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AFCAT Numerical Ability · Formula sheet

Mixture and Alligation formulas

16 formulas and 23 common traps for AFCAT Numerical Ability Mixture and Alligation, grouped by subtopic.

Full notes with worked examples

Adding One Liquid to a Mixture

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Splitting a mixture, then adding to it

Milk in a mixture

milk=mm+w×T\text{milk} = \dfrac{m}{m + w} \times T

Adding water to reach a ratio: the milk is fixed

Water to add

add=sr×M−W\text{add} = \dfrac{s}{r} \times M - W

Adding milk to reach a ratio: the water is fixed

Milk to add

add=rs×W−M\text{add} = \dfrac{r}{s} \times W - M

Unknown amounts: when the unchanged liquid keeps its term

One part

k=litres addedextra partsk = \dfrac{\text{litres added}}{\text{extra parts}}

When the unchanged liquid's term differs: rescale first

One part after rescaling

k=litres addedextra parts after matching the fixed termk = \dfrac{\text{litres added}}{\text{extra parts after matching the fixed term}}

Common traps

Add litres to litres, not to ratio terms

A 40-litre mixture with milk and water 3:13 : 1 has 30 litres of milk and 10 of water. With 10 more litres of water it is 30:20=3:230 : 20 = 3 : 2. Adding the 10 to the ratio term gives 3:113 : 11, which is meaningless.

Do not split the old total in the new ratio

To turn 40 litres of a 3:13 : 1 milk and water mixture into 1:11 : 1, halving 40 gives 20 litres of water and an answer of 10. But that quietly cuts the milk to 20. The milk stays 30, so the water must reach 30: add 20 litres.

Answer with the litres added

The question asks how much water to add, not how much water there will be. Subtract the water already in the mixture before you choose an option.

Adding milk, scale from the water

When milk is poured in, the milk is the liquid that changes. Working from the old milk (as if it were fixed) answers a different question. Start from the water.

A reversed ratio is a fraction of the water, not of the new total

To reverse 3:53 : 5 by adding milk, the milk must become 53\tfrac 53 of the water. It is not 58\tfrac 58 of anything, because the new total is not yet known.

k is not the answer

Finding one part is half the work. With k=6k = 6 in a 2:32 : 3 mixture the milk is 12, the water 18 and the mixture 30 litres; the options usually hold all three. Read which one is asked.

Match the liquid that did not change

Milk and water 4:14 : 1 become 2:32 : 3 after water is added. The milk is fixed, so write 2:32 : 3 as 4:64 : 6: the water grew by 5 parts. Reading the water's 1 and 3 as 2 extra parts compares parts of different sizes.

Percentage Strength of a Mixture

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When the percentage is of the liquid that stays

New total

new total=fixed amountits new share\text{new total} = \dfrac{\text{fixed amount}}{\text{its new share}}

When the percentage is of the liquid you add

New total

new total=other liquid1−p\text{new total} = \dfrac{\text{other liquid}}{1 - p}

Common traps

Divide by the new share; do not subtract percentages

Taking 16 litres of 50% milk down to 40% does not need 10% of 16 litres of water. The 8 litres of milk must be 40% of the new mixture, which is 20 litres, so add 4 litres.

A percentage is of the whole mixture

A 4% impurity means 4 litres in every 100 litres of the mixture, impurity included. It is not 4 litres for every 100 litres of the pure liquid.

Using the target share for the liquid that changes

Raising 10 litres of 40% acid to 60% by adding acid: dividing the 4 litres of acid by 60% gives 6.67 litres, less than you started with. The acid is the liquid that changes. Divide the 6 litres of water by 40% instead: 15 litres, so add 5.

Mean Price and Mean Concentration

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Mean price of a mixture

Mean price

pˉ=ap+bq+cra+b+c\bar p = \dfrac{ap + bq + cr}{a + b + c}

Mean strength: add up the pure part

Mean strength

c=c1V1+c2V2V1+V2c = \dfrac{c_1 V_1 + c_2 V_2}{V_1 + V_2}

Common traps

Weight the prices; do not average them

₹45 and ₹75 a kg mixed 2:12 : 1 cost 90+753=55\dfrac{90 + 75}{3} = 55 rupees a kg. The plain average, 60, treats the two as used equally.

