What is Fractions: Addition, Subtraction, Multiplication, and Division

Fractions, often viewed as mathematical jigsaw pieces, are an essential part of our numerical world. They represent parts of a whole, enabling us to express values that fall between integers.

Understanding how to manipulate fractions—whether it's adding, subtracting, multiplying, or dividing—is fundamental in many areas, from basic arithmetic to advanced mathematics. In this blog post, we'll explore the operations of addition, subtraction, multiplication, and division with fractions, breaking down each process step by step.

Addition of Fractions:

Adding fractions involves combining different fractional parts to form a single fraction. The key to adding fractions is ensuring that they have a common denominator—the same number on the bottom part of the fraction.

1. Find a Common Denominator: Identify the least common multiple (LCM) of the denominators of the fractions.
2. Equivalent Fractions: Express each fraction with the common denominator found in step 1.
3. Add Numerators: Once the fractions have a common denominator, add their numerators together.
4. Simplify, if Necessary: Simplify the resulting fraction by finding the greatest common divisor (GCD) of the numerator and denominator.

Example:

Let's add $\frac{1}{3}$ and $\frac{1}{6}$

1. The least common multiple of 3 and 6 is 6.
2. $\frac{1}{3}=\frac{2}{6}$ and $\frac{1}{6}$ stays as $\frac{1}{6}$
   
3. 
   
4. $\frac{2}{6}+\frac{1}{6}=\frac{3}{6}$
   
5. 
   
6. $\frac{3}{6}$ simplifies to $\frac{1}{2}$
   

Subtraction of Fractions:

Subtracting fractions follows a similar process to addition, with the key difference being that we subtract the numerators instead of adding them.

1. Find a Common Denominator: Just like in addition, identify a common denominator.
2. Equivalent Fractions: Rewrite each fraction with the common denominator.
3. Subtract Numerators: Subtract the numerators of the fractions.
4. Simplify, if Necessary: Simplify the resulting fraction.

Example: Let's subtract $\frac{1}{3}$ from $\frac{1}{2}$.

1. The common denominator for $\frac{1}{2}$ and $\frac{1}{3}$ is 6
2. 
   
3. $\frac{1}{2}=\frac{3}{6}$ and $\frac{1}{3}=\frac{2}{6}$
   
4. 
   
5. $\frac{3}{6}-\frac{2}{6}=\frac{1}{6}$
   

Multiplication of Fractions:

Multiplying fractions involves multiplying their numerators together and their denominators together.

1. Multiply Numerators: Multiply the numerators of the fractions.
2. Multiply Denominators: Multiply the denominators of the fractions.
3. Simplify, if Necessary: Simplify the resulting fraction.

Example: Let's multiply $\frac{2}{3}$ by $\frac{3}{4}$

1. $2\times 3=6$
2. $3\times 4=12$
3. 
   
4. $\frac{2}{3}\times \frac{3}{4}=\frac{6}{12}$
   
5. 
   
6. $\frac{6}{12}$ simplifies to $\frac{1}{2}$
   

Division of Fractions:

Dividing fractions is akin to multiplying, but with one additional step: taking the reciprocal of the second fraction (the divisor) and then proceeding with multiplication.

1. Take Reciprocal: Flip the second fraction upside down.
2. Multiply: Multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor).
3. Simplify, if Necessary: Simplify the resulting fraction.

Example: Let's divide $\frac{2}{3}$ by $\frac{3}{4}$

1. Reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$
   
2. 
   
3. $\frac{2}{3}\times \frac{4}{3}=\frac{8}{9}$
   

Fractions are foundational in mathematics and find applications in various fields, including science, engineering, and everyday life. Understanding how to manipulate fractions through addition, subtraction, multiplication, and division empowers us to solve a wide array of problems with precision and accuracy. So, the next time you encounter fractions, remember these fundamental operations—they'll serve as your guide through the numerical maze.