Addition and Subtraction of Fractions
For many, fractions can feel a bit daunting. But adding and subtracting fractions is simpler than it seems once you understand the process.
How to Add Fractions
When adding fractions, the first thing to check is whether they have the same denominator. Let’s look at the two possible scenarios:
A] Fractions with the Same Denominator
If the denominators are already the same, all you need to do is add the numerators while keeping the denominator unchanged.
For example, these two fractions share the same denominator (5):
$$ \frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5} $$
So, you simply add the numerators (2 + 1) and keep the denominator (5) the same:
$$ \frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5} $$
The result is $ \frac{3}{5} $.
B] Fractions with Different Denominators
When the denominators are different, you need to find a common denominator—specifically, the least common multiple (LCM)—and rewrite the fractions accordingly.
For instance, take these two fractions with different denominators:
$$ \frac{2}{3} + \frac{1}{4} $$
The least common multiple (LCM) of 3 and 4 is 12:
$$ LCM(3,4)=12 $$
Next, rewrite the fractions with 12 as the common denominator:
$$ \frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}, \quad \frac{1}{4} = \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12} $$
Now, rewrite the sum using these equivalent fractions:
$$ \frac{8}{12} + \frac{3}{12} $$
Since the denominators are now the same, you can add the numerators and keep the denominator unchanged:
$$ \frac{8}{12} + \frac{3}{12} = \frac{8+3}{12} = \frac{11}{12} $$
The final result is $ \frac{11}{12} $.
Can a fraction be negative? Absolutely. Fractions can be positive or negative depending on the context. However, in practical scenarios—such as dividing a cake into equal parts—we often only consider positive fractions. This is a context-specific limitation and doesn’t apply to subtraction in general.
How to Subtract Fractions
The rules for subtracting fractions are almost identical to those for addition. The only difference is that you subtract the numerators instead of adding them.
A] Fractions with the Same Denominator
When the denominators are the same, simply subtract the numerators.
For example, consider these two fractions:
$$ \frac{5}{8} - \frac{3}{8} $$
Subtract the numerators while keeping the denominator unchanged:
$$ \frac{5}{8} - \frac{3}{8} = \frac{5-3}{8} = \frac{2}{8} $$
This result, $ \frac{2}{8} $, can be simplified by dividing both the numerator and denominator by 2:
$$ \frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4} $$
So, the simplified result is $ \frac{1}{4} $.
If the numerator and denominator of a fraction share a common factor, it’s always a good idea to simplify the fraction to its lowest terms. Simplified fractions make calculations easier and results clearer. However, in certain situations (like intermediate steps), it might be practical to leave fractions unsimplified temporarily.
B] Fractions with Different Denominators
When the denominators differ, the process starts by finding a common denominator.
For example, consider these fractions:
$$ \frac{7}{10} - \frac{2}{5} $$
The least common multiple (LCM) of 10 and 5 is 10:
$$ LCM(10,5)=10 $$
Rewrite the fractions with 10 as the denominator:
$$ \frac{7}{10} = \frac{7}{10}, \quad \frac{2}{5} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10} $$
Now subtract the numerators:
$$ \frac{7}{10} - \frac{4}{10} = \frac{7-4}{10} = \frac{3}{10} $$
The final result is $ \frac{3}{10} $.
A Practical Example
Let’s try adding these two fractions:
$$ \frac{5}{6} + \frac{3}{8} $$
Since the denominators are different, we need to find the LCM of 6 and 8, which is 24:
$$ LCM(6,8) = 24 $$
Rewrite each fraction with a denominator of 24. Start with the first fraction by multiplying its numerator and denominator by 4 (since $ 6 \cdot 4 = 24 $):
$$ \frac{5 \cdot 4}{6 \cdot 4} + \frac{3}{8} $$
$$ \frac{20}{24} + \frac{3}{8} $$
For the second fraction, multiply its numerator and denominator by 3 (since $ 8 \cdot 3 = 24 $):
$$ \frac{20}{24} + \frac{3 \cdot 3}{8 \cdot 3} $$
Now that both fractions have the same denominator (24), add the numerators:
$$ \frac{20}{24} + \frac{9}{24} = \frac{20+9}{24} = \frac{29}{24} $$
The result is an improper fraction because the numerator is larger than the denominator.
You can rewrite it as a mixed number:
$$ \frac{29}{24} = \frac{24+5}{24} = \frac{24}{24} + \frac{5}{24} = 1 + \frac{5}{24} $$
The final result is:
$$ \frac{29}{24} = 1 + \frac{5}{24} $$
Improper fractions are often converted into mixed numbers (whole number + proper fraction) to make the value easier to interpret. This format clearly separates the integer and fractional parts, simplifying comparisons and highlighting the relationship between fractions and division: integer quotient + remainder.
With a little practice, what once seemed challenging will soon become a skill you can be proud of.
Remember: math is not just about numbers—it’s also about logic and patience!
