GMAT mixture problems fall into four families. Recognizing which one you’re facing is half the battle:
All four reduce to the same core idea: a mixture’s overall property is a weighted average of its components’ properties, weighted by quantity.
For two components A and B mixed into mixture M:
(Quantity_A × Concentration_A) + (Quantity_B × Concentration_B) = Quantity_M × Concentration_MWhere Quantity_M = Quantity_A + Quantity_B.
This single equation solves nearly every basic mixture problem. Everything else (alligation, etc.) is a shortcut derived from this.
Alligation is a shortcut for two-component mixture ratio problems. It avoids setting up algebra.
Setup:
A B
\ /
\ /
Mix (M)
/ \
/ \
(B − M) (M − A)Rule: The ratio of quantities of A to B equals the ratio of the differences (cross-subtracted), taken so both differences are positive:
Quantity_A : Quantity_B = (Value_B − Value_M) : (Value_M − Value_A)Example: Mix a 20% solution with a 50% solution to get a 30% solution. - A = 20, B = 50, M = 30 - Ratio A:B = (50−30) : (30−20) = 20:10 = 2:1 - So you need 2 parts of the 20% solution for every 1 part of the 50% solution.
Why it works: It’s just the master equation rearranged — alligation is algebra in disguise, but much faster under time pressure. This is the single highest-leverage technique for this topic; memorize the diagram, not just the formula.
Visual/intuitive check: The mixture’s value always sits closer to the component with more quantity — like a seesaw balancing around a fulcrum. If M is much closer to A than to B, then A makes up the bigger share.
Many “mixture” problems on GMAT aren’t chemistry at all — they’re disguised weighted averages: - Average price per item when buying at two different prices - Average score combining two class sections - Average speed for two legs of a trip (weighted by time, not distance — common trap) - Combined population/employee ratios (e.g., % of men in a combined company after merger)
Key insight: Whenever you see “X% of group 1 has property P, Y% of group 2 has property P, what % of the combined group has property P?” — this is alligation with concentrations P over each group’s size.
Common trap: For average speed, weighting is by time spent, not distance — unless equal distances are traveled, the simple average of two speeds is wrong. Use:
Average speed = Total distance / Total timeSetup: A container has volume V of pure liquid (or mixture). You repeatedly remove a fixed amount/fraction and replace it with water (or another substance), n times.
Key Formula:
Final amount of original substance = V × (1 − x/V)^nwhere x = amount removed/replaced each time, n = number of repetitions.
Example: A 10-liter container is full of pure milk. Each time, 2 liters are removed and replaced with water, repeated 3 times. How much milk remains?
Milk remaining = 10 × (1 − 2/10)^3 = 10 × (0.8)^3 = 10 × 0.512 = 5.12 litersNotes: - This formula only applies when the same fraction is removed each time (not necessarily the same absolute amount each time, but GMAT almost always uses constant amount + constant total volume, making fraction constant too). - If the problem only does this once, you don’t need the exponent formula — just track amounts directly; reserve the formula for 2+ iterations. - Always double check whether the question asks for the substance remaining or the substance removed/replaced (1 − remaining fraction).
These involve combining discrete items (not liquids) — e.g., “a bag has red and blue marbles in ratio 3:5, how many of each to add to change the ratio to 1:1.”
Approach: 1. Let original quantities be 3k and 5k (using the given ratio with a variable multiplier) 2. Set up an equation based on what’s added/removed 3. Solve for k, then answer the specific question
Cost-based version: “Coffee A costs $4/lb, Coffee B costs $6/lb. How many lbs of each to make 10 lbs worth $5.20/lb total?” - This is alligation again, with price as the “concentration”: - Ratio A:B = (6 − 5.20) : (5.20 − 4) = 0.8 : 1.2 = 2:3 - Total parts = 5, total weight = 10 lbs → A = 4 lbs, B = 6 lbs
| Trap | Why it happens | Fix |
|---|---|---|
| Averaging percentages directly | Treating 20% and 50% mix as 35% regardless of quantity | Only true if equal quantities — otherwise weighted average required |
| Speed mixtures averaged by distance instead of time | Misapplying “average of two numbers” | Use total distance ÷ total time |
| Forgetting “remaining” vs “removed” in dilution | Question asks for amount removed, you compute amount remaining | Re-read final question line carefully before bubbling answer |
| Applying exponent formula to a single removal | Overcomplicating one-step problems | Use exponent formula only for n ≥ 2 repeated replacements |
| Mixing up which difference goes with which quantity in alligation | Cross-multiplication direction confusion | Remember: quantity of A pairs with the farther difference (B − M), not (M − A) |
| Ratio problems where “parts” ≠ actual units | Assuming k = 1 | Always solve for k using a given total or given quantity before finalizing the answer |
Since your diagnosed bottleneck is end-of-section rushing rather than knowledge gaps, the highest-leverage move for mixture problems specifically is to default to alligation first before writing algebra — it’s typically 2-3 steps faster than setting up the master equation from scratch, which directly protects your pacing on Quant. Reserve full algebraic setup only for replacement/dilution problems or multi-component (3+) mixtures where alligation doesn’t directly apply.