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Mixtures And Alligations

    • Theory
    • Set 1
    • Set 2
    • Set 3
    • Set 4
    • Set 1
    • Set 2
  1. Aptitude
  3. Quantitative
  5. Mixtures and Alligations
  7. Theory

Mixtures and Alligations: Concepts, Formulas, and Clear Examples

Alligation finds mixing ratios from two prices/strengths to hit a mean. The cross rule is faster than forming equations for weighted average mixtures.

Replacement problems (jar partly swapped each time) use (1 − f)^n on pure quantity after nnn equal replacements of fraction fff.

Always clarify whether price, concentration, or milk:water ratio is being mixed.

Weighted mean setup

Mean price

pˉ=n1p1+n2p2n1+n2\bar{p} = \frac{n_1 p_1 + n_2 p_2}{n_1+n_2}pˉ​=n1​+n2​n1​p1​+n2​p2​​

Alligation cross rule

Cheaper ccc, dearer ddd, mean mmm. Ratio cheaper : dearer = (d−m):(m−c)(d-m):(m-c)(d−m):(m−c).

Cross rule

ncnd=d−mm−c\frac{n_c}{n_d} = \frac{d-m}{m-c}nd​nc​​=m−cd−m​

Example — Tea blending

Mix ₹40/kg and ₹60/kg to mean ₹52/kg. Ratio = (60−52):(52−40)=8:12=2:3(60-52):(52-40)=8:12=2:3(60−52):(52−40)=8:12=2:3.

Pure vs diluted

  • Milk + water: treat water as 0% milk or ₹0 component in alligation.

Replacement formula

n replacements of fraction f

Pure left=P(1−f)n\text{Pure left} = P\left(1 - f\right)^nPure left=P(1−f)n

Successive mixing

  • Two-step mix: compute intermediate mean, then alligate again.

Alcohol/water classic

  • Initial pure quantity × concentration after mix = sum of contributions.
  • Check total volume after mix.

Step-by-step method for exam questions

  1. Identify two components and target mean.
  2. Apply cross rule or weighted mean equation.
  3. For replacement, use (1−f)n(1-f)^n(1−f)n on pure part.
  4. Convert ratio to absolute liters/kg if asked.
  5. Verify mean lies between component values.
  6. Pick option.

Formula cheat sheet

SituationFormulaWhen to use
Two-price mixncnd=d−mm−c\frac{n_c}{n_d}=\frac{d-m}{m-c}nd​nc​​=m−cd−m​Alligation
Weighted avgpˉ=n1p1+n2p2n1+n2\bar{p}=\frac{n_1p_1+n_2p_2}{n_1+n_2}pˉ​=n1​+n2​n1​p1​+n2​p2​​Direct mix
Replace fraction fP(1−f)nP(1-f)^nP(1−f)nRepeated draw-off
Mean between?c<m<dc<m<dc<m<d requiredFeasibility check
Water as componentPrice or strength 0Milk-water

Frequently asked questions

Mean outside cheaper-dearer range?

Impossible mix—recheck numbers.

Ratio order?

Cheaper : dearer = (dearer−mean):(mean−cheaper).

Partial replacement vs add?

Replacement removes same amount before adding; read stem.

Three ingredients?

Use successive alligation or system of two equations.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.

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