How to Solve Alligation of Three Quantities
Solve alligation-of-three aptitude problems by chaining two-term alligation rules, with a worked example and practice questions.
Expected Interview Answer
Alligation of three quantities extends the standard two-way alligation rule by mixing three ingredients of known values to hit a target mean, solved by pairing the mean against each extreme in turn and combining the resulting ratios into a single three-way ratio.
The classic two-term alligation rule states that the ratio of quantities of two ingredients is inverse to the difference between each ingredient’s value and the mean value. With three ingredients, first sort the values from lowest to highest, then apply cross-differences: pair the cheapest with the mean, the mean with the dearest, and where the middle ingredient overlaps both pairings, split its contribution so the two ratios remain consistent through a common scaling factor. The final answer is a single ratio a:b:c that, when weighted against each ingredient’s value, reproduces exactly the target mean. Always verify by computing the weighted average back from the ratio.
- Extends the fast two-term shortcut instead of solving three simultaneous equations
- Works for any number of ingredients if applied pairwise and scaled consistently
- The final ratio can be verified quickly by recomputing the weighted mean
AI Mentor Explanation
A coach blends players with strike rates 80, 100, and 140 to hit a target team strike rate of 110 in a T20 lineup, choosing how many batters from each tier to include. Pairing the 80-tier against the mean gives one ratio, pairing the mean against the 140-tier gives another, and the 100-tier batters bridge both pairings since their rate sits between the target and one extreme. Alligation of three works exactly this way: chain two two-way alligations together and scale them onto one consistent ratio.
Worked example
Sort values
- 20, 30, 40; mean = 26
Cross-differences
- |40−26|=14, |20−26|=6
- ratio 14:6 = 7:3
Verify
- (7×20 + 3×40)/10 = 26
Step-by-Step Explanation
Step 1
Sort the three values
Arrange the ingredient values from lowest to highest around the target mean.
Step 2
Pair extremes against the mean
Compute |value − mean| for the lowest and highest ingredients.
Step 3
Cross the differences
The ratio of the two extreme quantities is the inverse of their differences from the mean.
Step 4
Reconcile the middle term
Scale the middle ingredient consistently so the full three-way ratio reproduces the target mean when checked.
What Interviewer Expects
- Correct sorting of the three ingredient values relative to the target mean
- Accurate cross-difference computation for the two extremes
- Sound handling of the middle value, whether equal to the mean or between two pairings
- Verification of the final ratio by recomputing the weighted average
Common Mistakes
- Applying two-term alligation directly to three terms without reconciling a common scale
- Forgetting to verify the final ratio by recomputing the weighted mean
- Mishandling the case where the middle value equals the target mean exactly
- Sorting the values incorrectly, which flips the direction of the cross-differences
Best Answer (HR Friendly)
“I treat it as two chained two-term alligations: pair the cheapest ingredient against the target mean, then pair the priciest ingredient against the same mean, and cross the differences to get each side of the ratio. The middle ingredient either equals the mean, in which case it can be added freely, or it bridges both pairings through a common scale factor. I always double check by plugging the final ratio back in and confirming it reproduces the target mean.”
Follow-up Questions
- How would you extend alligation to four or more ingredients?
- What happens if the middle value exactly equals the target mean?
- How does alligation relate to the concept of a weighted average in reverse?
- How would you solve an alligation-of-three problem where quantities, not just values, are constrained?
MCQ Practice
1. Three grades of rice cost 15, 20, and 30 per kg. To get a blend averaging 20 per kg using only the 15 and 30 grades, the ratio of 15-grade to 30-grade is?
Cross-differences: |30−20|=10, |20−15|=5, ratio 10:5 = 2:1.
2. In alligation of three quantities, if the middle value exactly equals the target mean, what can be said about its quantity?
A quantity valued exactly at the mean can be added in any proportion since it does not pull the average in either direction.
3. Which step is essential before applying cross-differences in alligation of three?
Sorting establishes which values are the extremes and which is the middle, so the cross-difference pairing is applied correctly.
Flash Cards
Two-term alligation rule? — Ratio of quantities is inverse to the difference of each value from the mean.
How to extend to three terms? — Pair each extreme against the mean, then reconcile the middle term through a common scale.
Middle value equal to the mean? — It can be added in any amount without shifting the blended average.
How to verify the final ratio? — Recompute the weighted average from the ratio and confirm it equals the target mean.