Averages, Ratio & Percentage

Averages, Ratio & Proportion

The connective tissue of arithmetic. Averages summarise a group; ratios compare parts; weighted averages join the two — and feed mixtures, DI, and word problems alike.

Core results

• Average $=\dfrac{\text{sum}}{\text{count}}$. Changing one value by $\Delta$ shifts the average by $\dfrac{\Delta}{n}$.
• Weighted average: for groups of sizes $w_i$ and means $\bar x_i$,

$\bar X=\frac{\sum w_i\,\bar x_i}{\sum w_i}.$

• Ratio scaling: $a:b=ka:kb$ — introduce a common multiplier $k$ to turn a ratio into actual numbers.
• Proportion: $\dfrac ab=\dfrac cd \iff ad=bc$ (cross-multiply).

A worked example

Class A has 30 students averaging 60; class B has 20 students averaging 70. What is the combined average?

It is not 65 — the larger class pulls the average toward 60. Weight by size:

$\bar X=\frac{30\cdot60+20\cdot70}{30+20}=\frac{1800+1400}{50}=\frac{3200}{50}=\mathbf{64}.$

Alligation view: the means 60 and 70 in the ratio of class sizes $30:20=3:2$ land the combined mean at $60+\frac{3}{5}\cdot10$… careful — it sits $\frac{2}{5}$ of the way from 60 to 70, i.e. $60+4=64$. Same answer, no fractions.

A ratio example

Divide ₹4,400 among A, B, C in the ratio 2 : 3 : 5.

Common multiplier $k=\dfrac{4400}{2+3+5}=440$. So A $=2k=880$, B $=3k=1320$, C $=5k=2200$.

Common traps

• Averaging averages. Never average two group means directly unless the groups are equal in size — weight them.
• Mixing ratio units with totals. Find $k$ first, then convert to actual amounts.
• "Increased in the ratio." $3:4$ means multiply by $\frac43$, not add.

Checklist

• Weight every average by its group size
• Convert ratios to numbers via a common multiplier $k$
• Cross-multiply for proportions
• Sanity-check: the combined average lies between the two group means, nearer the bigger group