Ratio & Average-Based Sets

At a glance 10 rules

1. 1Average to total$\text{total} = \text{average} \times \text{count}$$\text{missing value} = \text{total} - \text{known values}$
2. 2Split by a ratio$\text{total } T \text{ in } a : b \Rightarrow \dfrac{a}{a + b}\,T$$\text{the other part} = \dfrac{b}{a + b}\,T$
3. 3Ratio and a gap$a : b,\ \text{gap } d \Rightarrow \text{part} = \dfrac{d}{a - b}$$\text{total} = (a + b) \times \text{part}$
4. 4Joining two ratiosTo join A : B with B : C, scale both so B gets the same number: 4 : 5 and 3 : 2 become 12 : 15 and 15 : 10, so A : B : C = 12 : 15 : 10.
5. 5Weighted average$\bar{x} = \dfrac{\sum n x}{\sum n},\ n \text{ items at } x$$\text{base } A:\ \bar{x} = A + \dfrac{\sum n (x - A)}{\sum n}$
6. 6Shared memberTwo group averages that share one value: add the two group totals and subtract the grand total. What is left is the value counted twice.
7. 7Totals by ratio$\dfrac{T_1}{T_2} = \dfrac{n_1}{n_2} \times \dfrac{a_1}{a_2}$$T = n \times a:\ \text{total} = \text{count} \times \text{average}$
8. 8Balance the surplusWhen groups in different ratios combine into a known ratio, write each group's surplus (boys minus girls, say). The surpluses must add up to what the combined ratio allows.
9. 9Whole multiplesA group in the ratio a : b has a multiple of $a + b$ members, so a hidden total must be built from those multiples. Often only one way fits.
10. 10Largest one valueTo make one value as large as possible, make every other value as small as the conditions allow: whole, all different, on the right side of the median.