Average, Mixture & Alligation
Average is just total ÷ count, but the questions are rarely that direct — they change one member of a group and ask what happens to the average. The key insight: when the average changes by d for n members, the total changes by n × d. Mixtures then extend the same idea using alligation.
Don't read this page — work through it. Each block asks you to commit to an answer before it tells you anything.
Start here: what do you already know?
Average & mixture — true or false?
6 statements · 12s each
The results worth memorising
| Quantity | Formula |
|---|---|
| Average | Total ÷ count |
| Total | Average × count |
| First n natural numbers | (n + 1) / 2 |
| First n odd numbers | n |
| First n even numbers | n + 1 |
| Change in total | n × (change in average) |
The last row solves the "when a new person joins, the average rises by 2" family of questions in a single line.
Alligation: the cross method
To mix a cheaper item at price c and a dearer at d to get a mean price m, the quantities are in the ratio:
(d − m) : (m − c)
Note the cross: the cheaper quantity pairs with the dearer difference. Get that backwards and every alligation answer inverts.
Sort by what happens to the average
Does the average rise or fall?
0/6Every number is halved
Which group does it belong to?
Commit to the answer
Compute, then check
1/4What is the average of the first 9 natural numbers?
8
Method, in order
Match the set to its average
Quick reference
Exam tip: for "the average of n people increases by d when someone joins", the newcomer's value is old average + (n+1) × d. That one line replaces a page of algebra.