Quick reference
| Topic | Rule |
|---|---|
| Percent of | X% of Y = X ÷ 100 × Y |
| Percent change | (new − old) ÷ old × 100 |
| Fractions | a/b + c/d = (ad + bc) / bd |
| Mean | sum ÷ count |
| Sample standard deviation | √( Σ(x − mean)² ÷ (n − 1) ) |
| Quadratic roots | (−b ± √(b² − 4ac)) ÷ 2a |
| Pythagorean theorem | a² + b² = c² |
Which tool for which job
| You want to… | Use |
|---|---|
| Work out a discount, tip or price rise | Percentage |
| Add or simplify fractions from a recipe or plan | Fraction |
| Summarise test scores or survey answers | Mean, Median, Mode, then Standard Deviation |
| Resize an image or scale a recipe | Ratio |
| Find a common denominator | GCF & LCM |
| Check a right angle or a diagonal | Pythagorean Theorem |
| Run a fair draw | Random Number |
Common mistakes the tools catch
Percent change from the wrong base. Change is always measured from the starting value: 80 to 100 is a 25% rise, but 100 to 80 is a 20% fall.
Sample versus population. Most real data is a sample, which needs n − 1 when calculating standard deviation.
Adding fractions straight across. 1/2 + 1/3 is 5/6, not 2/5: fractions need a common denominator first.
Why every answer shows its working
A number on its own is hard to trust and impossible to learn from. Each tool here shows the method, the intermediate steps and, where it helps, a second way to check the result, so you can use it for homework, teaching or double-checking a spreadsheet.