Section 10, arithmetic

Mixtures & Alligation

Mixture questions reduce to two tools: alligation for mixing two things, and the replacement formula for “remove and refill” cycles. Alligation also speeds up averages, profit and interest questions, so it pays twice.

How CAT asks it

Two solutions mixed in a ratio; repeated draw-and-replace; water added or evaporated to reach a ratio; multi-step mixing (A with B, then more A); alloys of gold and copper; and alligation disguised as speeds or profits.

Concepts you must know

Alligation: mixing quantity q₁ of value c₁ with q₂ of value c₂ gives mean m when

\frac{q_1}{q_2} = \frac{c_2 - m}{m - c_1}

In words: each part’s weight is the distance of the other part’s value from the mean. The mean always lies between c₁ and c₂.
The ratio is in the unit the value is measured per: litres for concentration per litre, kg for price per kg, hours for speed in km per hour.
Replacement: from V litres of pure liquid, remove x litres and refill with water, n times. The pure liquid left is

V\left(1 - \frac{x}{V}\right)^{n}

Adding or evaporating water changes the total but not the solute. Track the component that does not change.
With three or more components, track each component’s quantity separately.

Thought process

Decide what is being averaged (concentration? price per kg?) and per what unit.
Identify the component that stays constant.
Two-part mix → alligation. Multi-step mix → track each component in a small table.
Sanity check: the mix’s value must lie between the two inputs.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

1Two solutions mixed: find the result’s concentration, or the ratio that hits a target.
2Repeated replacement: the amount left, or how many cycles before it falls below a level.
3Add water or evaporate to reach a target ratio.
4Multi-step mixing: A with B, then the result with more A.
5Alloys: ratios of metals; equal masses of two alloys melted together.
6Selling a mixture at a profit: find the cost of the mix first, then alligate.
7Alligation outside mixtures: average speeds (weights are times), profit percentages, interest rates, group averages.

Worked examples

Attempt each one before opening its solution, then compare methods.

Example 1, basic alligation
Rice at ₹40/kg and ₹60/kg is mixed so that the blend costs ₹52/kg. In what ratio?
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(60 − 52) : (52 − 40) = 8 : 12. Answer: 2 : 3 (cheaper : dearer).
Example 2, mix + profit
Tea at ₹300/kg and ₹400/kg is blended and sold at ₹420/kg for a 20% profit. In what ratio was it blended?
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Cost of the blend = 420 ÷ 1.2 = 350. (400 − 350) : (350 − 300) = 50 : 50. Answer: 1 : 1.
Example 3, adding water
30 L of a mixture has milk : water = 7 : 3. How much water must be added to make it 3 : 2?
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Milk stays at 21 L. For 3 : 2, water must be 21 × 2/3 = 14 L; there are 9 L now. Answer: 5 L.
Example 4, replacement
A 50 L container is full of juice. 10 L are removed and replaced with water; this is done 3 times in all. How much juice remains?
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50 × (4/5)³ = 50 × 64/125. Answer: 25.6 L.
Example 5, threshold
A 30 L jar of pure milk has 5 L removed and replaced with water each time. After how many operations is milk first less than half?
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Each cycle keeps 5/6 of the milk. (5/6)³ = 125/216 ≈ 0.58; (5/6)⁴ = 625/1296 ≈ 0.48. Answer: 4.
Example 6, two-step mixing
Solution A is 30% acid and B’s strength is unknown. A and B are mixed 1 : 2, then a volume of A equal to that mixture is added. The final mixture is 40% acid. Find B’s strength.
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Units: A 1 and B 2, then 3 more of A, so A 4 and B 2 out of 6. Acid: 4 × 0.3 + 2b = 6 × 0.4, so 1.2 + 2b = 2.4 and b = 0.6. Answer: 60%.
Example 7, alloys
Alloy P has gold : copper = 3 : 2 and alloy Q has 1 : 3. Equal masses are melted together. Find the new ratio.
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Take 20 kg of each (the LCM of 5 and 4). P gives 12 gold and 8 copper; Q gives 5 gold and 15 copper. Answer: 17 : 23.
Example 8, evaporation
40 kg of salt water is 4% salt. How much water must evaporate to make it 10% salt?
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Salt stays at 1.6 kg, so the new total is 1.6 ÷ 0.10 = 16 kg. Answer: 24 kg.
Example 9, alligation on speed
A man covers 300 km in 6 hours, partly by car at 60 km/h and partly by bus at 40 km/h. How far did he go by car?
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Average speed = 50 km/h. Speeds average by time, so car time : bus time = (50 − 40) : (60 − 50) = 1 : 1, or 3 hours each. Check: 180/60 + 120/40 = 6. Answer: 180 km.

Traps

Alligating speeds by distance. Average speed weights speeds by time.
Using the replacement formula when the amount removed changes between cycles, or the refill is not the pure “other” liquid.
Forgetting to recompute both components after adding more of a mixture.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

1
12 L of 25% alcohol is mixed with 8 L of 40% alcohol. Find the concentration.
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31%: (3 + 3.2) ÷ 20.
2
How much pure alcohol must be added to 30 L of a 20% solution to make it 40%?
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10 L: (6 + x)/(30 + x) = 0.4.
3
From an 80 L cask of wine, 8 L are drawn and replaced with water, twice. How much wine is left?
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64.8 L: 80 × 0.9².
4
Two vessels have milk : water = 3 : 1 and 5 : 3. In what ratio should they be mixed to get 2 : 1?
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1 : 2 (first : second): the milk fractions are 18/24 and 15/24 against a target of 16/24, so first : second = (16 − 15) : (18 − 16).
5
Pulses at ₹90/kg and ₹120/kg are mixed 2 : 1. At what price should the mix sell for a 25% profit?
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₹125/kg: the mix costs (180 + 120) ÷ 3 = 100.
6
Alloy A (copper : zinc = 3 : 2) and alloy B (4 : 1) are mixed 1 : 1 by mass. What percent is copper?
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70%: the average of 60% and 80%.

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