slices = 8 → denominator = 8
Step 3: Write as fraction → \(\frac{2}{8}\)

So you ate \(\frac{2}{8}\) of the pizza.

Meaning: A fraction shows how many parts you have compared to the total number of parts in one whole.

Types of Fractions

There are three main types of fractions. Let me explain each one step by step.

Proper Fraction: The numerator is smaller than the denominator. The value is less than 1.

Example: \(\frac{3}{5}\)
Step 1: Numerator = 3
Step 2: Denominator = 5
Step 3: 3 is smaller than 5 → This is a proper fraction
Step 4: Value is less than 1

Improper Fraction: The numerator is greater than or equal to the denominator. The value is 1 or more.

Example: \(\frac{7}{4}\)
Step 1: Numerator = 7
Step 2: Denominator = 4
Step 3: 7 is greater than 4 → This is an improper fraction
Step 4: Value is more than 1

Mixed Number: A whole number combined with a proper fraction.

Example: \(2 \frac{3}{4}\)
Step 1: Whole number = 2
Step 2: Fraction part = \(\frac{3}{4}\)
Step 3: Together = 2 whole parts plus \(\frac{3}{4}\) of another part

Tip: Mixed numbers are easier to understand in real life. \(2 \frac{1}{2}\) cups means "2 whole cups plus half a cup."

Meaning: Proper fractions are less than 1, improper fractions are 1 or more, and mixed numbers combine whole parts with fraction parts.

Changing Mixed Numbers to Improper Fractions

When you multiply or divide mixed numbers, you must change them to improper fractions first. Here is the step-by-step method.

To change a mixed number to an improper fraction:

• Step 1: Multiply the whole number by the denominator
• Step 2: Add the numerator
• Step 3: Put the result over the same denominator

Shortcut Formula: Whole × Denominator + Numerator = New Numerator. Keep the same denominator.

Example 1: Change \(2 \frac{3}{4}\) to an improper fraction.

Step 1: Multiply whole by denominator → \(2 \times 4 = 8\)
Step 2: Add the numerator → \(8 + 3 = 11\)
Step 3: Keep the same denominator → \(\frac{11}{4}\)

Answer: \(\frac{11}{4}\)

Example 2: Change \(3 \frac{1}{2}\) to an improper fraction.

Step 1: Multiply whole by denominator → \(3 \times 2 = 6\)
Step 2: Add the numerator → \(6 + 1 = 7\)
Step 3: Keep the same denominator → \(\frac{7}{2}\)

Answer: \(\frac{7}{2}\)

Meaning: Turn mixed numbers into improper fractions by multiplying the whole by the denominator, adding the numerator, and keeping the same denominator.

Simplifying Fractions

Simplifying means making the fraction smaller but keeping the same value. You divide both the numerator and denominator by the same number.

Step-by-step to simplify:

• Step 1: Find a number that divides both numerator and denominator evenly
• Step 2: Divide the numerator by that number
• Step 3: Divide the denominator by the same number
• Step 4: Repeat until you cannot divide anymore

Tip: Look for common factors like 2, 3, 5, or the greatest common factor (GCF).

Example 1: Simplify \(\frac{6}{8}\).

Step 1: Find a common number → Both 6 and 8 can be divided by 2
Step 2: Divide numerator → \(6 ÷ 2 = 3\)
Step 3: Divide denominator → \(8 ÷ 2 = 4\)
Step 4: Write new fraction → \(\frac{3}{4}\)

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

Example 2: Simplify \(\frac{12}{16}\).

Step 1: Find a common number → Both 12 and 16 can be divided by 4
Step 2: Divide numerator → \(12 ÷ 4 = 3\)
Step 3: Divide denominator → \(16 ÷ 4 = 4\)
Step 4: Write new fraction → \(\frac{3}{4}\)

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

Adding Fractions with the Same Denominator

When the denominators are the same, this is very easy. You only add the top numbers.

Step-by-step:

• Step 1: Check if denominators are the same
• Step 2: Add the numerators (top numbers)
• Step 3: Keep the denominator the same
• Step 4: Simplify if needed

Shortcut: Same denominator: add numerators only.

Example: \(\frac{2}{7} + \frac{3}{7}\)

Step 1: Denominators are the same (both 7) → OK
Step 2: Add numerators → \(2 + 3 = 5\)
Step 3: Keep denominator → 7
Step 4: Write result → \(\frac{5}{7}\)

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

Meaning: With the same denominator, just add the top numbers and keep the bottom number.

Adding Fractions with Different Denominators

When denominators are different, you must make them the same first. This is called finding the LCD (Least Common Denominator).

Step-by-step to add \(\frac{1}{4} + \frac{1}{6}\):

• Step 1: Find the LCD of 4 and 6
• Step 2: Rewrite each fraction with the LCD
• Step 3: Add the numerators
• Step 4: Keep the LCD as denominator
• Step 5: Simplify if needed

Shortcut for LCD: List multiples of the larger denominator and check if the smaller denominator divides into it.

