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Mixture & Alligation

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Quick recall

  • Use the lower-value quantity first in the alligation ratio.
  • Replacement assumes the mixture is thoroughly mixed and the total volume is restored each time.

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Formula Cards

1. Mixture value

M=q1V1+q2V2q1+q2M=\frac{q_1V_1+q_2V_2}{q_1+q_2}

q = quantity; V = component price or concentration.

2. Alligation

q1:q2=(V2−M):(M−V1)q_1:q_2=(V_2-M):(M-V_1)

Requires V₁ < M < V₂. Ratio is lower-value : higher-value quantity.

3. Repeated replacement

Original substance left=P(1−xP)n\text{Original substance left}=P\left(1-\frac{x}{P}\right)^n

Initially P units of pure substance; withdraw x units and restore total volume after each of n well-mixed replacements.

Worked Examples

Example 1

Question: Mix sugar costing Rs. 100/kg and Rs. 160/kg to obtain Rs. 130/kg. Find the ratio.

Step 1: Subtract diagonally from the target price.

160−130=30,130−100=30160-130=30,\qquad130-100=30

Step 2: Form lower-price : higher-price quantity ratio.

q1:q2=30:30=1:1q_1:q_2=30:30=1:1

Answer: 1:1.

Example 2

Question: A container holds 40 L of milk. Replace 4 L with water three times. Find milk remaining.

Step 1: Find the retained fraction after each replacement.

1−440=0.91-\frac{4}{40}=0.9

Step 2: Apply it three times.

40(0.9)3=29.1640(0.9)^3=29.16

Answer: 29.16 L of milk.

Example 3

Question: Mix 20 L of 20% acid solution with 30 L of 40% acid solution. Find the final concentration.

Step 1: Find the acid quantity.

20(0.20)+30(0.40)=16 L20(0.20)+30(0.40)=16\text{ L}

Step 2: Divide by the total mixture volume.

1620+30×100=32%\frac{16}{20+30}\times100=32\%

Answer: 32% acid.

Drill

Try each question before opening its answer.

Question 1

Mix tea at Rs.60/kg and Rs.90/kg to get Rs.75/kg. Find ratio.

  • Show answer

    1:1

Question 2

A 60L mixture of milk & water has ratio 2:1. How much water is added to make ratio 1:2?

  • Show answer

    60L

Question 3

10L of a 25% acid solution mixed with 15L of 40% acid solution

  • Show answer

    find final %. — 34%

Question 4

A vessel has 50L milk. 5L replaced by water twice. Milk left?

  • Show answer

    40.5L

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