Averages

The average (or mean) is the ratio of the sum of all observations to the total number of observations. Mathematically:

$$\text{Average} = \frac{\text{Sum of Observations}}{\text{Number of Observations}}$$

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Key Concepts

1. Basic Formula for Average

$$\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}}$$

Example:
Find the average weight of three boys whose weights are 46 kg, 54 kg, and 53 kg.

$$\text{Average} = \frac{46 + 54 + 53}{3} = \frac{153}{3} = 51 \, \text{kg}$$

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2. Properties of Averages

1. If each value in a dataset is increased by a constant $a$, the average increases by $a$.
2. If each value in a dataset is decreased by a constant $a$, the average decreases by $a$.
3. If each value in a dataset is multiplied by a constant $a$, the average is also multiplied by $a$.
4. If each value in a dataset is divided by a constant $a$, the average is also divided by $a$.

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3. Weighted Average

When different groups have different sizes or weights, the weighted average is used:

$$\text{Weighted Average} = \frac{\sum (\text{Weight} \times \text{Value})}{\sum (\text{Weights})}$$

Example:
A school trip consists of 160 girls, 40 boys, and 100 teachers. Find the average number of people per group:

$$\text{Average} = \frac{160 + 40 + 100}{3} = \frac{300}{3} = 100$$

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4. Average Speed

The formula for average speed depends on the conditions:

• Case 1: When traveling at two different speeds for equal time:

$$\text{Average Speed} = \frac{\text{Speed}_1 + \text{Speed}_2}{2}$$

• Case 2: When traveling at two different speeds for equal distances:

$$\text{Average Speed} = \frac{2 \times (\text{Speed}_1 \times \text{Speed}_2)}{\text{Speed}_1 + \text{Speed}_2}$$

• Case 3: When traveling at three different speeds for equal distances:

$$\text{Average Speed} = \frac{3}{\frac{1}{\text{Speed}_1} + \frac{1}{\text{Speed}_2} + \frac{1}{\text{Speed}_3}}$$

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Formulas Related to Numbers

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Rules and Tricks

• If the sum or total changes due to an addition or deletion, adjust the average accordingly:

  • New Average: $$\text{New Average} = \text{Old Average} ± (\Delta / n)$$ where $Δ =$ change in sum and $n =$ new total count.

• Combining two groups with different averages:

  • Combined Average:
  $$\text{Combined Average} = \frac{{(n_1 \times x_1) + (n_2 \times x_2)}}{{n_1 + n_2}}$$

  where $n_1, n_2 =$ group sizes and $x_1, x_2 =$ averages.

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Solved Examples

Question: The average of 10 numbers is 23. If each number increases by 4, what will the new average be?
Solution:
New sum = Old sum + Total increase
Old sum = $23 \times 10 = 230,$ Total increase = $4 \times 10 = 40,$ New sum = $230 + 40 = 270.$
New average:

$$270 \div 10 = 27$$

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Question: The average weight of seven boys is 56 kg. The weights of six boys are: 52, 57, 55, 60, 59, and 55 kg. Find the weight of the seventh boy.
Solution:
Total weight of seven boys:
$56 \times 7 = 392,$
Total weight of six boys:
$52 + 57 + 55 + 60 + 59 + 55 = 338.$
Weight of seventh boy:
$392 − 338 = 54\,kg.$

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Question: The mean of a group of students’ marks is calculated as follows: First group (13 students): mean is $32.$ Second group (12 students): mean is $39.$ Find the combined mean for all students.
Solution:
Total marks for Group A:
$32 \times 13 =416.$
Total marks for Group B:
$39 \times 12=468.$
Combined mean:

$$Combined Mean= Total Marks÷Total Count=900÷25=36.$$