Understanding Fractions

Concept Explanation

A fraction represents a part of a whole. It consists of two numbers: the numerator (the top number) and the denominator (the bottom number). The numerator shows how many parts we have, while the denominator indicates how many equal parts the whole is divided into. For example, in the fraction $\frac{3}{4}$, 3 is the numerator, and 4 is the denominator, meaning we have 3 out of 4 equal parts.

Fractions can also represent numbers between whole numbers, making them useful for measurements, dividing objects, and understanding ratios. Fractions can be proper (numerator less than the denominator), improper (numerator greater than or equal to the denominator), or even mixed numbers (a whole number combined with a fraction).

Worked Examples

Example 1

Problem: $\frac{1}{2} + \frac{1}{4}$

Solution: $\frac{3}{4}$

Explanation: Convert to a common denominator (4), then add: $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.

Example 2

Problem: $\frac{5}{8}$ - $\frac{1}{8}$

Solution: $\frac{4}{8} = \frac{1}{2}$

Explanation: Subtract numerators since denominators are the same: $5 - 1 = 4$. Simplify $\frac{4}{8}$ to $\frac{1}{2}$.

Example 3

Problem: $\frac{2}{3} \times \frac{3}{4}$

Solution: $\frac{6}{12} = \frac{1}{2}$

Explanation: Multiply numerators (2 × 3 = 6) and denominators (3 × 4 = 12), then simplify.

Common Errors

Practice Problems

1. Problem:

   $\frac{1} {3} + \frac{1} {3}$

   Solution:

   $\frac{2} {3}$

2. Problem:

   $\frac{4} {5} - \frac{2} {5}$

   Solution:

   $\frac{2} {5}$

3. Problem:

   $\frac{3} {7} \times \frac{2} {3}$

   Solution:

   $\frac{6} {21} = \frac{2} {7}$

4. Problem:

   $\frac{5} {6} \div \frac{1} {2}$

   Solution:

   $\frac{5} {6} \times \frac{2} {1} = \frac{10} {6} = \frac{5} {3}$

5. Problem:

   Compare $\frac{3} {4}$ and $\frac{5} {6}$

   Solution:

   $\frac{3} {4} = 0.75$ and $\frac{5} {6} \approx 0.833$, so $\frac{5} {6} > \frac{3} {4}$

Real-World Application Example

Imagine you have a pizza divided into 8 equal slices. If you eat 3 slices, you’ve eaten $\frac{3}{8}$ of the pizza. This helps visualize fractions in everyday life, showing parts of a whole in a relatable way.