Alligation Rule: The Cross Method

Q: What is the alligation rule in simple words?

When you mix a cheaper and a dearer ingredient to reach a target value, cheaper : dearer = (dearer − target) : (target − cheaper). Each ingredient's share is the other one's distance from the target, so you used more of whichever sits closer. It is a shortcut for the weighted-average equation.

Q: What is the difference between alligation and a weighted average?

Same idea, opposite directions. A weighted average starts with the quantities and finds the mean. Alligation starts with the mean and finds the ratio. Given the mix value and asked how much of each, cross; given the amounts and asked for the mix value, use the weighted average.

Q: Can I use alligation when the mixture is sold at a profit?

Yes, but convert first. Ingredient prices are costs, so the mean must be a cost too. Divide the selling price by (1 + profit%/100) to get the cost of the mix, then cross. Plugging in the selling price gives a wrong ratio that is usually sitting in the options.

Q: How do I use alligation for average speed questions?

Speed is distance per hour, so the cross gives the ratio of times, not distances. Find the overall average speed, cross it with the two speeds, split the total time in that ratio, then multiply each time by its speed. Splitting the distance directly is the most common mistake.

📑 On this page

⚡ The short answer

Alligation gives the ratio in which to mix two things to hit a target value: Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper). Each side's share is the other side's distance from the mean. It works for prices, concentrations, marks and speeds, if the mean lies between the two and is a cost, not a selling price.

🎬 The 30-second story

The Sunset Cooler crisis

“Mango juice ₹120 a litre, lemonade ₹60. Half-half, done!”

🥭

Aarav

Aarav and Zoya run a mocktail stall at the fest.

“We told the sponsor it costs ₹100 a litre. Half-half is ₹90.”

🧾

Zoya

“So 60% mango? 70%? I'll pour till it tastes expensive.”

🤯

Aarav

30 litres of stock, zero plan.

“Lemonade is 40 away from 100, mango is 20 away. Swap them: 20 : 40.”

✖️

Zoya

Zoya draws a cross on a napkin.

Tara's maths: Lemonade : mango = (120 − 100) : (100 − 60) = 20 : 40 = 1 : 2. So 30 L = 10 L lemonade + 20 L mango. Check: (10 × 60 + 20 × 120) ÷ 30 = ₹100.

What is alligation, really?

Forget the word for a second. You have a cheap ingredient and a dear one, and you want a mix that lands on a value in between. Alligation is just the question: how much of each?

Think of a seesaw. ₹60 lemonade on one end, ₹120 mango on the other. The target, ₹100, is where the pivot must sit. It sits nearer the mango, so the mango side must be heavier: more mango, less lemonade.

🎚️ The scale

Where the target sits tells you who dominates

← Cheap endDear end →

₹60 lemonade

40 away from the target

₹90 the 50:50 guess

Exactly midway: equal parts

₹100 target

Past the midpoint: mango wins

₹120 mango

Only 20 away from the target

📌 The seesaw rule

The closer an ingredient is to the mean, the more of it you used. Mango is half as far from ₹100 as lemonade, so you need twice as much mango. That inverse swap is the whole of alligation.

What is the alligation rule, and why does the cross work?

Write the cheaper value top-left, the dearer top-right, the mean in the middle. Subtract along each diagonal. The lines cross because each ingredient gets the other one's distance.

✖️ The alligation cross

Sunset Cooler: prices per litre

Lemonade : Mango = 1 : 2

Lemonade gets 120 − 100 = 20 parts; mango gets 100 − 60 = 40 parts. 20 : 40 = 1 : 2.

Why does it work? Use x litres of lemonade and y of mango. Each litre of lemonade is ₹40 below the target; each litre of mango is ₹20 above it. To land on ₹100, the shortfall must cancel the excess: 40x = 20y, so x : y = 20 : 40. In general, x × (Mean − Cheaper) = y × (Dearer − Mean). The cross is that one line of algebra, drawn as a picture.

How do you use the cross method step by step?

✍️ Worked example Warm-up ⏱ 30 sec

Two rices, one blend

Kabir mixes ₹42/kg rice with ₹54/kg rice to make 30 kg costing ₹50/kg. How many kg of each?

