Simplifying ratios using GCD
To simplify a ratio, divide both terms by their greatest common divisor (GCD). The GCD is the largest positive integer that divides both numbers without a remainder. Dividing 12:8 by their GCD of 4 gives 3:2. The simplified ratio has the same proportional relationship as the original but uses the smallest possible whole numbers.
The Euclidean algorithm is an efficient method for finding the GCD. Repeatedly replace the larger number with the remainder when the larger is divided by the smaller, until the remainder is zero. The last non-zero value is the GCD. For 48 and 36: 48 mod 36 = 12; 36 mod 12 = 0; GCD = 12. So 48:36 simplifies to 4:3.
Decimal values in a ratio cannot be directly simplified using integer GCD. The standard approach is to multiply both terms by a power of 10 sufficient to clear all decimal places, then apply the integer GCD. For example, 1.5:2.5 becomes 15:25, which simplifies to 3:5.
GCD(a, b) via Euclidean algorithm: while b ≠ 0, [a, b] = [b, a mod b]; GCD = a
Repeatedly replace (a, b) with (b, remainder of a ÷ b) until b equals zero. The remaining a is the GCD.
Simplified ratio = (a / GCD) : (b / GCD)
Divide each term of the ratio by the GCD to reduce it to its lowest terms.
A as % of total = a / (a + b) x 100
Express the first quantity as a share of the combined total by dividing a by (a + b) and multiplying by 100.