For 4 and 6:
Multiples of 6: 6, 12, 18, 24...
Check: 4 divides into 12 → LCD = 12

Example: \(\frac{1}{4} + \frac{1}{6}\)

Step 1: Find LCD → LCD of 4 and 6 is 12
Step 2: Rewrite \(\frac{1}{4}\): \(12 ÷ 4 = 3\), so \(\frac{1}{4} = \frac{3}{12}\)
Step 3: Rewrite \(\frac{1}{6}\): \(12 ÷ 6 = 2\), so \(\frac{1}{6} = \frac{2}{12}\)
Step 4: Add numerators → \(3 + 2 = 5\)
Step 5: Keep denominator → 12
Step 6: Write result → \(\frac{5}{12}\)

Answer: \(\frac{5}{12}\)

Meaning: Change fractions to have the same denominator first, then add the top numbers.

Subtracting Fractions

Subtraction works like addition. The steps are almost the same.

Step-by-step:

• Step 1: If denominators are different, find the LCD first
• Step 2: Rewrite fractions with the same denominator
• Step 3: Subtract the numerators
• Step 4: Keep the denominator
• Step 5: Simplify if needed

Tip: When subtracting mixed numbers, sometimes you need to borrow from the whole number.

Example 1: \(\frac{5}{7} - \frac{2}{7}\) (same denominator)

Step 1: Denominators are the same → OK
Step 2: Subtract numerators → \(5 - 2 = 3\)
Step 3: Keep denominator → 7

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

Example 2: \(\frac{3}{4} - \frac{1}{6}\) (different denominators)

Step 1: Find LCD → LCD of 4 and 6 is 12
Step 2: Rewrite \(\frac{3}{4}\): \(12 ÷ 4 = 3\), \(3 × 3 = 9\), so \(\frac{9}{12}\)
Step 3: Rewrite \(\frac{1}{6}\): \(12 ÷ 6 = 2\), \(1 × 2 = 2\), so \(\frac{2}{12}\)
Step 4: Subtract numerators → \(9 - 2 = 7\)
Step 5: Keep denominator → 12

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

Meaning: Subtract the top numbers after making the bottom numbers the same.

Adding and Subtracting Mixed Numbers

For addition and subtraction, handle the whole numbers and fractions separately.

Step-by-step for \(2 \frac{1}{4} + 1 \frac{2}{4}\):

• Step 1: Add the whole numbers
• Step 2: Add the fractions
• Step 3: Combine the results
• Step 4: Simplify if needed

Tip: Handle whole numbers and fractions separately for addition and subtraction.

Example: \(2 \frac{1}{4} + 1 \frac{2}{4}\)

Step 1: Add whole numbers → \(2 + 1 = 3\)
Step 2: Add fractions → \(\frac{1}{4} + \frac{2}{4} = \frac{3}{4}\)
Step 3: Combine → \(3 \frac{3}{4}\)

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

Meaning: Add or subtract the whole parts, then add or subtract the fraction parts.

Multiplying Fractions

Multiplying is easier than adding because you do not need the same denominator.

Step-by-step for \(\frac{2}{3} \times \frac{3}{4}\):

• Step 1: Multiply the numerators (top numbers)
• Step 2: Multiply the denominators (bottom numbers)
• Step 3: Write the result as a fraction
• Step 4: Simplify if needed

Shortcut: Cross-cancel before multiplying to make the numbers smaller and easier.

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

Step 1: Multiply numerators → \(2 × 3 = 6\)
Step 2: Multiply denominators → \(3 × 4 = 12\)
Step 3: Write fraction → \(\frac{6}{12}\)
Step 4: Simplify → Divide both by 6 → \(\frac{1}{2}\)

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

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

Step 1: Change to improper fractions
\(2 \frac{1}{2} = \frac{5}{2}\)
\(1 \frac{1}{3} = \frac{4}{3}\)
Step 2: Multiply numerators → \(5 × 4 = 20\)
Step 3: Multiply denominators → \(2 × 3 = 6\)
Step 4: Write fraction → \(\frac{20}{6}\)
Step 5: Simplify → Divide both by 2 → \(\frac{10}{3}\)
Step 6: Change back to mixed → \(3 \frac{1}{3}\)

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

Meaning: Multiply top numbers together and bottom numbers together. Simplify the result.

Dividing Fractions

Division has one special rule: flip the second fraction and change division to multiplication.

Step-by-step for \(\frac{3}{4} \div \frac{1}{2}\):

• Step 1: Flip the second fraction (find reciprocal)
• Step 2: Change division to multiplication
• Step 3: Multiply the fractions
• Step 4: Simplify if needed

Shortcut: "Flip and multiply." The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

Reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\)

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

Step 1: Flip the second fraction → \(\frac{1}{2}\) becomes \(\frac{2}{1}\)
Step 2: Change ÷ to × → \(\frac{3}{4} × \frac{2}{1}\)
Step 3: Multiply numerators → \(3 × 2 = 6\)
Step 4: Multiply denominators → \(4 × 1 = 4\)
Step 5: Write fraction → \(\frac{6}{4}\)
Step 6: Simplify → Divide both by 2 → \(\frac{3}{2}\)
Step 7: Change to mixed → \(1 \frac{1}{2}\)

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

Meaning: Turn division into multiplication by flipping the second fraction, then multiply normally.

