Arithmetic Mean
The arithmetic mean is the sum of the values divided by how many there are. The most useful reframing for the GRE is the reverse: sum = average × count. If six numbers average 15, their total is 90 — and now any question about one missing value, or about what happens when you add a seventh number, becomes simple. A second powerful idea is the deviation view: the mean is the balance point where the amounts above it exactly cancel the amounts below it, so if most values are clustered you can guess-and-adjust from a convenient anchor. For evenly spaced sets (consecutive integers, arithmetic sequences), the mean is just the average of the first and last terms, so the mean of 1 through 99 is (1 + 99)/2 = 50 without any summing. Keep the count honest — "the average of the integers from 1 to 10 inclusive" has 10 terms, not 9.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Averages
| Arithmetic mean | average = sum ÷ count |
|---|---|
| Sum from average | sum = average × count |
| Weighted average | (n₁a₁ + n₂a₂) / (n₁ + n₂) |
| New average after adding a value | (old sum + new value) / (count + 1) |
Mixtures & alligation
| Alligation ratio | n₁ : n₂ = (a₂ − M) : (M − a₁) (M = mixture mean) |
|---|---|
| Concentration | pure component ÷ total amount |
| Replacement (repeated) | remaining = original × (1 − r/V)ⁿ |
| Mean of two equal groups | (a₁ + a₂) / 2 (weights equal) |