How to Apply the Alligation Rule for Mean Price
Apply the alligation rule to find mixing ratios from a mean price, with a cross-diagram method, worked example and practice questions.
Expected Interview Answer
The alligation rule finds the ratio in which two ingredients of different prices (or concentrations) must be mixed to achieve a given mean price, using the shortcut: ratio = (dearer price − mean price) : (mean price − cheaper price).
Alligation is a compressed version of the weighted-average equation: if quantity q1 at price p1 and quantity q2 at price p2 mix to mean price m, then q1(m−p1)... rearranges to q1:q2 = (p2−m):(m−p1). The rule is typically drawn as a cross diagram with cheaper price and dearer price on the left, mean price in the middle, and the two differences taken diagonally to form the ratio. It applies equally to prices, percentages (like milk purity or alcohol concentration), and speeds, wherever a weighted average is being reverse-engineered into a mixing ratio. The mean price must always lie between the cheaper and dearer values, otherwise the problem has no valid mixture.
- One cross-diagram shortcut replaces solving a full weighted-average equation
- Generalizes beyond prices to concentrations, purities, and speeds
- Quickly sanity-checks whether a stated mean value is even possible
AI Mentor Explanation
Two batters with career averages of 30 and 50 need to combine innings counts so the team’s combined average lands at exactly 42 — alligation gives the innings ratio directly as (50−42):(42−30) = 8:12 = 2:3, without solving a full weighted-average equation from scratch. The cross-diagram trick places 30 and 50 on the ends, 42 in the middle, and takes the diagonal differences as the ratio — a direct shortcut equivalent to the alligation rule for mean price.
Worked example
Cheaper : Dearer
- 60 rupees/kg
- 90 rupees/kg
Mean price
- 70 rupees/kg
Ratio (cheaper:dearer)
- (90−70):(70−60) = 2:1
Step-by-Step Explanation
Step 1
Identify cheaper and dearer values
Label the lower price/value p1 and higher price/value p2.
Step 2
Draw the cross diagram
Place p1 and p2 on the ends, mean price m in the middle.
Step 3
Take diagonal differences
Ratio of cheaper:dearer quantity = (p2−m):(m−p1).
Step 4
Verify the mean lies between
Confirm p1 < m < p2 (or the reverse); otherwise the mixture is impossible.
What Interviewer Expects
- Correct setup of the alligation cross diagram
- Accurate diagonal-difference ratio computation
- Recognizing when alligation applies beyond price (concentration, speed, rating)
- Checking the mean value is achievable between the two extremes
Common Mistakes
- Inverting the ratio (dearer:cheaper instead of cheaper:dearer)
- Using average speed alligation on distance instead of time
- Applying alligation when the mean lies outside the two extreme values
- Confusing the mean price with one of the ingredient prices
Best Answer (HR Friendly)
“Alligation is a shortcut for reverse-engineering a mixing ratio from a target mean. I set the cheaper price and dearer price on either side of a cross diagram with the mean price in the middle, then take the diagonal differences: dearer-minus-mean over mean-minus-cheaper gives the ratio of cheaper to dearer quantity. It works for percentages and speeds too, not just prices, and the mean always has to sit between the two extremes for the mixture to make sense.”
Follow-up Questions
- How does alligation apply to milk-water purity problems specifically?
- How would you handle alligation with three or more ingredients?
- Why must the mean value lie strictly between the two extremes?
- How is alligation related to the general weighted-average formula?
MCQ Practice
1. Rice at 40 rupees/kg is mixed with rice at 60 rupees/kg to get a mean price of 52 rupees/kg. The ratio of cheaper to dearer rice is?
Ratio = (60−52):(52−40) = 8:12, which simplifies to 2:3, so cheaper:dearer = 2:3.
2. For the alligation rule, if p1 is cheaper and p2 is dearer, the ratio of quantity of p1 to quantity of p2 is?
The alligation rule states quantity ratio p1:p2 = (p2−m):(m−p1), the diagonal cross-difference.
3. Which condition must hold for an alligation mean-price problem to have a valid solution?
The mean price must lie between the two ingredient prices, otherwise no valid mixing ratio exists.
Flash Cards
Alligation ratio formula? — Cheaper:Dearer quantity = (Dearer price − Mean) : (Mean − Cheaper price).
What does alligation shortcut avoid? — Solving the full weighted-average equation from scratch.
Where else does alligation apply besides price? — Percentages, concentrations, purities, ratings, and speeds.
Required condition for a valid alligation solution? — The mean value must lie strictly between the two extreme values.