Review of Basic Fraction Operations

TL;DR

This note covers the four basic operations with fractions: addition, subtraction, multiplication, and division. You'll learn the key steps for each, focusing on finding common denominators for adding and subtracting. Understanding these operations is crucial for more advanced fraction work.

1. The Mental Model

Think of fractions as parts of a whole. When you add or subtract, you need those parts to be the same size. When you multiply, you're finding a "fraction of a fraction." When you divide, you're seeing how many times one fraction fits into another.

2. The Core Material

Fractions represent a part of a whole. The top number is the numerator (how many parts you have), and the bottom number is the denominator (how many total parts make up the whole).

Adding and Subtracting Fractions

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To add or subtract fractions, you must have a common denominator. This means both fractions need to represent parts of the same size.

1. Find the Least Common Denominator (LCD): This is the smallest number that both denominators divide into evenly.
2. Convert Fractions: Change each fraction to an equivalent fraction with the LCD as its new denominator. Do this by multiplying both the numerator and the denominator by the same number.
3. Add or Subtract Numerators: Once the denominators are the same, you just add or subtract the numerators. The denominator stays the same.
4. Simplify: Reduce the resulting fraction to its simplest form if possible.

Example: Add $\frac{1}{3} + \frac{1}{2}$
* LCD of 3 and 2 is 6.
* $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
* $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$
* $\frac{2}{6} + \frac{3}{6} = \frac{2+3}{6} = \frac{5}{6}$

Multiplying Fractions

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Multiplying fractions is generally simpler than adding or subtracting. You don't need a common denominator!

1. Multiply Numerators: Multiply the top numbers together.
2. Multiply Denominators: Multiply the bottom numbers together.
3. Simplify: Reduce the resulting fraction to its simplest form. You can often cross-simplify before multiplying to make the numbers smaller.

Example: Multiply $\frac{2}{3} \times \frac{4}{5}$
* Numerators: $2 \times 4 = 8$
* Denominators: $3 \times 5 = 15$
* Result: $\frac{8}{15}$ (cannot be simplified)

Dividing Fractions

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Dividing fractions involves a neat trick: "keep, change, flip."

1. Keep: Keep the first fraction as it is.
2. Change: Change the division sign to a multiplication sign.
3. Flip: Flip the second fraction (find its reciprocal – swap its numerator and denominator).
4. Multiply: Now, multiply the fractions as you learned above.

Example: Divide $\frac{3}{4} \div \frac{1}{2}$
* Keep $\frac{3}{4}$
* Change $\div$ to $\times$
* Flip $\frac{1}{2}$ to $\frac{2}{1}$
* Now multiply: $\frac{3}{4} \times \frac{2}{1} = \frac{3 \times 2}{4 \times 1} = \frac{6}{4}$
* Simplify: $\frac{6}{4} = \frac{3}{2}$ or $1\frac{1}{2}$

Here's a diagram summarizing the core operations:

graph TD
    Start["Begin Fraction Operation"] --> DecideOp{"Which Operation?"}

    DecideOp -- "Addition/Subtraction" --> AddSubSteps["1. Find LCD (Least Common Denominator)
                                                 2. Convert Fractions to LCD
                                                 3. Add/Subtract Numerators
                                                 4. Keep Denominator
                                                 5. Simplify"]

    DecideOp -- "Multiplication" --> MultiplySteps["1. Multiply Numerators
                                                  2. Multiply Denominators
                                                  3. Simplify (Cross-simplify first if possible)"]

    DecideOp -- "Division" --> DivideSteps["1. Keep First Fraction
                                              2. Change ÷ to ×
                                              3. Flip Second Fraction (Reciprocal)
                                              4. Multiply Fractions (as above)
                                              5. Simplify"]

    AddSubSteps --> End["Operation Complete"]
    MultiplySteps --> End
    DivideSteps --> End

3. Worked Example

Let's solve: $\frac{5}{6} - \frac{1}{4} + \frac{2}{3}$

1. Work with subtraction first: $\frac{5}{6} - \frac{1}{4}$

   • LCD of 6 and 4 is 12.
   • $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
   • $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
   • $\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$

2. Now add the remaining fraction: $\frac{7}{12} + \frac{2}{3}$

   • LCD of 12 and 3 is 12. (The first fraction already has the LCD!)
   • $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
   • $\frac{7}{12} + \frac{8}{12} = \frac{15}{12}$

3. Simplify the result: $\frac{15}{12}$ can be divided by 3.

   • $\frac{15 \div 3}{12 \div 3} = \frac{5}{4}$ or $1\frac{1}{4}$.

So, $\frac{5}{6} - \frac{1}{4} + \frac{2}{3} = \frac{5}{4}$.

4. Key Takeaways

• Always find a common denominator before adding or subtracting fractions.
• To multiply fractions, multiply the numerators together and the denominators together.
• To divide fractions, "keep, change, flip" then multiply.
• Simplifying fractions to their lowest terms is a final step for all operations.
• Mixed numbers should usually be converted to improper fractions before performing operations.
• Don't confuse the rules for adding/subtracting with those for multiplying/dividing.

Common Mistakes to Avoid:
- Adding or subtracting denominators (e.g., $\frac{1}{2} + \frac{1}{3}
eq \frac{2}{5}$).
- Forgetting to find a common denominator for addition/subtraction.
- Not flipping the second fraction when dividing.
- Forgetting to simplify the final answer.

5. Now Try It

Calculate the following: $(\frac{3}{5} \times \frac{1}{6}) \div (\frac{2}{3} + \frac{1}{2})$.

What to do: First, solve the multiplication inside the first parenthesis. Next, solve the addition inside the second parenthesis. Finally, perform the division using your two results.

What success looks like: You should arrive at a single fraction in its simplest form. (The correct answer is $\frac{3}{35}$.)