Tips and Shortcuts for Fractions

Fractions can be fast if you use shortcuts. Here is the best order to follow.

Tips:

• Addition/Subtraction with same denominator: add or subtract numerators only.
• Addition/Subtraction with different denominators: find LCD first.
• Multiplication: multiply numerators and denominators directly.
• Division: flip the second fraction and multiply.
• Mixed numbers for × or ÷: change to improper first.
• Cross-cancel before multiplying to simplify.
• Simplify at the end as the final step.

Shortcut: When multiplying a whole number by a fraction, divide the whole number by the denominator first (if it divides evenly), then multiply by the numerator.

Example: A baker needs 12 cakes. Each cake requires \(\frac{3}{4}\) cup of sugar. How many cups of sugar total?

Without shortcut: \(12 × \frac{3}{4} = \frac{12}{1} × \frac{3}{4} = \frac{36}{4} = 9\)
With shortcut: \(12 ÷ 4 = 3\), then \(3 × 3 = 9\)

Answer: 9 cups (same result, but much faster!)

More Practice Examples

Example 1: \(\frac{2}{5} + \frac{1}{5}\)
Step 1: Same denominator → Add numerators → \(2 + 1 = 3\)
Answer: \(\frac{3}{5}\)

Example 2: \(\frac{1}{3} + \frac{1}{6}\)
Step 1: LCD = 6
Step 2: \(\frac{1}{3} = \frac{2}{6}\)
Step 3: \(\frac{2}{6} + \frac{1}{6} = \frac{3}{6}\)
Step 4: Simplify → \(\frac{1}{2}\)
Answer: \(\frac{1}{2}\)

Example 3: \(3 \frac{1}{4} - 1 \frac{2}{4}\)
Step 1: Borrow from 3 → \(2 \frac{5}{4}\)
Step 2: Subtract whole → \(2 - 1 = 1\)
Step 3: Subtract fraction → \(\frac{5}{4} - \frac{2}{4} = \frac{3}{4}\)
Answer: \(1 \frac{3}{4}\)

Example 4: \(\frac{2}{3} \times \frac{3}{5}\)
Step 1: Multiply numerators → \(2 × 3 = 6\)
Step 2: Multiply denominators → \(3 × 5 = 15\)
Step 3: \(\frac{6}{15}\)
Step 4: Simplify → \(\frac{2}{5}\)
Answer: \(\frac{2}{5}\)

Example 5: \(\frac{3}{4} \div \frac{3}{8}\)
Step 1: Flip second → \(\frac{3}{8}\) becomes \(\frac{8}{3}\)
Step 2: \(\frac{3}{4} × \frac{8}{3}\)
Step 3: Multiply → \(\frac{24}{12}\)
Step 4: Simplify → 2
Answer: 2

What To Remember

• A fraction shows parts of a whole: numerator on top, denominator on bottom.
• Proper fraction: numerator less than denominator (less than 1).
• Improper fraction: numerator greater than or equal to denominator (1 or more).
• Mixed number: whole number plus a proper fraction.
• Change mixed to improper for multiplication or division.
• Same denominator: add or subtract numerators only.
• Different denominators: find LCD first.
• Multiplication: multiply numerators and denominators.
• Division: flip the second fraction and multiply.
• Simplify your answer at the end.

Multiple Choice Questions

Answer these in the comment section. Choose the best answer for each item.

1. What is \(\frac{2}{5} + \frac{1}{5}\)?

A. \(\frac{1}{5}\)
B. \(\frac{3}{5}\)
C. \(\frac{2}{10}\)
D. \(\frac{4}{5}\)

2. What is \(\frac{1}{2} \times \frac{3}{4}\)?

A. \(\frac{3}{8}\)
B. \(\frac{4}{6}\)
C. \(\frac{3}{6}\)
D. \(\frac{1}{4}\)

3. What is \(2 \frac{1}{3} + 1 \frac{1}{3}\)?

A. \(3 \frac{1}{3}\)
B. \(3 \frac{2}{3}\)
C. 4
D. 3

4. What is \(\frac{3}{4} \div \frac{1}{2}\)?

A. \(\frac{3}{8}\)
B. \(\frac{3}{2}\)
C. \(\frac{1}{2}\)
D. 2

5. Change \(3 \frac{2}{5}\) to an improper fraction.

A. \(\frac{15}{5}\)
B. \(\frac{17}{5}\)
C. \(\frac{13}{5}\)
D. \(\frac{16}{5}\)