Averages and Mixtures
Averages and mixtures are two sides of the same coin — both are solved fastest by tracking total sums rather than recalculating an average from scratch after every change.
10 questions · 5 minutes · instant scoring
What this topic actually tests
An average is simply the sum of all values divided by the count of values, and the single fastest CLAT technique is to work with the total sum, not the average itself, whenever a value is added, removed, or replaced. If a class of 30 students has an average weight of 45 kg, the total weight is 30 x 45 = 1,350 kg — a fixed number that stays put in your working even as the problem changes. When a new member joins and the average shifts, find the new total first (new count x new average) and subtract the old total to isolate the new member's individual value, rather than trying to reason about the change in average directly. Mixture problems (combining two quantities of different concentration, price, or speed to hit a target average) are best solved with alligation: for two components with values a and b mixed to produce average m, the ratio in which they must be combined is (b-m):(m-a). Worked example: tea worth Rs 60/kg is mixed with tea worth Rs 90/kg to produce a blend worth Rs 70/kg. By alligation, the ratio of the Rs 60 tea to the Rs 90 tea is (90-70):(70-60) = 20:10 = 2:1 — meaning twice as much of the cheaper tea is needed, which matches intuition since the target price (70) sits closer to 60 than to 90. For 'replace part of a solution with water' questions, remember that the concentration of what remains is unchanged by removal; only the water added afterward dilutes it, so calculate the acid or milk quantity in the remaining volume first, then add the diluting water to the total volume before recomputing the percentage.
The common trap on this topic
The most frequent averages trap is recomputing the average by intuition rather than by the sum-and-count method when a value is added or removed, especially in 'weight of the new person' or 'age of the excluded member' questions — the correct approach always requires converting back to total sums first, because averages themselves cannot be added, subtracted, or otherwise manipulated the way totals can. A second, closely related trap in mixture problems is applying alligation directly to volumes without checking whether the quantities being averaged are prices, concentrations, or speeds — alligation works for any weighted average, but students sometimes mix up which of the two ratio terms corresponds to which component, reversing the final ratio. A third specific trap appears in 'replace X litres of solution with water' questions: many students assume the concentration of the remaining solution changes before dilution, when in fact removing a portion of a uniform mixture does not change its concentration at all — only the subsequent addition of pure water dilutes it — so the acid or milk amount must be calculated on the reduced volume that remains, not the original volume.
Take the micro-test
The average of 5 numbers is 42. If one number is excluded, the average of the remaining 4 numbers becomes 40. Find the excluded number.
The average weight of 30 students in a class is 45 kg. When the class teacher's weight is included, the average rises by 1 kg. Find the teacher's weight.
Find the average of the first 50 natural numbers.
Find the total marks scored by the remaining 25 students.
Find the average marks scored by the remaining 25 students.
A 40-litre mixture of milk and water is in the ratio 3:1. How many litres of water must be added to change the ratio of milk to water to 3:2?
In what ratio should a shopkeeper mix tea worth Rs 60/kg with tea worth Rs 90/kg so that the resulting mixture is worth Rs 70/kg?
The average of 11 results is 50. The average of the first 6 results is 49, and the average of the last 6 results is 52. Find the value of the 6th result.
A cistern contains 100 litres of a solution that is 40% acid. 20 litres of this solution is drawn off and replaced with 20 litres of pure water. Find the new concentration of acid in the mixture.
20 kg of rice priced at Rs 40/kg is mixed with 30 kg of rice priced at Rs 60/kg. Find the average price per kg of the resulting mixture.
FAQ
What is the fastest way to solve "a new member changes the average" questions?
Convert both the old and new averages into total sums (count x average), then subtract the old total from the new total to isolate the value of the person or item that was added or removed.
What is alligation and when should I use it?
Alligation is a shortcut for mixture problems: for two components with values a and b combined to give an average m, they must be mixed in the ratio (b-m):(m-a). Use it whenever two quantities are blended to hit a target average, price, or concentration.
Does removing part of a mixture change its concentration?
No — removing a portion of a uniform (well-mixed) solution leaves the concentration of what remains unchanged; only adding a diluting substance afterward, like water, changes the concentration.
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