When the bottom numbers (denominators) are identical, you simply add or subtract the top numbers (numerators) and keep the denominator the same.
If the denominators are different, you must find a Least Common Multiple (LCM) to make them the same before you can solve the problem.
To solve $\frac{5}{4} + \frac{7}{6}$, we find the LCM of 4 and 6, which is 12.
Convert the fractions: Multiply the numerator and denominator so both denominators become 12.
$\frac{5 \times 3}{4 \times 3} = \frac{15}{12}$
$\frac{7 \times 2}{6 \times 2} = \frac{14}{12}$
Add them up:
This is often faster for smaller numbers. You multiply the denominators together to get a common denominator, then cross-multiply the numerators.
Example: $\frac{5}{3} + \frac{8}{7}$
Common Denominator: $3 \times 7 = 21$
Cross-Multiply: $(5 \times 7) + (8 \times 3) = 35 + 24$
Result:
| Operation | Scenario | Rule |
| Addition | Same Denominator | Add numerators, keep denominator. |
| Subtraction | Same Denominator | Subtract numerators, keep denominator. |
| Mixed | Different Denominators | Find LCM or Cross-Multiply first. |
To solve $\frac{5}{3} + 1$:
Rewrite 1 as $\frac{1}{1}$.
Use the cross-multiplication method: $\frac{5 + 3}{3}$.
Your result is $\frac{8}{3}$.
To solve $5 – \frac{7}{6}$:
Rewrite 5 as $\frac{5}{1}$.
Find the common denominator: $\frac{30 – 7}{6}$.
Your result is $\frac{23}{6}$.
If you have more than two fractions, the rule remains the same: if the denominators are identical, you can perform all operations across the single shared denominator.
Example:
Try solving these based on the methods we’ve covered. I’ve grouped them by type to help you practice!
$\frac{7}{9} + \frac{5}{9} = $ _____
$\frac{7}{9} – \frac{5}{9} = $ _____
$\frac{5}{6} + \frac{3}{8} = $ _____
$\frac{5}{6} – \frac{3}{8} = $ _____
$\frac{5}{3} + \frac{7}{8} = $ _____
$\frac{5}{3} – \frac{7}{8} = $ _____
$\frac{5}{3} + 6 = $ _____
$\frac{5}{3} – 6 = $ _____
$3 + \frac{5}{7} = $ _____
$3 – \frac{5}{7} = $ _____
To solve $\frac{5}{6} + \frac{7}{8} – \frac{3}{12}$, you first find the LCM of 6, 8, and 12:
Find the LCM: By dividing the numbers by common factors (2, 3, etc.), we find the LCM is $2 \times 2 \times 3 \times 2 = 24$.
Adjust the Numerators:
$6 \times 4 = 24$, so $5 \times 4 = 20$
$8 \times 3 = 24$, so $7 \times 3 = 21$
$12 \times 2 = 24$, so $3 \times 2 = 6$
Combine and Solve: $\frac{20 + 21 – 6}{24} = \frac{35}{24}$.
When a problem includes negative fractions in brackets, remember that subtracting a negative is the same as adding a positive.
Example: $\frac{16}{9} – (-\frac{5}{12}) + (-\frac{7}{18})$ simplifies to $\frac{16}{9} + \frac{5}{12} – \frac{7}{18}$.
LCM of 9, 12, 18: 36.
Adjusted Numerators: $\frac{64 + 15 – 14}{36}$.
Final Result: $\frac{65}{36}$.