Question

The average age of 10 students in a class is 20 years, if a new student is also included, then the new average age of all the students increases by 1 year. The age of the new student is : (a) 21 years (b) 30 years (c) 31 years (d) none of these

Short answer

Answer: The age of the new student is 31 years.

Step-by-step solution

Calculate the combined ages of the initial students

To find the combined ages of the 10 initial students, we multiply the average age by the number of students. We are given that the average age is 20 years, so the combined ages of the 10 students is: combined_ages_initial_students = 10 * 20 = 200 years

Calculate the new average age for all the students

With the new student joining the class, we are told that the average age increases by 1. Therefore, the new average age for all the students is 21 years.

Calculate the combined ages of all students, including the new student

With the new student, there are now 11 students in the class. We need to find the combined age of all the students, using the new average age. To do this, we multiply the new average age by the total number of students: combined_ages_all_students = 11 * 21 = 231 years

Find the age of the new student

To find the age of the new student, we subtract the combined ages of the initial students from the combined ages of all the students (including the new student): new_student_age = combined_ages_all_students - combined_ages_initial_students new_student_age = 231 - 200 = 31 years The age of the new student is 31 years, so the answer is (c) 31 years.

Key concepts

Quantitative Aptitude

Quantitative aptitude is a crucial skill for anyone engaged in problem-solving or any analytical task that demands numerical reasoning. It encompasses the ability to understand and manipulate numbers, often underpinning the strategies used in tackling various mathematical problems.

For students grappling with the average age problem, enhancing quantitative aptitude involves understanding not just the arithmetic operations but also the reasoning behind them. When a student is asked to calculate the age of a new member in a group based on the change in average age, it's about more than multiplication and subtraction; it's about comprehending how averages work and how the inclusion of a single data point can shift a set's mean value.

Problem Solving

Problem solving is a systematic approach that takes us from an initial situation to a desired goal. It is not only about the solution but about the path we take to reach it. In the context of the average age problem, the steps taken to understand and solve the problem are as critical as finding the correct age of the new student.

Effective problem-solving in math employs a clear sequence of steps:

• Understanding the problem,
• Devising a plan,
• Carrying out the plan, and
• Reviewing the solution for accuracy.

The present problem required students to carefully read and decode the task, form an equation based on the arithmetic mean principle, execute the necessary calculations, and finally verify that the answer made logical sense within the context of the problem.

Arithmetic Mean

Arithmetic mean, often known simply as the average, is a fundamental concept in statistics and everyday mathematics. It is defined as the sum of a set of values divided by the number of values. In real-life scenarios, the arithmetic mean can provide a measure of central tendency for a dataset, allowing for a better understanding of the distribution of values.

In our textbook problem, the arithmetic mean helps to distribute the total age evenly across students and illustrates how the addition of one new age can change the overall average. To calculate the impact of a new number on an established average, one must recognize that the average represents an equal share of the 'whole' for each member of the set. When this 'whole' increases, and the set gains a member, recalculating the mean reflects the inclusion and gives insight into the scale of the new number relative to the original mean.