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Average

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Introduction to Average

In everyday life, we often want to find a single value that represents a group of numbers. This value is called the average. The average gives us an idea of the "central" or "typical" value among a set of numbers. For example, if you want to know the average marks scored by students in a class or the average price of fruits bought, the concept of average helps us summarize the data into one meaningful number.

Understanding averages is crucial not only in daily life but also in competitive exams, where questions often test your ability to quickly and accurately calculate averages in various contexts.

The basic idea behind average is simple: it is the total of all values divided by how many values there are.

Definition and Formula of Average

The average of a set of numbers is defined as the sum of all the numbers divided by the total count of those numbers.

Mathematically, if you have numbers \(x_1, x_2, x_3, \ldots, x_n\), their average \(A\) is given by:

Average

\[A = \frac{\sum_{i=1}^n x_i}{n}\]

Sum of all observations divided by number of observations

A = Average
\(x_i\) = Each observation
n = Number of observations

This formula can be rearranged to find the sum if the average and number of terms are known:

Sum of Observations

\[\text{Sum} = A \times n\]

Multiply average by number of observations to get total sum

A = Average
n = Number of observations

Or to find the number of observations if the sum and average are known:

Number of Observations

\[n = \frac{\text{Sum}}{A}\]

Divide sum by average to find number of terms

n = Number of observations
Sum = Total sum of observations
A = Average
Sum Number of Terms Average Divide

This diagram shows how the sum and number of terms relate to the average: Average = Sum / Number of Terms.

Weighted Average

Sometimes, not all observations are equally important. For example, if you buy 3 kg of apples at one price and 2 kg at another price, the average price per kg is not just the simple average of the two prices. Instead, you must consider the quantity (weight) of each purchase. This is where weighted average comes in.

The weighted average gives different importance (weights) to different observations.

If \(x_1, x_2, \ldots, x_n\) are the values and \(w_1, w_2, \ldots, w_n\) are their respective weights, then the weighted average \(A_w\) is:

Weighted Average

\[A_w = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}\]

Sum of (weight x value) divided by sum of weights

\(w_i\) = Weight of ith observation
\(x_i\) = Value of ith observation
Example: Weighted Average Price of Apples
Quantity (kg) Price per kg (Rs.) Total Cost (Rs.)
3 120 360
2 150 300
5 (Total) 660 (Total)

The weighted average price per kg is:

Weighted Average Price

\[A_w = \frac{360 + 300}{3 + 2} = \frac{660}{5} = 132\]

Total cost divided by total quantity

660 = Total cost
5 = Total quantity (kg)

Worked Examples

Example 1: Simple Average of Marks Easy
Calculate the average marks of a student who scored 75, 80, 85, 90, and 95 in five subjects.

Step 1: Add all the marks:

75 + 80 + 85 + 90 + 95 = 425

Step 2: Count the number of subjects:

Number of subjects = 5

Step 3: Use the average formula:

\( \text{Average} = \frac{\text{Sum of marks}}{\text{Number of subjects}} = \frac{425}{5} = 85 \)

Answer: The average marks are 85.

Example 2: Finding Missing Value from Average Medium
A student's average score in 5 tests is 72. If the scores of four tests are 70, 75, 68, and 74, find the score of the fifth test.

Step 1: Calculate the total sum of all 5 tests using the average:

\( \text{Total sum} = \text{Average} \times \text{Number of tests} = 72 \times 5 = 360 \)

Step 2: Add the known scores:

70 + 75 + 68 + 74 = 287

Step 3: Find the missing score:

\( \text{Missing score} = \text{Total sum} - \text{Sum of known scores} = 360 - 287 = 73 \)

Answer: The score of the fifth test is 73.

Example 3: Weighted Average of Prices Medium
A person buys 3 kg of apples at Rs.120/kg and 2 kg of apples at Rs.150/kg. Find the average price per kg.

