Fractions can feel intimidating for students of all ages, but mastering the foundational steps turns uncertainty into confidence.
By Tuesday Fuller
Updated Aug 30, 2022
Start with the expression 3/6 + 1/8. Because the denominators differ—sixths and eighths—you cannot add them directly. They must share a common denominator.
List the multiples of 6: 12, 18, 24, 30, 36, …
List the multiples of 8: 16, 24, 32, 40, 48, …
Identify the smallest number that appears in both lists. Here it is 24.
Convert the first fraction to a denominator of 24 by multiplying both numerator and denominator by 4 (since 6 × 4 = 24):
3/6 = 12/24.
Convert the second fraction similarly, using a factor of 3 (because 8 × 3 = 24):
1/8 = 3/24.
Rewrite the expression with the new common denominator: 12/24 + 3/24. Now you can add the numerators.
Consider the problem 3/4 + 2/4. Since the denominators match, you can proceed directly.
Add the numerators: 3 + 2 = 5.
Write the sum over the shared denominator: 5/4. This improper fraction can be left as is or converted to a mixed number: 5 ÷ 4 = 1 with a remainder of 1, so 1 1/4.
Now examine 5/8 – 3/8, which also has matching denominators.
Subtract the numerators: 5 – 3 = 2.
Express the difference: 2/8. Reduce it by dividing numerator and denominator by 2: 1/4.
For multiplication, denominators need not match. Take 5/7 × 3/4 as an example.
Multiply the numerators (5 × 3) and the denominators (7 × 4) to obtain 15/28.
Thus, 5/7 × 3/4 = 15/28.
Division requires a slightly different approach. Consider 4/5 ÷ 2/3—a so‑called complex fraction.
Invert the divisor and convert the operation to multiplication: 4/5 × 3/2.
Multiply across: 4 × 3 = 12 and 5 × 2 = 10, giving 12/10. Reduce by dividing numerator and denominator by 2 to get 6/5. If you prefer an in‑problem reduction, cross‑cancel the 2’s before multiplying: 4 ↘ 2 = 2, 3 ↘ 2 = 1, then 2/5 × 3/1 = 6/5.
The final result of the division is 6/5 (or 1 1/5 in mixed‑number form).