Divide by the sum of the ratio

Three kinds mixed 1:2:31 : 2 : 3 make 6 kg in one round, so divide the total cost by 6, not by 3, the number of kinds.

Do not add the ratio terms

Equal amounts of a 1:11 : 1 and a 1:41 : 4 mixture are not 2:52 : 5. Milk is 12\tfrac 12 of one and 15\tfrac 15 of the other, so the mixture is 720\tfrac{7}{20} milk: 7:137 : 13.

The Rule of Alligation

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The alligation cross

Rule of alligation

cheaperdearer=d−mm−c\dfrac{\text{cheaper}}{\text{dearer}} = \dfrac{d - m}{m - c}

From the ratio to a quantity, a price, or any average

Scaling the ratio

cheaper=d−mm−c×dearer\text{cheaper} = \dfrac{d - m}{m - c} \times \text{dearer}

Common traps

The cheaper kind gets the dearer kind's distance

Mixing ₹10 and ₹20 to make ₹12: the cheaper gets 20−12=820 - 12 = 8 and the dearer 12−10=212 - 10 = 2, so cheaper : dearer =4:1= 4 : 1. Pairing each price with its own distance gives 1:41 : 4, the ratio turned round.

The mean must lie between the two

A 40% and a 60% solution can never make a 70% one. If the target is outside the two values, a difference comes out negative: re-read the stem.

Scale from the right side

If cheaper : dearer is 3:23 : 2 and 12 kg of the dearer kind is used, one part is 6 kg and the cheaper kind is 18 kg. Multiplying 12 by 23\tfrac 23 attaches the known amount to the wrong term.

Profit and Loss on Mixtures

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Water sold at the price of milk

Cost of one litre of the mixture

CP=SP×100100+gain\text{CP} = \text{SP} \times \dfrac{100}{100 + \text{gain}}

Profit on a mixture, and mixed profit rates

Profit on a mixture

profit=SP−mean costmean cost×100%\text{profit} = \dfrac{\text{SP} - \text{mean cost}}{\text{mean cost}} \times 100\%

Common traps

Gain is measured on the milk, not on the mixture

1 litre of water in 6 litres of mixture is 1 litre of water to 5 of milk: a gain of 20%. Dividing by the whole 6 litres gives 16.7%.

The selling price is not the mean

With a profit in the question, the cross needs the cost of a litre of the mixture. Divide the selling price by 1+gain1 + \text{gain} before you cross it with the price of milk.

Weight the profit rates

A quarter of the stock sold at 20% and the rest at 12% gives 0.25×20+0.75×12=14%0.25 \times 20 + 0.75 \times 12 = 14\% overall, not the plain average 16%.

Removal and Replacement

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One draw and one top-up: track each liquid

What a draw leaves

left=(1−xT)×amount before\text{left} = \left(1 - \dfrac{x}{T}\right) \times \text{amount before}

Topping up with water: the milk keeps a fixed fraction

Fraction the milk keeps

share aftershare before=T−xT\dfrac{\text{share after}}{\text{share before}} = \dfrac{T - x}{T}

Repeated replacement

Milk after n rounds

milk left=V(1−xV)n\text{milk left} = V\left(1 - \dfrac{x}{V}\right)^n

Common traps

A draw takes some of every liquid

Removing 10 litres of a 4:14 : 1 milk and water mixture removes 8 litres of milk and 2 of water. Taking all 10 from the milk, or all from the water, changes the ratio by the wrong amount.

The percentage is multiplied, not reduced by a subtraction

Drawing a quarter of a 90% milk mixture and topping it up with water leaves 90%×34=67.5%90\% \times \tfrac 34 = 67.5\% milk. Subtracting 25 points gives 65%, which is wrong.

Count the rounds

'Done once and then repeated twice more' is three rounds. 'Repeated once more' after the first is two. Write the number of rounds before raising the fraction to a power.

Do not subtract the same amount each round

Two rounds of drawing 10 litres from 50 litres of milk leave 50×0.82=3250 \times 0.8^2 = 32 litres, not 50−20=3050 - 20 = 30. The second draw takes milk and water, so it removes less than 10 litres of milk.

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