Question

what is the definition of subtractring fractions, multiplying fractons, and dividing fractions

Answer 1

Here are short, precise definitions and rules plus examples for each operation.

1) Subtracting fractions
- Rule: To subtract fractions you must have a common denominator. If denominators are the same, subtract the numerators and keep the denominator:
a/b − c/b = (a − c)/b.
- If denominators differ, convert to a common denominator (often the least common denominator or simply the product of denominators), then subtract:
a/b − c/d = (ad − bc)/(bd) (or use LCM for a smaller denominator), provided b and d ≠ 0.
- Always simplify the result.
- Example: 3/4 − 1/6. Convert to twelfths: 9/12 − 2/12 = 7/12.

2) Multiplying fractions
- Rule: Multiply numerators and multiply denominators:
(a/b) × (c/d) = (ac)/(bd), with b and d ≠ 0.
- You can simplify by canceling common factors between any numerator and any denominator before or after multiplying.
- Example: (2/3) × (3/5) = (2×3)/(3×5) = 6/15 = 2/5 (cancel the 3).

3) Dividing fractions
- Rule: Division by a fraction is multiplication by its reciprocal:
(a/b) ÷ (c/d) = (a/b) × (d/c), provided c ≠ 0.
- Then multiply and simplify as with multiplication.
- Example: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8 = 1 7/8.

Notes:
- Denominators must never be zero.
- For mixed numbers, convert to improper fractions to multiply or divide (or borrow/convert when subtracting).
- Always reduce your final answer to lowest terms (divide numerator and denominator by their gcd).