How to Solve Mixture and Alligation Problems
Solve mixture and alligation aptitude problems using the cross-rule for weighted averages — with a worked example and practice questions.
Expected Interview Answer
Mixture and alligation problems find the ratio in which two ingredients of different values (price, concentration, etc.) must be mixed to hit a target average, using the rule that the ratio of quantities equals the ratio of the distances from the target to each ingredient’s value, cross-wise.
The alligation rule states: (quantity of cheaper) : (quantity of dearer) = (dearer value − mean value) : (mean value − cheaper value). This works because the mean is a weighted average, and the weights must be inversely proportional to how far each ingredient sits from the mean — a value closer to the mean needs a smaller quantity to balance a value further away. The same rule applies to concentrations (like acid mixtures) and speeds/scores, not just prices, since it is really just solving a weighted-average equation for the mixing ratio.
- One cross-rule replaces solving simultaneous equations
- Extends to concentration, price and any weighted-average scenario
- Fast mental-math shortcut for competitive exams
AI Mentor Explanation
Two batters with averages 30 and 50 need to be picked in a ratio so the team’s combined average across their innings hits 35 — since 35 is much closer to 30 than to 50, far more innings should come from the 30-average batter. The alligation rule gives this directly: ratio = (50−35) : (35−30) = 15:5 = 3:1, meaning three times as many innings from the closer-average batter, cross-multiplying the distances from the target.
Worked example (alligation cross-rule)
Cheaper (20/kg)
- Distance from mean: 24 − 20 = 4
Dearer (30/kg)
- Distance from mean: 30 − 24 = 6
Ratio (cross)
- Cheaper : Dearer = 6 : 4 = 3 : 2
Step-by-Step Explanation
Step 1
Identify cheaper, dearer, mean values
Set up the two ingredient values and the target mean value.
Step 2
Apply the cross rule
Ratio = (dearer − mean) : (mean − cheaper), giving cheaper : dearer.
Step 3
Simplify the ratio
Reduce to lowest terms, same as any ratio problem.
Step 4
Scale to actual quantities
Use the common-multiplier technique if a total quantity is given.
What Interviewer Expects
- Correct cross-rule setup (dearer−mean : mean−cheaper)
- Understanding why the rule works (inverse distance weighting)
- Ability to reduce and scale the resulting ratio
- Recognizing when alligation applies (any weighted average)
Common Mistakes
- Reversing the cross-rule (getting dearer:cheaper backwards)
- Using simple average instead of weighted alligation
- Applying alligation when quantities, not values, are what is unknown
- Sign errors when the mean is outside the range of the two values
Best Answer (HR Friendly)
“Alligation is a shortcut for weighted averages. If you are mixing two things with different values to hit a target average, the ratio you need is the distance from the target to the dearer value, over the distance from the target to the cheaper value, cross-multiplied. It works because whichever value is closer to the target needs the bigger share to pull the average there.”
Follow-up Questions
- How would you extend alligation to three or more ingredients?
- How is alligation used in acid/water concentration problems?
- What happens if the target mean equals one of the two values?
- How does alligation relate to weighted averages in general?
MCQ Practice
1. Mix rice at 15/kg and rice at 25/kg to get a mixture at 18/kg. The ratio of cheaper to dearer rice is?
Ratio = (25−18):(18−15) = 7:3.
2. In the alligation rule, the ratio of cheaper to dearer quantity equals?
The cross rule: cheaper:dearer = (dearer − mean) : (mean − cheaper).
3. A 40% acid solution and a 60% acid solution are mixed to get 50%. The mixing ratio is?
Ratio = (60−50):(50−40) = 10:10 = 1:1.
Flash Cards
Alligation cross-rule? — Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper).
Why does the closer value need a bigger share? — Because the ratio is inversely proportional to distance from the mean.
What kinds of problems use alligation? — Any weighted-average scenario: prices, concentrations, scores, speeds.
After finding the ratio, how do you get actual quantities? — Apply the common-multiplier ratio technique using the given total.