Question

The average age of 6 servants in my farm house is 28 years. new and young servant replaces an old servant, then the new average reduces by 1 year, the age of the new servant is: (a) 26 years (b) 22 years (c) \(-35\) years (d) can't be determined

Short answer

Answer: (d) can't be determined.

Step-by-step solution

Calculate the sum of the ages of the original 6 servants

The average age of the 6 servants is 28 years. To find the sum of their ages, we multiply the average age by the total number of servants: Sum of ages = Average age × Number of servants Sum of ages = 28 × 6 = 168

Calculate the sum of the ages of the servants after the replacement

When the young servant replaces the old one, the new average age reduces by 1 year. Therefore, the new average age is 28 - 1 = 27 years. We can now find the sum of the ages after the replacement: New sum of ages = New average age × Number of servants New sum of ages = 27 × 6 = 162

Create an equation to find the age of the new servant

Let x represent the age of the old servant and y represent the age of the new servant. We can write an equation as follows: Sum of ages - x + y = New sum of ages 168 - x + y = 162

Solve for the age of the new servant

From the equation in Step 3, we just need the information about the age of the old servant. Since we don't have that information, the age of the new servant can't be determined. Therefore, the correct answer is (d) can't be determined.

Key concepts

Problem Solving

Problem-solving abilities are essential in mathematics and real-life scenarios alike. It involves understanding the problem, identifying the key information provided, and devising a systematic approach to find the solution.

In the context of the given average age problem, problem solving begins with comprehending the situation: calculating the sum of ages before and after a change in the group composition. Effective problem-solving in this scenario involves algebraic thinking and the ability to translate words into numerical expressions and equations.

Breaking Down the Problem

Understanding that the 'average age' mentioned is a clue to finding the total sum of ages is the first step. Following that, knowing that an 'average' changes when one member of the group is replaced provides a path to formulate an equation. Additionally, the savvy problem solver checks if the available information is sufficient for finding a unique solution or if there are multiple possibilities that can fulfill the given conditions.

Quantitative Aptitude

Quantitative aptitude is all about the ability to handle numerical and mathematical calculations and to make a logical deduction from quantitative information. This skill is tested in various competitive exams and is a foundational element required for a wide range of academic and professional fields.

In applying quantitative aptitude to the average age problem, one must be comfortable manipulating numbers and variables. This includes capabilities like:

• Multiplying to find the sum of ages.
• Understanding the implication of changing averages.
• Setting up and simplifying algebraic expressions.

While the problem might seem straightforward, it requires finding the aggregate of the ages before and after the replacement, and creating an equation to explore the relationship between the ages of the old and new servant. Here the quantity (sum of ages) is pivotal to solving the problem, despite the fact that the actual problem lacks sufficient information to find a unique solution.

Age-Related Algebra

Age-related algebra problems are a classic case of translating a word problem into algebraic expressions and equations. These problems typically involve using average ages, sum of ages, and the relationship between ages at different points in time.

For the given exercise, understanding age-related algebra involves knowing how to calculate an average (the total sum of ages divided by the number of individuals) and how an average changes with the introduction or removal of group members.

Algebraic Representation

Representing the problem algebraically, the original sum of ages is expressed as the product of the average age and the number of servants. When an individual is replaced, a new equation must be created to account for the addition of the new servant's age and the subtraction of the old servant's age. This illustrates the dynamic nature of algebraic equations in representing real-world scenarios. Challenges arise when there isn't enough information about the variables in the equation, leading to an indeterminate solution—a common occurrence in age-related algebra problems.