Lesson 1.4: Average, Mixture, Alligation

Introduction:
This lesson covers averages, mixture problems, and the alligation method, which are frequently tested in SSC exams. Mastering these concepts helps solve arithmetic and reasoning problems quickly.

1. Average

• Definition: Average (or mean) is the sum of observations divided by the number of observations.

Average=Sum of observationsNumber of observations\text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}}Average=Number of observationsSum of observations​

• Example 1:
  Marks of 5 students = 60, 70, 80, 90, 100
  Sum = 400 → Average = 400 / 5 = 80

• Shortcut Tips:

  • If the average of n numbers = A, then total sum = n × A

  • If one number is replaced, new average = old average + (difference / n)

2. Mixture Problems

• Definition: Problems where two or more ingredients with different costs or concentrations are mixed.

• Key Formula:

New Value=Sum of (Quantity × Rate)Total Quantity\text{New Value} = \frac{\text{Sum of (Quantity × Rate)}}{\text{Total Quantity}}New Value=Total QuantitySum of (Quantity × Rate)​

• Example 1:
  10 kg of sugar at ₹40/kg mixed with 20 kg of sugar at ₹50/kg. Find average price per kg.

Average Price=(10×40)+(20×50)10+20=400+100030=₹46.67\text{Average Price} = \frac{(10×40) + (20×50)}{10+20} = \frac{400+1000}{30} = ₹46.67Average Price=10+20(10×40)+(20×50)​=30400+1000​=₹46.67

3. Alligation Method

• Definition: A shortcut to calculate the ratio in which two or more quantities at different rates are mixed to get a desired mean rate.

• Rule:

Quantity 1 : Quantity 2=Mean – Rate of Quantity 2Rate of Quantity 1 – Mean\text{Quantity 1 : Quantity 2} = \frac{\text{Mean – Rate of Quantity 2}}{\text{Rate of Quantity 1 – Mean}}Quantity 1 : Quantity 2=Rate of Quantity 1 – MeanMean – Rate of Quantity 2​

• Example 1:
  Mixing milk at ₹40/litre with milk at ₹60/litre to get milk at ₹50/litre.

$$\text{Ratio}=(50-40):(60-50)=10:10=1:1$$

4. SSC Exam Tips

• For averages, quickly find sum using the formula: Sum = Average × n

• In mixture problems, always calculate total cost first.

• For alligation, use the difference method to find the ratio instantly.

• Practice word problems carefully; these appear in CGL Tier 1 and CHSL exams.

5. Practice Questions

1. Find the average of 15, 20, 25, 30, 35.

2. 30 litres of milk at ₹40/litre is mixed with 50 litres of milk at ₹50/litre. Find the cost per litre of mixture.

3. Two liquids are mixed: 60 litres at ₹12/litre and 40 litres at ₹20/litre. Find the ratio of quantities to get ₹16/litre.

4. The average weight of 5 students is 60 kg. If one student weighing 70 kg is replaced by another weighing 50 kg, find the new average.

5. Milk at ₹30/litre is mixed with milk at ₹50/litre to get 80 litres at ₹40/litre. Find quantities of each milk.