Solving problems involving fractions requires understanding a few key operations: adding, subtracting, multiplying, and dividing. Each operation has its own set of rules.
Adding and Subtracting Fractions
The fundamental rule for adding and subtracting fractions is that they must have a common denominator.
- Find the Least Common Denominator (LCD): Determine the smallest number that is a multiple of both denominators. This is your common denominator.
- Convert Fractions: Multiply the numerator and denominator of each fraction by a number that will make the denominator equal to the LCD.
- Add or Subtract Numerators: Once the fractions have a common denominator, add or subtract the numerators. The denominator stays the same.
- Simplify: Reduce the resulting fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF).
Example:
Calculate $\frac{1}{4} + \frac{2}{3}$.
- The LCD of 4 and 3 is 12.
- Convert: $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ and $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$.
- Add: $\frac{3}{12} + \frac{8}{12} = \frac{3+8}{12} = \frac{11}{12}$.
- Simplify: $\frac{11}{12}$ is already in its simplest form.
Multiplying Fractions
Multiplying fractions is straightforward:
- Multiply Numerators: Multiply the numerators of the two fractions.
- Multiply Denominators: Multiply the denominators of the two fractions.
- Simplify: Reduce the resulting fraction to its simplest form.
Example:
Calculate $\frac{2}{5} \times \frac{3}{4}$.
- Multiply Numerators: $2 \times 3 = 6$.
- Multiply Denominators: $5 \times 4 = 20$.
- Result: $\frac{6}{20}$.
- Simplify: Both 6 and 20 are divisible by 2, so $\frac{6}{20} = \frac{3}{10}$.
Dividing Fractions
Dividing fractions involves a simple trick: "Keep, Change, Flip".
- Keep: Keep the first fraction as it is.
- Change: Change the division sign to a multiplication sign.
- Flip: Flip (reciprocal) the second fraction (the divisor). This means swapping its numerator and denominator.
- Multiply: Multiply the fractions as described above.
- Simplify: Reduce the resulting fraction to its simplest form.
Example:
Calculate $\frac{1}{2} \div \frac{3}{4}$.
- Keep: $\frac{1}{2}$
- Change: $\div$ becomes $\times$
- Flip: $\frac{3}{4}$ becomes $\frac{4}{3}$
- Multiply: $\frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6}$
- Simplify: Both 4 and 6 are divisible by 2, so $\frac{4}{6} = \frac{2}{3}$.
In summary, solving with fractions requires understanding the specific rules for addition/subtraction (common denominator required), multiplication (straight across), and division (keep, change, flip). Always simplify your final answer!
