Question

In a village the average age of \(n\) people is 42 years. But after the verification it was found that the age of a person had been considered 20 years less than the actual age, so the teis average, after the correction, increased by \(1 .\) The value of nis: (a) 21 (b) 20 (g) 22 (d) none of these

Short answer

Answer: There are 20 people in the village group.

Step-by-step solution

Understand the given data

We know that the average age of n people is 42 years, which implies that the total age of these n people is 42n years. We are also informed that after a correction (adding 20 years to one person's age), the average age increases by 1 year, making it 43 years.

Determine the correct total age of n people

Since the corrected average age of n people is 43 years, we can calculate the total correct age of these n people as the product of the corrected average age and the number of people, which is 43n years.

Find the difference between the incorrect and correct total ages

Now, we will subtract the incorrect total age (42n) from the correct total age (43n) to find the difference, i.e., 43n - 42n = n years. This difference is equal to the 20 years added to the age of one person.

Determine the value of n

Since the difference between the incorrect and correct total ages is equal to the 20 years added to the age of one person, we can conclude that n = 20. Hence, the value of n is 20 (option b).

Key concepts

Understanding Arithmetic Mean

When we talk about the average age of a group of people, we often refer to the *arithmetic mean*. This is a mathematical way to find a central value in data. To calculate the arithmetic mean of ages, for example, you sum up all the individual ages and then divide by the total number of people. If you have 5 people with ages 10, 20, 30, 40, and 50, the arithmetic mean is calculated as follows: - Total of ages = 10 + 20 + 30 + 40 + 50 = 150- Number of people = 5- Arithmetic mean = \( \frac{150}{5} = 30 \)
This simple measure gives us a quick snapshot of the average age within the group and is helpful when summarizing and comparing data sets in everyday scenarios.

Approaching Problem-Solving with Averages

Problem-solving involves breaking down a question into manageable parts. In this scenario, we're given an average age, but with an error in the data. Start by identifying the information given: - Initial average age: 42 years - One person's age underestimated by 20 years - Corrected average increases by 1 year
From these details, we know that the total age of the group can be calculated using the formula for arithmetic mean: Total age = Average age \( \times \) Number of people \( n \). Use reasoning to determine the difference between the initial incorrect and the corrected total ages, then relate this to the actual correction.

Utilizing Mathematical Reasoning

Mathematical reasoning consists of logical thinking to solve complex problems. In our exercise, reasoning helps in understanding how a change in data affects averages. - Originally, the total age was \( 42n \) based on 42 years as the mean.- After adding 20 to one person's age, the mean turned out to be 43 years, calculated as \( 43n \). - This switch from 42n to 43n illustrates a discrepancy of \( 43n - 42n = n \).
By linking this to the 20-year difference, we logically conclude that the number of people in the group must be 20. Exercising mathematical reasoning is crucial in deducing these relationships and confirming the solution accurately.