Solving Fractions Quiz

Study Notes

Solving Fractions

Addition of Fractions

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} )

Subtraction of Fractions

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} )

Multiplication of Fractions

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.

Division of Fractions

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.

Simplifying Fractions

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).

Addition of Fractions

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} ).

Subtraction of Fractions

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} ).

Multiplication of Fractions

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

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.

Simplifying Fractions

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.

Description

Test your skills in solving fractions with this quiz on addition, subtraction, and multiplication. You'll tackle problems involving both same and different denominators to master the concepts of fraction operations.