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CLAT Quantitative Techniques★ 2–3 Q within Quantitative Techniques

Averages, Mixtures and Alligation

Averages appear in almost every CLAT data caselet, and mixtures bring in the neat alligation shortcut. This chapter covers the average, the effect of adding or removing a value, the weighted average, and the rule of alligation for combining two prices or concentrations.

⏱ 18 min readModerate difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

CLAT PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Compute an average and recover the sum from average and count
2Adjust an average when a value is added or removed
3Apply the weighted average when group sizes differ
4Read a mixture's components as fractions of the whole
5Use alligation to combine two prices or concentrations at a mean

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Why this chapter matters in CLAT

The average is the single most-used idea in CLAT data caselets, and mixtures bring the elegant alligation shortcut that solves a blend in one line. Both are elementary, but students stumble on the weighted average (mistaking it for a simple mean) and on add-or-remove questions (forgetting to work through the sum). This chapter fixes the average and its variations and the alligation rule, converting a recurring block of the section into fast marks.

Before you start — revise these

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Percentage, Ratio and Proportion

Mixtures and alligation build directly on ratio.

Averages, Mixtures and Alligation — CLAT Quantitative Techniques

The average is the most-used single idea in CLAT data caselets — scores, incomes, quantities. Close behind sit mixtures, where the elegant alligation rule lets you combine two prices or concentrations in your head. This chapter builds the average and its variations — adding, removing, weighting — and then the alligation shortcut that turns a mixture problem into a one-line ratio.

1. The average

$average = \frac{sum of values}{number of values}$

Average of $10, 20, 30, 40$ is $\frac{100}{4} = 25$ .
The key rearrangement: $sum = average \times count$ — this unlocks most caselet questions.

Whenever a question gives an average and a count, first recover the sum — then work with it.

2. Adding or removing a value

Because the average rides on the sum, changing the data changes the sum first.

Average of 5 numbers is 20; remove the number 30. New average of the 4?
- Sum $= 20 \times 5 = 100$ ; after removing 30, sum $= 70$ ; new average $= \frac{70}{4} = 17.5$ .
Average age of 4 people is 25; a person aged 35 joins. New average?
- Sum $= 100$ ; new sum $= 135$ ; new average $= \frac{135}{5} = 27$ .

Method: convert average → sum, adjust the sum, then divide by the new count.

3. The weighted average

When groups have different sizes, weight each value by its count.

$weighted average = \frac{\sum ( value \times weight )}{\sum weight}$

3 kg at ₹40 and 2 kg at ₹50: average price $= \frac{3 \times 40 + 2 \times 50}{5} = \frac{220}{5} = 44$ , i.e. ₹44.
Note it is not the simple mean ₹45 — the cheaper item weighs more.

4. Mixtures — ratio of components

A mixture in ratio $a : b$ has components in fractions $\frac{a}{a + b}$ and $\frac{b}{a + b}$ of the whole.

A 40-litre mixture of milk and water in $5 : 3$ has water $= \frac{3}{8} \times 40 = 15$ litres (milk $= 25$ ).

5. Alligation — combining two at a mean

To mix two ingredients of prices (or concentrations) $c$ (cheaper) and $d$ (dearer) into a mean $m$ , the ratio in which they must be mixed is:

$\frac{quantity of cheaper}{quantity of dearer} = \frac{d - m}{m - c}$

Rice at ₹30 mixed with rice at ₹45 to make ₹40:
- $\frac{30 -rice}{45 -rice} = \frac{45 - 40}{40 - 30} = \frac{5}{10} = \frac{1}{2}$ , i.e. 1 : 2.

Read the cross: the distance of the other price from the mean gives each ingredient's share. The mean lies nearer the ingredient present in greater quantity.

6. Exam protocol

Convert every average to a sum before manipulating the data.
To add/remove a value, adjust the sum, then divide by the new count.
Use the weighted average when group sizes differ — never a plain mean.
Read a mixture's components as fractions of the whole.
For two-ingredient blends, apply alligation: $\frac{d - m}{m - c}$ .
Sanity-check: the mean sits nearer the larger quantity.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Average

average = \frac{sum of values}{number of values}

Rearrange to sum = average × count — the key to most caselets.

Weighted average

weighted average = \frac{\sum ( value \times weight )}{\sum weight}

Weight each value by its group size, not a plain mean.

Alligation

\frac{cheaper}{dearer} = \frac{d - m}{m - c}

Cross the distances from the mean; the mean sits nearer the larger quantity.

