The average income of six friends A, B, C, D, E and F is Rs. 6,500. The total income of B, D, E and F is Rs. 25,000. If A's income is Rs. 4,000 less than C's income, then what is the income of C?
Rs. 9,000
Understanding the Income Problem
This problem involves calculating individual incomes based on the average income of a group and the total income of a subset of that group. We are given the average income of six friends, the total income of four of them, and a relationship between the incomes of two remaining friends. We need to find the income of one specific friend.
Key Concepts: Average and Total Income
The average of a set of values is calculated by dividing the sum of all values by the number of values. Conversely, the total sum of values can be found by multiplying the average by the number of values.
Mathematically, the formula for average is:
\(\text{Average} = \frac{\text{Sum of Values}}{\text{Number of Values}}\)
From this, we can derive the formula for the sum:
\(\text{Sum of Values} = \text{Average} \times \text{Number of Values}\)
Step-by-Step Solution to Find C's Income
Let the incomes of the six friends be A, B, C, D, E, and F.
Step 1: Calculate the Total Income of All Six Friends
We are given the average income of the six friends (A, B, C, D, E, F) is Rs. 6,500.
Number of friends = 6
Average income = Rs. 6,500
Using the formula for total income:
\(\text{Total Income of A, B, C, D, E, F} = \text{Average Income} \times \text{Number of Friends}\)
\(\text{Total Income} = 6500 \times 6 = 39000\)
The total income of the six friends is Rs. 39,000.
Step 2: Find the Combined Income of A and C
We are given the total income of B, D, E, and F is Rs. 25,000.
The total income of all six friends is the sum of the income of A, C, and the total income of B, D, E, F.
\(\text{Total Income of (A + C + B + D + E + F)} = \text{Income of A} + \text{Income of C} + \text{Total Income of (B + D + E + F)}\)
So, we can find the combined income of A and C by subtracting the total income of B, D, E, F from the total income of all six friends.
\(\text{Combined Income of A and C} = \text{Total Income of A, B, C, D, E, F} - \text{Total Income of B, D, E, F}\)
\(\text{Combined Income of A and C} = 39000 - 25000 = 14000\)
The combined income of A and C is Rs. 14,000.
Step 3: Use the Relationship between A's and C's Income to Solve for C
We are told that A's income is Rs. 4,000 less than C's income.
Let C's income be represented by \(C\).
Then, A's income can be represented by \(C - 4000\).
We know that the combined income of A and C is Rs. 14,000. So, we can write an equation:
\(\text{A's Income} + \text{C's Income} = 14000\)
\((C - 4000) + C = 14000\)
Now, solve the equation for \(C\):
\(2C - 4000 = 14000\)
Add 4000 to both sides of the equation:
\(2C = 14000 + 4000\)
\(2C = 18000\)
Divide both sides by 2:
\(C = \frac{18000}{2}\)
\(C = 9000\)
Thus, C's income is Rs. 9,000.
We can also find A's income to verify: A's income = \(C - 4000 = 9000 - 4000 = 5000\). The sum of A and C's income is \(5000 + 9000 = 14000\), which matches our calculation in Step 2.
Summary of Incomes
| Friend(s) | Income/Total Income |
|---|---|
| A + B + C + D + E + F | Rs. 39,000 (Calculated) |
| B + D + E + F | Rs. 25,000 (Given) |
| A + C | Rs. 14,000 (Calculated: 39000 - 25000) |
| A | Rs. 5,000 (Calculated: C - 4000) |
| C | Rs. 9,000 (Calculated) |
Therefore, the income of C is Rs. 9,000.
Revision Table: Average and Income Calculations
| Concept | Formula | Application in Problem |
|---|---|---|
| Average | \( \frac{\text{Sum}}{\text{Count}} \) | Given average of 6 friends. |
| Total Sum | \( \text{Average} \times \text{Count} \) | Used to find total income of 6 friends. |
| Finding Subset Sum | Total Sum - Sum of Other Subsets | Used to find the combined income of A and C. |
| Algebraic Equation | Represent unknowns with variables and form equations based on given relationships. | Used the relationship \(A = C - 4000\) and \(A + C = 14000\) to solve for C. |
Additional Information: Understanding Averages
An average (also called the mean) is a single value that represents the center or typical value of a set of numbers. It helps in summarizing large amounts of data into a single figure.
- Averages are widely used in statistics, economics (like average income, GDP per capita), and everyday life.
- Understanding averages is crucial for interpreting data and solving problems involving distributions of values.
- While the mean is the most common type of average, other types exist, such as the median (the middle value in a sorted list) and the mode (the most frequent value).
- In this problem, we used the mean to find the total sum of incomes, which was essential for isolating the combined income of A and C.
This problem is a good example of how combining concepts like average, total sum, and basic algebraic equations can solve quantitative aptitude questions effectively.
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