1
Cheaper = 42, dearer = 54, mean = 50.
2
Cheaper's share = 54 − 50 = 4; dearer's share = 50 − 42 = 8. Ratio = 1 : 2.
3
30 kg ÷ 3 parts = 10 kg per part → 10 kg and 20 kg.
4
Check: (10 × 42 + 20 × 54) ÷ 30 = 1,500 ÷ 30 = ₹50.

Answer: 10 kg of the ₹42 rice and 20 kg of the ₹54 rice.

⚡ The exam-hall shortcut

₹50 is past the ₹48 midpoint, so the dearer rice must be the bigger share. Kill any option with more cheap rice.

The bigger unlock: a concentration is also just a value, and water is an ingredient at 0%. Dilution becomes alligation.

✖️ The alligation cross

Diluting sanitiser: alcohol concentration

Water : Sanitiser = 1 : 4

Water gets 75 − 60 = 15 parts; sanitiser gets 60 − 0 = 60 parts. 15 : 60 = 1 : 4.

✍️ Worked example CLAT level ⏱ 40 sec

Watering down the hostel sanitiser

Meher has 24 litres of 75%-alcohol sanitiser. How much water brings it down to 60%?

1
Water is a 0% ingredient. Cheaper = 0, dearer = 75, mean = 60.
2
Water : sanitiser = (75 − 60) : (60 − 0) = 15 : 60 = 1 : 4.
3
Sanitiser is 4 parts = 24 L, so 1 part = 6 L of water.
4
Check: alcohol = 0.75 × 24 = 18 L; 18 ÷ 30 = 60%.

Answer: 6 litres of water.

⚡ The other 10-second route

Fix what never changes: 18 L of alcohol. At 60% the total must be 18 ÷ 0.6 = 30 L, so add 6 L.

✍️ Worked example Tricky ⏱ 60 sec

Selling price is not cost price

Kabir mixes dal at ₹90/kg and ₹120/kg, and sells the mix at ₹126/kg to make a 20% profit. In what ratio did he mix them?

1
The cross needs the cost of the mix, not its selling price.
2
Cost of mix = 126 ÷ 1.2 = ₹105/kg.
3
Ratio = (120 − 105) : (105 − 90) = 15 : 15 = 1 : 1. (₹105 is the exact midpoint.)

Answer: 1 : 1

⚡ The exam-hall shortcut

Plug ₹126 straight in and it sits outside 90–120. A mean outside the range is your alarm bell that a profit is hiding in the question.

⚠️ The selling-price trap

'Sells the mixture at ₹X and gains Y%' is the classic setup. The ingredients are costs, so the mean must be a cost: CP = SP ÷ (1 + profit%/100). Skip this and the wrong ratio is waiting in the options.

Can alligation solve problems that aren't about mixing?

Yes, and this is where CLAT gets sneaky. The cross works whenever two groups with their own averages combine into one. Only one thing changes: what the ratio counts.

🧩 At a glance

Same cross, different costumes

🛒Price per kg

Ratio counts kg.

🧪Concentration %

Ratio counts litres. Water is 0%, pure stuff 100%.

📝Average marks

Ratio counts students. Sections at 62 and 80, class at 68 → 2 : 1.

💹Profit / loss %

Ratio counts cost. 5% loss and 15% profit, 9% overall → 3 : 7.

🛵Speed in km/h

Ratio counts hours, not km. 40 and 60 km/h averaging 46 → 7 : 3 in time.

❓The test

Whatever sits in the value's denominator (per kg, per student, per hour) is what the ratio counts.

The speed tile bites. If a question asks for the distance at 40 km/h, find the time first, then multiply. Splitting the distance 7 : 3 directly is the most common wrong answer in this topic.

🎛️ Try it yourself — change any number

The Sunset Cooler mixer

Lemonade cost (₹/L) Mango juice cost (₹/L) Target cost of the mix (₹/L)

Cheaper : Dearer

1 : 2

(120 − 100) : (100 − 60) = 1 : 2

It opens on the story: 1 : 2. Set the target to 90: 1 : 1. Try 115, then 65, and watch the nearer ingredient take over. Now type 130. No mix of ₹60 and ₹120 can cost ₹130.

