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Have you ever wondered how to find a single value that best represents a group of numbers? In everyday life and in competitive exams, this value is known as the average. For example, if you want to find the average marks scored by students in a test or the average price of different fruits you bought, you use the concept of average.
The word aggregate means the total or sum of all parts. When you combine different groups or sets of values, knowing their total sum (the aggregate) helps you understand the whole picture better. This chapter will help you master the concepts of average and aggregate, enabling you to solve a wide range of problems efficiently, especially those common in competitive exams.
Understanding average and aggregate is crucial not just in academics but in real-world situations like calculating average speed during a trip, average salary in a company, or average expenditure. Let's begin this journey by exploring these concepts from the ground up.
The simple average, also called the arithmetic mean, gives a measure of the central value in a set of numbers. It is found by adding all the values together and then dividing the total by the number of values.
Why do we use simple average? It helps to find a representative number that summarizes the entire data set. For example, if five students score marks of 75, 80, 85, 90, and 70, the average gives a single number indicating the general performance of the group.
Formula:
Let's visualize this:
Imagine five students scored marks as follows: 75, 80, 85, 90, and 70. What is the average mark?
Solution:
Answer: The average mark is 80.
Sometimes, not all values contribute equally. A weighted average takes into account the weights or importance of each value. For instance, if we buy 3 kg of apples at one price and 5 kg at another, the average price per kg should reflect the quantities purchased.
Why use weighted average? Because some values matter more due to their frequency, size, or importance. Simply averaging the values without accounting for these differences could give incorrect results.
Formula:
To better understand, see the following table:
| Item | Value (\(x_i\)) | Weight (\(w_i\)) | Product (\(w_i \times x_i\)) |
|---|---|---|---|
| Apples (3 kg) | Rs.60 per kg | 3 | 180 |
| Apples (5 kg) | Rs.50 per kg | 5 | 250 |
| Sum | 8 | 430 | |
The weighted average price per kg of apples is therefore:
\[\frac{(3 \times 60) + (5 \times 50)}{3 + 5} = \frac{180 + 250}{8} = \frac{430}{8} = 53.75\]
So, you pay an average price of Rs.53.75 per kg for the apples bought.
The term aggregate simply means the total sum of the quantities considered. It is useful when combining data from different groups or categories.
For example, if a class of 30 students has an average score of 75 and another class of 20 students has an average score of 80, to find the combined average, you calculate the aggregate marks first (by multiplying each average with its group size), then find the total average.
Aggregate formula:
Class A has 30 students with an average of 75 marks. Class B has 20 students with an average of 80 marks. Find the combined average marks of the two classes.
Solution:
Answer: The combined average is 77 marks.
Step 1: Add all the marks: \(75 + 80 + 85 + 90 + 70 = 400\).
Step 2: Count the number of students: 5.
Step 3: Divide the sum by the count: \(\frac{400}{5} = 80\).
Answer: The average mark is 80.
Step 1: Calculate total cost: \[(3 \times 60) + (5 \times 50) = 180 + 250 = 430\]
Step 2: Calculate total quantity: \[3 + 5 = 8 \text{ kg}\]
Step 3: Find weighted average price: \[\frac{430}{8} = 53.75\]
Answer: Average price per kg is Rs.53.75.
Step 1: Calculate aggregate totals: \[30 \times 75 = 2250,\quad 20 \times 80 = 1600\]
Step 2: Sum total marks: \[2250 + 1600 = 3850\]
Step 3: Sum number of students: \[30 + 20 = 50\]
Step 4: Divide aggregate by total students: \[\frac{3850}{50} = 77\]
Answer: The combined average is 77 marks.
Step 1: Find time taken for each part: \[ t_1 = \frac{40}{60} = \frac{2}{3} \text{ hours}, \quad t_2 = \frac{60}{90} = \frac{2}{3} \text{ hours} \]
Step 2: Total distance: \[ 40 + 60 = 100 \text{ km} \]
Step 3: Total time: \[ \frac{2}{3} + \frac{2}{3} = \frac{4}{3} \text{ hours} \]
Step 4: Average speed: \[ \frac{100}{\frac{4}{3}} = 100 \times \frac{3}{4} = 75 \text{ km/h} \]
Answer: Average speed for the entire journey is 75 km/h.
Step 1: Compute total salary for each department: \[ 50 \times 40000 = 2000000, \quad 30 \times 60000 = 1800000 \]
Step 2: Calculate total salary budget: \[ 2000000 + 1800000 = 3800000 \]
Step 3: Total number of employees: \[ 50 + 30 = 80 \]
Step 4: Calculate average salary: \[ \frac{3800000}{80} = 47500 \]
Answer: The average monthly salary is Rs.47,500.
When to use: When combining groups or quantities with different sample sizes or weights.
When to use: To find totals or verify averages from combined sets.
When to use: In any problem involving distances, weights, or speeds.
When to use: Problems involving salaries, prices, profit or loss averages.
When to use: During timed exams for quick verification.
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