Fractions: Simplifying, Comparing, and Operations

Study Notes

To simplify a fraction, divide the numerator and the denominator by their greatest common divisor (GCD).
Steps to simplify:
1. Find the GCD of the numerator and denominator.
2. Divide both by the GCD.
Example: Simplifying 8/12
- GCD is 4.
- 8 ÷ 4 = 2, 12 ÷ 4 = 3 → simplified fraction is 2/3.

To compare fractions, they must have a common denominator or be converted to decimals.
Steps to compare:
1. Find a common denominator.
2. Convert fractions to equivalent fractions.
3. Compare numerators.
For ordering, arrange fractions from least to greatest or vice versa.
Example: Comparing 1/4 and 2/5
- Common denominator is 20: 1/4 = 5/20, 2/5 = 8/20 → 5/20 < 8/20 (1/4 < 2/5).

When adding or subtracting fractions:
1. Ensure the fractions have a common denominator.
2. Add or subtract the numerators.
3. Keep the common denominator.
4. Simplify if necessary.
Example: Adding 1/3 and 1/6
- Common denominator is 6: 1/3 = 2/6 → 2/6 + 1/6 = 3/6 → simplified to 1/2.

Multiplication:
- Multiply the numerators together and the denominators together.
- Example: (2/3) * (4/5) = (2 * 4) / (3 * 5) = 8/15.
Division:
- Multiply by the reciprocal of the second fraction.
- Example: (2/3) ÷ (4/5) → (2/3) * (5/4) = (2 * 5) / (3 * 4) = 10/12 → simplified to 5/6.

Key strategies:
1. Identify the fractions involved in the problem.
2. Determine the operation needed (addition, subtraction, multiplication, division).
3. Set up the equation using the fractions.
4. Solve the equation, simplifying when necessary.
Example: If 2/5 of a pizza is eaten and then 1/10 is eaten, how much pizza is left?
- Total eaten: (2/5) + (1/10) → common denominator of 10 → (4/10) + (1/10) = 5/10 → remaining = 1 - 5/10 = 5/10 → simplified to 1/2.

To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD).
Steps for simplification:
- Find the GCD of the numerator and denominator.
- Divide both by the GCD to achieve the simplest form.
Example: Simplifying 8/12 yields a GCD of 4; thus, 8 ÷ 4 = 2 and 12 ÷ 4 = 3, resulting in the simplified fraction 2/3.

Comparing fractions requires a common denominator or conversion to decimals.
Steps for comparison:
- Identify a common denominator for both fractions.
- Convert fractions to equivalent fractions.
- Compare the numerators to determine which fraction is larger.
For ordering fractions, arrange them from least to greatest, or the reverse.
Example: To compare 1/4 and 2/5, find a common denominator of 20; so 1/4 converts to 5/20 and 2/5 to 8/20, leading to the conclusion that 1/4 < 2/5.

To add or subtract fractions:
- Ensure a common denominator exists.
- Combine or subtract the numerators while keeping the common denominator.
- Simplify the resulting fraction if necessary.
Example: To add 1/3 and 1/6, convert 1/3 to a common denominator of 6 (resulting in 2/6); then 2/6 + 1/6 = 3/6, which simplifies to 1/2.

For multiplication of fractions:
- Multiply the numerators together and the denominators together.
Example: (2/3) * (4/5) results in (2 * 4) / (3 * 5) = 8/15.
For division of fractions:
- Multiply by the reciprocal of the second fraction.
Example: (2/3) ÷ (4/5) is converted to (2/3) * (5/4), yielding (2 * 5) / (3 * 4) = 10/12, which simplifies to 5/6.

Key strategies for solving word problems involving fractions:
- Identify the fractions involved in the scenario.
- Determine the necessary operation: addition, subtraction, multiplication, or division.
- Set up an equation using the identified fractions.
- Solve the equation, simplifying when required.
Example: If 2/5 of a pizza is consumed, followed by 1/10, the total consumed can be calculated: (2/5) + (1/10) requires a common denominator of 10, resulting in (4/10) + (1/10) = 5/10; thus, the remaining amount is 1 - 5/10 = 5/10, which simplifies to 1/2.