Add, subtract, multiply, divide — simplified
Add, subtract, multiply, divide — simplified
Addition and subtraction require a common denominator. The reliable method is to cross-multiply:
a/b + c/d = (ad + cb) / bd
a/b − c/d = (ad − cb) / bd
For 3/4 + 5/6: (3×6 + 5×4) / (4×6) = (18 + 20) / 24 = 38/24, which simplifies to 19/12, or 1 7/12.
Multiplication is the easiest — multiply across the top and across the bottom:
a/b × c/d = ac / bd
Division is multiplication by the reciprocal — flip the second fraction and multiply:
a/b ÷ c/d = a/b × d/c = ad / bc
This rule is usually memorised without explanation, which is a shame because the reason is simple. Dividing by a number means asking how many of it fit into the first quantity.
Take 1/2 ÷ 1/4. How many quarters fit into a half? Two. And indeed 1/2 × 4/1 = 4/2 = 2. Dividing by a fraction smaller than one produces a larger answer, because you are counting how many small pieces fit into something bigger.
More formally, dividing by c/d is the same as multiplying by its multiplicative inverse d/c, since (c/d) × (d/c) = 1. The rule is not a trick; it follows from what division means.
A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. To simplify, divide both by their greatest common divisor.
For 38/24: the GCD is 2, so dividing both gives 19/12. For 24/36: the GCD is 12, giving 2/3.
The Euclidean algorithm finds the GCD efficiently — repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero. For 38 and 24: 38 mod 24 = 14, then 24 mod 14 = 10, then 14 mod 10 = 4, then 10 mod 4 = 2, then 4 mod 2 = 0. The GCD is 2.
An improper fraction has a numerator larger than its denominator, such as 19/12. A mixed number expresses the same value as a whole part plus a proper fraction: 1 7/12.
To convert, divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the new numerator: 19 ÷ 12 = 1 remainder 7, giving 1 7/12.
Going the other way, multiply the whole number by the denominator and add the numerator: 2 3/5 becomes (2×5 + 3)/5 = 13/5.
Mixed numbers are easier to read; improper fractions are easier to compute with. Convert to improper form before doing arithmetic, then back at the end if the context calls for it.
Cooking. Scaling a recipe that serves 4 to serve 6 means multiplying every quantity by 3/2. Two-thirds of 3/4 cup is 1/2 cup exactly — a calculation decimals make messier, not cleaner.
Construction and woodworking. Imperial measurements are fraction-based throughout: 3/8 inch, 5/16 inch, 11/16 inch. Adding board thicknesses and cut allowances is fraction arithmetic.
Music. Time signatures and note durations are fractions, and rhythmic subdivision is fraction arithmetic performed in real time.
Exact values. One third is exactly 1/3. As a decimal it is 0.333... and any truncation introduces error that compounds through subsequent calculations. This is why fractions persist in mathematics rather than being replaced by decimals.
Cross-multiply: a/b + c/d equals (ad + cb) divided by bd. Then simplify by dividing the result by the greatest common divisor of the numerator and denominator.
Because dividing by a number is the same as multiplying by its reciprocal. Dividing by c/d is identical to multiplying by d/c, since those two multiply to 1.
Divide both the numerator and denominator by their greatest common divisor. For 38/24 the GCD is 2, giving 19/12.
One where the numerator is larger than the denominator, such as 19/12. It represents the same value as the mixed number 1 and 7/12, and is easier to use in calculations.