What is the fastest way to do alligation in the exam?

On the exam screen you rarely need the full cross. Find the midpoint, see which way the mean leans, and kill options.

✓Midpoint check. Mean exactly midway? 1 : 1, no arithmetic.
✓Lean check. Mean above the midpoint → more of the dearer, so cheaper : dearer is below 1. This alone often kills two options.
✓Reverse alligation. Mixed x : y (cheaper : dearer)? Mean = Cheaper + (Dearer − Cheaper) × y ÷ (x + y). ₹40 and ₹80 tea in 3 : 1 → 40 + 40 × 1/4 = ₹50.
✓Back-substitute. Options give quantities? Test one against the total.

How does alligation hide in a CLAT caselet?

CLAT rarely says 'mix two teas'. It gives two prices, a total count and a total amount, and asks how many of each. Total amount ÷ total count is a hidden average price. That's your mean.

✍️ Worked example CLAT level ⏱ 50 sec

The cloud-kitchen Saturday

Kabir's cloud kitchen sells two biryani boxes: Regular at ₹180 and Family at ₹300. On Saturday it sold 120 boxes and collected ₹27,600. How many Family boxes were sold?

1
Hidden mean: 27,600 ÷ 120 = ₹230 per box.
2
Regular : Family = (300 − 230) : (230 − 180) = 70 : 50 = 7 : 5.
3
120 ÷ 12 parts = 10 per part → 70 Regular, 50 Family.
4
Check: 70 × 180 + 50 × 300 = 12,600 + 15,000 = 27,600.

Answer: 50 Family boxes (and 70 Regular).

⚡ The exam-hall shortcut

₹230 is below the ₹240 midpoint, so Regular must outsell Family.

✖️ The alligation cross

Biryani boxes: the hidden average price

Regular : Family = 7 : 5

Regular gets 300 − 230 = 70 parts; Family gets 230 − 180 = 50 parts. 70 : 50 = 7 : 5.

📊 The data

Saturday's sales, rebuilt from two totals

Regular (70 boxes) · 70 × ₹180

₹12,600

Family (50 boxes) · 50 × ₹300

₹15,000

Fewer Family boxes, more revenue: 15,000 ÷ 27,600 ≈ 54% of the takings.

Which alligation traps cost the most marks?

Trap	What it looks like	The fix
Selling price as the mean	'Sells at ₹126, gains 20%'	Convert to cost first
Flipped ratio	2 : 1 when asked cheaper : dearer	Cheaper gets the dearer's distance
Mean outside the range	₹60 and ₹120 to make ₹130	Impossible: recheck the question
Wrong weights	Speeds split by distance	Speeds mix by time
Ratio, no total	Only the average is given	Quantities: cannot be determined

→ scroll to see more

The last row matters. The cross only ever gives a ratio. To get litres or boxes, the passage must give one quantity. If it gives none, 'cannot be determined' is the right answer, not a cop-out.

How has CLAT asked alligation?

📝 Asked in CLAT 2019 · Q55 Price alligation

A stand-alone question in the 2019 paper.

A grocer mixes coffee powder of 2 types, one of which is priced at Rs. 60 and the other at Rs. 90. What should be the ratio of combining the two, to sell the blended mix coffee powder of the two types at Rs. 80?

A2:1

B2:3

C1:2

D3:2

Reveal answer + reasoning

Answer: C

Our solution (2019 has no official key). No profit is mentioned, so ₹80 is the cost of the blend. Lean check: the midpoint is ₹75 and ₹80 is above it, so more of the ₹90 coffee; only 1 : 2 and 2 : 3 survive. Cross: (90 − 80) : (80 − 60) = 10 : 20 = 1 : 2. Time: 15 seconds.

Why the others fail

A — The right numbers, flipped: 2 : 1 is dearer : cheaper.
B — That's 60 : 90, the ratio of the prices. Prices are not quantities.
D — 90 : 60, the price ratio flipped.

