Mixtures and Alligations: Practice Questions & Notes

Mixture and Alligation deals with the mixing of two or more entities (like liquids, grains, or prices) to find the ratio, concentration, or cost of the resulting mixture. It combines both algebraic logic and a powerful visual shortcut called Alligation Rule.

Intuitive Understanding

Mixture: When two or more items (like milk and water, acid and base, tea and sugar) are mixed, the result is a mixture. You’re usually given quantities, percentages, or ratios and are asked to compute the value or ratio of a component in the result.

Alligation: A shortcut technique to quickly find the ratio in which two ingredients must be mixed to achieve a desired mean concentration or cost.

Key Formulas and Concepts

1. Average Quantity or Concentration (Weighted Mean)

When mixing two types with quantities and percentages:

\text{Average} = \frac{(Q_1 \times C_1) + (Q_2 \times C_2)}{Q_1 + Q_2}

Where $Q$ = quantity, and $C$ = concentration or price.

2. Alligation Rule

Used to find the ratio of mixing two components when the mean value is known.

\text{Ratio} = (D - M) : (M - C)

Where:

$D$ = concentration or price of the dearer (higher) item
$C$ = concentration or price of the cheaper (lower) item
$M$ = mean value of the mixture

Visual Setup:

        C (cheaper)
          \
           \
            \       D - M
             > M <
            /       M - C
           /
        D (dearer)

3. Replacing Mixtures (Repeated Replacement Formula)

If a mixture has quantity $V$ , and each time $R$ is removed and replaced with a new substance, then after $n$ operations:

\text{Final Quantity of Original} = V \left(1 - \frac{R}{V}\right)^n

Conceptual Examples

Q1. A 60 L solution contains milk and water in 3:2. How much water should be added to make the ratio 1:1?

Milk = $\frac{3}{5} \times 60 = 36$ L
Water = $\frac{2}{5} \times 60 = 24$ L
Let $x$ be added to water:

\frac{36}{24 + x} = 1 \Rightarrow x = 12

Q2. Two kinds of tea worth ₹30/kg and ₹45/kg are mixed. If the resulting mixture is ₹36/kg, find the ratio.

Use alligation:

       30
        \       45 - 36 = 9
         \     /
          36
         /     \ 
        /       36 - 30 = 6
       45

So, ratio = 9 : 6 = 3:2

Q3. A container has 40 L of milk. 8 L is removed and replaced with water. This process is repeated 3 times. Find final quantity of milk.

Using formula:

40 \left(1 - \frac{8}{40}\right)^3 = 40 \left(\frac{4}{5}\right)^3 = 40 \times \frac{64}{125} = 20.48 \text{ L}

Tricks and Shortcuts

Alligation is fastest when:

You're mixing two items with known individual values and a target average.
Ratio of quantities is asked, not total value.

Weighted Mean is best when:

More than 2 components.
Total value is needed (not ratio).

Visual Summary

Mixture → Combining different concentrations
Alligation → Shortcut to find ratio of mixing

- Alligation Triangle:
        C
         \      D - M
          > M <
         /      M - C
        D

- Replacement formula:
    Final = V × (1 - R/V)^n

Common Mistakes and Conceptual Tips

Mistake	Why it happens	Fix
Applying Alligation to non-uniform units	Price per kg vs %	Use only same types (price with price, % with %)
Forgetting proportions when replacing	Misusing percentage logic	Use the exponential formula for repeated replacement
Misplacing values in triangle	Cheaper and dearer swapped	Always place higher value below, lower above
Mixing more than two items with alligation	Not possible	Use weighted average for 3+ items

Real-Life Applications

Mixing chemicals or solutions (medicine, labs)
Blending oils, fuels, or grains
Pricing strategies when combining products
Replacement problems in tanks and containers

Practice Examples

Q1. A solution has 40% acid. How much water must be added to 20 L of it to make the acid concentration 25%?

Let water added = $x$
Acid = 40% of 20 = 8 L
Total = 20 + x

\frac{8}{20 + x} = 0.25 \Rightarrow 8 = 0.25(20 + x) \Rightarrow x = 12

Q2. A vessel has 60 L of 70% alcohol. 20 L is removed and replaced with water. What's final alcohol percentage?

First:
Alcohol = 70% of 60 = 42 L
Removing 20 L removes $\frac{1}{3}$
→ Alcohol left = $\frac{2}{3} \times 42 = 28$
→ Water = 20 L + 18 L already = 32 L
→ Final = 28 / 60 = 46.67%

Q3. In what ratio should water (₹0/litre) and syrup (₹60/litre) be mixed to get a mixture worth ₹20/litre?

Alligation:

     0
      \       60 - 20 = 40
       > 20 <
      /       20 - 0 = 20
     60

→ Ratio = 40 : 20 = 2:1