🔢 Basic Math Calculators

Percentages: Three Core Problems

Every percentage calculation is a variation of one formula: Part = Percent × Whole ÷ 100. The three common problem types are: (1) Find the percentage — What percent of 80 is 20? Answer: 20 ÷ 80 × 100 = 25%. (2) Find the part — What is 35% of 240? Answer: 35 ÷ 100 × 240 = 84. (3) Find the whole — 15 is 25% of what number? Answer: 15 ÷ 25 × 100 = 60. Percentage change is: ((New − Old) ÷ Old) × 100. A price rising from $80 to $100 is a 25% increase; falling from $100 to $80 is a 20% decrease. Note that percentage increases and decreases are not symmetric: a 50% increase followed by a 50% decrease returns to 75% of the original value, not 100%.

Fractions: Addition, Subtraction, and Simplification

To add or subtract fractions, find a common denominator, convert both fractions, then add or subtract the numerators: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. To multiply fractions, multiply numerators and denominators directly: 2/3 × 3/5 = 6/15 = 2/5. To divide fractions, multiply the first by the reciprocal of the second: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. To simplify a fraction, divide both numerator and denominator by their greatest common divisor (GCD). The GCD of 12 and 18 is 6, so 12/18 simplifies to 2/3. Mixed numbers (like 2 3/4) convert to improper fractions by multiplying the whole number by the denominator and adding the numerator: 2 3/4 = (2×4+3)/4 = 11/4.

Ratios and Proportions

A ratio compares two quantities and can be expressed as a:b, a/b, or "a to b." Ratios are simplified the same way as fractions, by dividing both parts by their GCD. A ratio of 15:25 simplifies to 3:5. Proportions state that two ratios are equal: a/b = c/d, or equivalently a:b = c:d. To solve for the missing value in a proportion, use cross-multiplication: if 3/4 = x/20, then 3×20 = 4×x, so x = 60/4 = 15. Proportions are used in scaling recipes, converting units, calculating map distances, and solving similar triangle problems. Scale factor problems — "a model is 1:48 scale, the real object is 96 inches, how long is the model?" — are a direct application of proportions.

Averages and Weighted Averages

The arithmetic mean (average) is the sum of all values divided by the count. For the set 18: sum = 50, count = 5, mean = 10. A weighted average assigns different weights to each value based on its importance or frequency. Weighted mean = Σ(value × weight) ÷ Σ(weights). For example, a student with exam scores of 75 (30% weight), 82 (30% weight), and 90 (40% weight): weighted average = (75×0.3 + 82×0.3 + 90×0.4) = 22.5 + 24.6 + 36 = 83.1. GPA calculations are weighted averages where each course grade is weighted by credit hours. Course averages are often weighted by assessment type (homework, midterm, final).