📝 Asked in CLAT 2018 · Q142 Reverse alligation

A stand-alone question in the 2018 paper.

In a 120 litre of solution of Acid and water, acid is 75%. A person takes out 20 litres of this solution and added 16.2 litres of acid and 3.8 litres of water in the remaining solution. What is the percentage of water in the final solution?

A22

B24

C25

D28

Reveal answer + reasoning

Answer: B

Track water. After removing 20 L, 100 L remain with 25 L water (25%). The 20 L added holds 3.8 L water (19%). So you mix 100 L at 25% with 20 L at 19%, a 5 : 1 mix. Reverse alligation: the mean sits 1/6 of the way from 25 to 19: 25 − 6 × 1/6 = 24%. Check: 28.8 ÷ 120 = 24%. Time: 40 seconds.

Why the others fail

A — 22 is the simple average of 25% and 19%, as if the parts were equal. They aren't.
C — Assumes the top-up kept the old strength. It was only 19% water.
D — 28.8 L of water read as a percentage of 100 L. The final volume is 120 L.

💥 Myth vs fact

Myths that cost marks

🙅 Myth

Alligation is only for prices of rice and tea.

Tap to bust it →

✅ Fact

Any two averages that combine: concentrations, marks, profit %, speeds.

🙅 Myth

The cheaper ingredient gets the cheaper distance.

Tap to bust it →

✅ Fact

Each ingredient gets the other one's distance from the mean. That's why the lines cross.

🙅 Myth

Any target price can be reached by mixing.

Tap to bust it →

✅ Fact

The mean must sit strictly between the two values.

🎯 Quick quiz

Five quick crosses

0 / 5

Rice at ₹36/kg is mixed with rice at ₹48/kg to cost ₹44/kg.

Q1. Ratio (cheaper : dearer)?

Zoya has 30 litres of 80%-fruit concentrate and wants it at 60%.

Q2. How much water must she add?

A class of 40 averages 76. Boys average 70, girls 85.

Q3. How many girls?

Tea at ₹200/kg and ₹280/kg is blended and sold at ₹270/kg for a 20% profit.

Q4. Ratio (cheaper : dearer)?

Aarav rides 5 hours, partly at 40 km/h and partly at 60 km/h, covering 230 km.

Q5. How far did he ride at 40 km/h?

🎯 Alligation in a nutshell

Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper).
It's the weighted average solved for the ratio.
Water is 0%; the pure stuff is 100%.
Selling price and profit given? Convert to cost first.
The ratio counts the value's denominator: kg, litres, students, hours.
Ratio but no quantity: cannot be determined.

📎 Formula card

Alligation: Cheaper : Dearer = (d − m) : (m − c), with c < m < d.
Reverse: mixed x : y, mean = (x × c + y × d) ÷ (x + y).
Midpoint: m = (c + d)/2 means 1 : 1.
Profit hidden: cost of mix = SP ÷ (1 + p/100).
Dilution: water : solution = (strength − target) : target.
Weights: ₹/kg → kg · % → litres · marks → students · km/h → hours.

Frequently asked questions

What is the alligation rule in simple words?

When you mix a cheaper and a dearer ingredient to reach a target value, cheaper : dearer = (dearer − target) : (target − cheaper). Each ingredient's share is the other one's distance from the target, so you used more of whichever sits closer. It is a shortcut for the weighted-average equation.

What is the difference between alligation and a weighted average?

Same idea, opposite directions. A weighted average starts with the quantities and finds the mean. Alligation starts with the mean and finds the ratio. Given the mix value and asked how much of each, cross; given the amounts and asked for the mix value, use the weighted average.

Can I use alligation when the mixture is sold at a profit?

Yes, but convert first. Ingredient prices are costs, so the mean must be a cost too. Divide the selling price by (1 + profit%/100) to get the cost of the mix, then cross. Plugging in the selling price gives a wrong ratio that is usually sitting in the options.

How do I use alligation for average speed questions?

Speed is distance per hour, so the cross gives the ratio of times, not distances. Find the overall average speed, cross it with the two speeds, split the total time in that ratio, then multiply each time by its speed. Splitting the distance directly is the most common mistake.