Operations with Fractions (Addition, Subtraction)

TL;DR

Adding and subtracting fractions requires a common denominator before you can combine the numerators. Think of finding a common denominator as finding a shared unit of measurement for your fractions. Once you have that, the operation is straightforward addition or subtraction of the top numbers.

1. The Mental Model

Imagine you have different sized pieces of cake – one is 1/2 of a cake, another is 1/4. You can't easily add them until you cut the 1/2 piece into 1/4s, making it 2/4. This "cutting into smaller, equal pieces" is what finding a common denominator does.

2. The Core Material

To add or subtract fractions, you must make sure the bottom numbers (denominators) are the same. If they aren't, you need to find a common denominator. The easiest common denominator to find is the least common multiple (LCM) of the original denominators.

Finding a Common Denominator

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1. List Multiples: Write out multiples of each denominator until you find a number they both share.
2. Multiply to Match: Once you find the LCM, figure out what you need to multiply each original denominator by to get that LCM.
3. "Do to the Top What You Do to the Bottom": Multiply the numerator (top number) of each fraction by the same number you multiplied its denominator by. This keeps the fraction's value the same, just in a different "form" or "unit."

Adding Fractions

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Once you have a common denominator:
1. Add the numerators.
2. Keep the common denominator the same.
3. Simplify the resulting fraction if possible.

Example: $1/3 + 1/2$
* Multiples of 3: 3, 6, 9...
* Multiples of 2: 2, 4, 6, 8...
* LCM is 6.
* $1/3$ becomes $(1 \times 2) / (3 \times 2) = 2/6$
* $1/2$ becomes $(1 \times 3) / (2 \times 3) = 3/6$
* Now add: $2/6 + 3/6 = 5/6$

Subtracting Fractions

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This process is nearly identical to addition:
1. Find a common denominator.
2. Subtract the numerators.
3. Keep the common denominator the same.
4. Simplify the resulting fraction if possible.

Example: $3/4 - 1/8$
* Multiples of 4: 4, 8, 12...
* Multiples of 8: 8, 16...
* LCM is 8.
* $3/4$ becomes $(3 \times 2) / (4 \times 2) = 6/8$
* $1/8$ stays $1/8$
* Now subtract: $6/8 - 1/8 = 5/8$

graph TD
    A["Start with Fractions (e.g., a/b + c/d)"] --> B{"Denominators Different?"}
    B -- Yes --> C["Find LCM of Denominators"]
    C --> D["Convert Fractions to Common Denominator"]
    D --> E["Perform Operation (Add or Subtract Numerators)"]
    E --> F["Keep Common Denominator"]
    F --> G["Simplify Resulting Fraction (if needed)"]
    G --> H["End"]
    B -- No --> E

3. Worked Example

Let's subtract these mixed numbers: $3 \frac{1}{2} - 1 \frac{2}{3}$

1. Convert to improper fractions:
   • $3 \frac{1}{2} = (3 \times 2 + 1) / 2 = 7/2$
   • $1 \frac{2}{3} = (1 \times 3 + 2) / 3 = 5/3$
2. Find a common denominator for 2 and 3: The LCM of 2 and 3 is 6.
3. Convert fractions:
   • $7/2 = (7 \times 3) / (2 \times 3) = 21/6$
   • $5/3 = (5 \times 2) / (3 \times 2) = 10/6$
4. Subtract the numerators: $21/6 - 10/6 = 11/6$
5. Convert back to a mixed number (optional, but good practice):
   • $11 \div 6 = 1$ with a remainder of $5$. So, $1 \frac{5}{6}$.

4. Key Takeaways

• You must have a common denominator to add or subtract fractions.
• The least common multiple (LCM) is usually the most efficient common denominator.
• Whatever you multiply the denominator by, you must multiply the numerator by the same number.
• Always simplify your final fraction to its lowest terms.
• Mixed numbers should often be converted to improper fractions before operating.

Common Mistakes to Avoid:
- Adding or subtracting numerators without a common denominator first.
- Adding or subtracting denominators (the denominator stays the same after conversion).
- Forgetting to multiply the numerator when you adjust the denominator.
- Not simplifying the final answer.

5. Now Try It

Work through the following problem: $2/5 + 3/10 - 1/4$. Show each step, including finding common denominators and simplifying your final answer. Success looks like arriving at the correct simplified fraction by following all the steps.