📏 Averages

Day 1 | ⏱️ Part of 4-Hour Quant Block | TCS Interview Prep

📌 Subtopics

• Basic average calculation
• Weighted average
• Average of consecutive numbers
• Average speed problems
• Adding/removing elements from a group
• Change in average when a value is replaced

📖 Comprehensive Notes

1. Basic Average Calculation

An average (specifically, the arithmetic mean) represents the central value of a set of data. $\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$ $\text{Sum} = \text{Average} \times \text{Number of observations}$

Example: Find the average of $12, 15, 21, 18,$ and $24$. $\text{Sum} = 12 + 15 + 21 + 18 + 24 = 90$. $\text{Average} = \frac{90}{55} = 18$.

2. Weighted Average

When different groups have different sizes (or weights), their combined average is calculated as a weighted average rather than a simple average of averages. $\text{Weighted Average} = \frac{n_1 A_1 + n_2 A_2 + \dots + n_k A_k}{n_1 + n_2 + \dots + n_k}$ where $n_i$ represents the size/weight of group $i$ and $A_i$ represents the average of group $i$.

Example: The average weight of $20$ boys is $50\text{ kg}$ and $30$ girls is $45\text{ kg}$. Find the average weight of the entire class. $\text{Weighted Average} = \frac{(20 \times 50) + (30 \times 45)}{20 + 30} = \frac{1000 + 1350}{50} = \frac{2350}{50} = 47\text{ kg}$.

3. Average of Consecutive Numbers

For any arithmetic progression (AP), including consecutive natural numbers, even numbers, odd numbers, or multiples: $\text{Average} = \frac{\text{First Term} + \text{Last Term}}{2}$

• If the number of terms $n$ is odd, the average is the middle term.
• If the number of terms $n$ is even, the average is the mean of the two middle terms.

Example: Find the average of first $50$ natural numbers ($1$ to $50$). $\text{Average} = \frac{1 + 50}{2} = 25.5$.

4. Average Speed Problems

• Equal Distances: If a body covers a distance with speed $a$ and the same distance with speed $b$, the average speed for the whole journey is the harmonic mean of the speeds: $\text{Average Speed} = \frac{2ab}{a+b}$
• Unequal Distances: Use the base formula: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

Example: A car travels from city A to B at $40\text{ km/h}$ and returns at $60\text{ km/h}$. Find the average speed. $\text{Average Speed} = \frac{2 \times 40 \times 60}{40 + 60} = \frac{4800}{100} = 48\text{ km/h}$.

5. Adding/Removing Elements from a Group

• Adding a new member: $\text{New Value} = \text{Old Average} + (\text{Change in Average} \times \text{New Number of Members})$
• Removing a member: $\text{Removed Value} = \text{Old Average} - (\text{Change in Average} \times \text{New Number of Members})$

Example: The average weight of $15$ students is $40\text{ kg}$. If a teacher is added, the average increases by $1\text{ kg}$. Find the teacher’s weight. $\text{Teacher's Weight} = 40 + (1 \times 16) = 56\text{ kg}$.

6. Change in Average when a Value is Replaced

If one member is replaced by another: $\text{New Value} - \text{Old Value} = \text{Change in Average} \times \text{Total Members}$

• If the average increases, the new value is greater than the old value.
• If the average decreases, the new value is smaller than the old value.

Example: The average age of $10$ people is increased by $2$ years when one person aged $25$ is replaced by a new person. Find the age of the new person. $\text{New Age} - 25 = 2 \times 10 \implies \text{New Age} = 25 + 20 = 45$ years.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

Average = Sum of observations / Number of observations
Weighted Avg = Σ(weight × value) / Σ(weights)
Average Speed = 2ab / (a + b)  [for equal distances]

Tip: If a new member joins a group and the average increases by x, the new member’s value = Old Average + x × (New count)

🔗 Related Topics

• Statistics — Mean is a central measure in statistics
• Time Speed and Distance — Average speed uses the harmonic mean formula
• Percentages — Percentage change in averages
• Ages — Average age problems
• Data Interpretation — Calculating averages from data sets