Solving Fractions Quiz

Study Notes

Same Denominator: Add the numerators; keep the denominator.
- Example: ( \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} )
Different Denominators:
1. Find a common denominator (LCM of the denominators).
2. Adjust numerators accordingly.
3. Add the adjusted numerators; keep the common denominator.
- Example: ( \frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + b \cdot c}{b \cdot d} )

Same Denominator: Subtract the numerators; keep the denominator.
- Example: ( \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} )
Different Denominators:
1. Find a common denominator.
2. Adjust numerators accordingly.
3. Subtract the adjusted numerators; keep the common denominator.
- Example: ( \frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - b \cdot c}{b \cdot d} )

Multiply the numerators together and the denominators together.
- Example: ( \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} )
Simplify before multiplying if possible.

Invert the second fraction and multiply.
- Example: ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c} )
Simplify before and after dividing if possible.

Finding GCD (Greatest Common Divisor):
1. Identify the GCD of the numerator and denominator.
2. Divide both by the GCD.
Reducing to Lowest Terms:
- A fraction is in simplest form when the numerator and denominator have no common factors (other than 1).
Example:
- ( \frac{8}{12} ) simplifies to ( \frac{2}{3} ) (GCD is 4).

For fractions with the same denominator, simply add the numerators while keeping the denominator unchanged. For instance, ( \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} ).
To add fractions with different denominators, determine the least common multiple (LCM) of the denominators, adjust the numerators to match this common denominator, and then add the resulting numerators. Example: ( \frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + b \cdot c}{b \cdot d} ).

For fractions that share the same denominator, subtract the numerators while keeping the denominator the same. For example, ( \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} ).
For fractions with different denominators, find a common denominator, adjust the numerators correspondingly, and subtract the adjusted numerators. Example: ( \frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - b \cdot c}{b \cdot d} ).

To multiply fractions, multiply the numerators together and the denominators together, yielding ( \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} ).
It is beneficial to simplify fractions before multiplication if possible to make calculations easier.

Division of fractions involves inverting the second fraction and then multiplying. This can be expressed as ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c} ).
As with multiplication, simplifying fractions both before and after division can ease the process.

To simplify a fraction, first find the greatest common divisor (GCD) of the numerator and denominator. Then, divide both the numerator and the denominator by the GCD.
A fraction is considered to be in its simplest form when there are no common factors between the numerator and denominator aside from 1.
An example of simplification: ( \frac{8}{12} ) can be reduced to ( \frac{2}{3} ) since the GCD is 4.