Averages & Mixtures
An average is nothing more than a fair-share redistribution: total ÷ count. That one definition — read backwards as total = average × count — solves the large majority of GRE average questions, because the test almost always gives you two of the three quantities and asks for the third. The wrinkle the GRE leans on is the weighted average: when groups have different sizes, the combined average is pulled toward the bigger group, never the plain midpoint of the two averages. Mixtures are weighted averages in disguise (blend two salt solutions and the result’s concentration is the weighted average of the two), and alligation is a lightning-fast shortcut for finding the ratio in which to mix them. The last piece is "change in average" — how one new value or one swap shifts the mean — which the GRE tests constantly because it rewards understanding over grinding. This chapter builds all four: the arithmetic mean, weighted averages, mixtures and alligation, and change-in-average reasoning.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Rewrite every average condition as a total: total = average × count. Two of the three are always given.
- For evenly spaced sets, the mean is just (first + last)/2 — no summing needed (1 to 99 averages 50).
- A weighted average always lands between the two averages, closer to the group with more weight — a fast Quantitative Comparison cue.
- Alligation gives the mixing ratio in one line: n₁ : n₂ = (a₂ − M) : (M − a₁), the cross differences around the target M.
- To change an average, track the change in the SUM: raising one value by d over n items lifts the mean by d/n.
- Diluting with water keeps the pure component fixed — solve from that constant amount, not the changing total.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Averaging two group averages directly when the groups are different sizes (ignoring the weights).
- Miscounting an inclusive range — 20 to 40 inclusive has 21 terms, not 20.
- In alligation, mixing up which difference belongs to which ingredient (pair each amount with the OTHER’s gap).
- Recomputing an entire average from scratch instead of adjusting the total for a change-in-average question.
- Forgetting that adding a value equal to the current mean leaves the average unchanged.
- In dilution, changing the amount of the pure component — only the total (with added water) changes.
📈 GRE exam insight & question patterns
Quantitative Comparison — class A (40 students) averages 60; class B (10 students) averages 90. Quantity A: combined average. Quantity B: 70. Which is greater?
Combined = (40·60 + 10·90)/50 = (2400 + 900)/50 = 66. Quantity B (70) is greater — the mean leans toward the larger class A.
Change in average — 6 numbers average 15. Adding a 7th makes the average 16. What is the 7th number?
New total = 16·7 = 112; old total = 90; 7th = 112 − 90 = 22.
Numeric Entry — in what ratio must 10% and 40% solutions be mixed to get 30%? Enter as weaker : stronger.
Alligation: (40 − 30) : (30 − 10) = 10 : 20 = 1 : 2.
Mixture — 8 litres of 25% acid are mixed with 2 litres of 75% acid. What is the concentration?
(8·25 + 2·75)/10 = (200 + 150)/10 = 35%.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Averages & Mixtures when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
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) |