CLAT test series

Ready to be tested on this?

Averages, Mixtures and Alligation shows up in every CLAT paper — sit a timed one and find out where you actually stand.

2–3Qs in paper

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Traps CLAT sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Taking a simple mean when groups differ in size.

✓ When quantities carry different weights, use the weighted average. Mixing 3 kg at ₹40 with 2 kg at ₹50 gives ₹44, not the simple mean of ₹45, because the cheaper item weighs more.

WATCH OUT

✗ Adjusting the average directly instead of the sum.

✓ The average rides on the sum, so convert average to sum first, change the sum by adding or removing the value, then divide by the new count. Trying to shift the average without the sum leads to errors.

WATCH OUT

✗ Reversing the alligation ratio.

✓ The quantity of the cheaper ingredient is proportional to (dearer − mean), and the dearer to (mean − cheaper) — the distance of the other price from the mean. Cross the differences; do not pair each price with its own distance.

WATCH OUT

✗ Misreading a mixture ratio as an amount.

✓ A 5 : 3 milk-to-water mixture means water is 3/(5+3) of the whole, not 3 litres. Convert the ratio term to a fraction of the total before finding the amount.

WATCH OUT

✗ Forgetting a shared value is counted twice.

✓ In overlapping-group averages, adding the two group sums double-counts the shared item. Subtract the total to recover it — a classic 'find the middle result' setup.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Averages, Mixtures and Alligation?

8 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

8 questions~6 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•average = sum ÷ count; and sum = average × count
•To add or remove a value, adjust the sum, then divide by the new count
•Weighted average weights each value by its group size — not a plain mean
•A mixture in ratio a : b has components a/(a+b) and b/(a+b) of the whole
•Alligation: cheaper : dearer = (d − m) : (m − c)
•The mean sits nearer the ingredient present in greater quantity
•In overlapping groups, the shared value is double-counted — subtract the total

CLAT question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: 3

Question style	Marks each	Typical count	What it tests
Basic & adjusted averages	~1–2 Q
Weighted average	~1 Q
Mixtures & alligation	~1 Q

Prep strategy

Practise converting average to sum and back on caselet data
Drill add/remove and overlapping-group average questions
Master the weighted average versus the plain mean
Learn alligation as a one-line cross for two-ingredient blends

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Convert every average to a sum before manipulating the data.
To add or remove a value, adjust the sum, then divide by the new count.
Use the weighted average when group sizes differ — never a plain mean.
Read a mixture's components as fractions of the whole.
For two-ingredient blends, apply alligation: (d − m) : (m − c).
Sanity-check: the mean sits nearer the larger quantity.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Costing and pricing

Blending goods of different costs to hit a target price is everyday commercial arithmetic that alligation solves instantly.

Interpreting aggregate figures

Understanding weighted averages guards against being misled by a simple mean of unequal groups.

Evidence and statistics

Averages of test results, incomes or damages appear constantly in legal and policy contexts.

Where else this topic is tested

Prepare once, score in every exam that asks it.

AILET (NLU Delhi)Averages & mixtures in quant

SLAT (Symbiosis)Arithmetic — averages

MH CET LawNumerical ability — averages/mixtures

LSAT—IndiaNo quant section

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

Work through the sum. Convert the average to a sum (sum = average × count), then add or subtract the value to get the new sum, and divide by the new count. For example, removing 30 from five numbers averaging 20 leaves a sum of 70 over 4 numbers, an average of 17.5. Never try to shift the average directly.

Whenever the groups being combined have different sizes or weights. Mixing 3 kg at ₹40 with 2 kg at ₹50 is not the simple mean of ₹45 but the weighted ₹44, because the cheaper rice contributes more weight. If the counts differ, weight each value by its count.

To blend two ingredients into a mean price m, the cheaper and dearer quantities are in the ratio (d − m) : (m − c) — each ingredient's share is the distance of the other price from the mean. For ₹30 and ₹45 rice at a ₹40 mean, that is 5 : 10 = 1 : 2. The mean always lies nearer the ingredient present in greater quantity, which is a handy check.

When two overlapping groups both include the same item — like the 6th of 11 results appearing in both the first six and the last six — adding their sums counts that item twice. Subtracting the grand total leaves exactly that double-counted value, so the 6th result = (first six + last six) − total of all eleven.

No. The same rule applies to any two-part blend measured on a common scale — prices, concentrations, percentages of purity, even average marks of two groups. As long as you are combining two quantities into a weighted mean, alligation gives the mixing ratio directly.