Step 1: Calculate total cost for each purchase:

3 kg x Rs.120 = Rs.360

2 kg x Rs.150 = Rs.300

Step 2: Calculate total quantity and total cost:

Total quantity = 3 + 2 = 5 kg

Total cost = Rs.360 + Rs.300 = Rs.660

Step 3: Calculate weighted average price:

\( \text{Average price} = \frac{\text{Total cost}}{\text{Total quantity}} = \frac{660}{5} = 132 \)

Answer: The average price per kg is Rs.132.

Example 4: Combined Average of Two Groups Hard
Class A has 40 students with an average of 70 marks, and Class B has 60 students with an average of 80 marks. Find the combined average marks of both classes.

Step 1: Calculate total marks for each class:

Class A total = 40 x 70 = 2800

Class B total = 60 x 80 = 4800

Step 2: Calculate combined total marks and total number of students:

Total marks = 2800 + 4800 = 7600

Total students = 40 + 60 = 100

Step 3: Calculate combined average:

\( \text{Combined average} = \frac{7600}{100} = 76 \)

Answer: The combined average marks are 76.

Example 5: Average Speed Problem Hard
A car travels 60 km at 40 km/h and then 60 km at 60 km/h. Find the average speed for the entire journey.

Step 1: Calculate time taken for each part:

Time for first 60 km = \( \frac{60}{40} = 1.5 \) hours

Time for next 60 km = \( \frac{60}{60} = 1 \) hour

Step 2: Calculate total distance and total time:

Total distance = 60 + 60 = 120 km

Total time = 1.5 + 1 = 2.5 hours

Step 3: Calculate average speed:

\( \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{120}{2.5} = 48 \) km/h

Answer: The average speed for the entire journey is 48 km/h.

Formula Bank

Simple Average
\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
where: Average = average value, Sum of observations = total sum of all values, Number of observations = count of values
Sum of Observations
\[ \text{Sum} = \text{Average} \times \text{Number of observations} \]
where: Sum = total sum, Average = average value, Number of observations = count of values
Weighted Average
\[ \text{Weighted Average} = \frac{\sum (w_i \times x_i)}{\sum w_i} \]
where: \(w_i\) = weight of ith observation, \(x_i\) = value of ith observation
Combined Average of Two Groups
\[ \text{Combined Average} = \frac{(n_1 \times a_1) + (n_2 \times a_2)}{n_1 + n_2} \]
where: \(n_1, n_2\) = number of observations in groups 1 and 2, \(a_1, a_2\) = averages of groups 1 and 2
Average Speed
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
where: Total Distance = sum of distances, Total Time = sum of times

Tips & Tricks

Tip: Use sum instead of average for intermediate calculations to avoid rounding errors.

When to use: When solving multi-step average problems.

Tip: For combined averages, multiply average by number of terms before adding.

When to use: When combining groups with different averages.

Tip: In weighted average problems, always identify weights clearly (like quantity or frequency).

When to use: When dealing with items bought in different quantities or frequencies.

Tip: Remember average speed is not the average of speeds unless time intervals are equal.

When to use: When solving average speed problems with different speeds over different distances or times.

Tip: Use shortcut: If one value is missing, use total sum = average x total number of values to find it quickly.

When to use: When a single value is unknown in a set.

Common Mistakes to Avoid

❌ Adding averages directly without considering the number of observations.
✓ Calculate total sums first, then find combined average by dividing by total number of observations.
Why: Students confuse average with sum and ignore the weight of each group.
❌ Using simple average formula for weighted average problems.
✓ Use weighted average formula considering weights (quantities or frequencies).
Why: Students overlook that different observations may have different importance.
❌ Calculating average speed as arithmetic mean of speeds.
✓ Calculate total distance and total time, then divide to get average speed.
Why: Students assume equal time spent at each speed, which is often not true.
❌ Ignoring units in problems involving measurements or currency.
✓ Always include units and convert if necessary to maintain consistency.
Why: Leads to incorrect answers and confusion.
❌ Rounding intermediate values too early.
✓ Keep intermediate calculations precise and round only final answer.
Why: Early rounding causes cumulative errors.
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