Question

The average age of 18 pupils of Dronacharya was 25 years. If the age of Dronacharya was also included, the average age of 19 people becomes 26 years. The average age of the Dronacharya at that time was : (a) 33 (b) 44 (c) 50 (d) 51

Short answer

Answer: The age of Dronacharya is 44 years.

Step-by-step solution

Find the total age of 18 students

Use the average formula to find the total age of the 18 students. The formula for the average is: Average = (Total age) / (Number of people) We are given that the average age of 18 students is 25 years. We can rearrange the formula to find the total age: Total age = Average * (Number of people) Total age = 25 * 18 Calculate the total age: Total age = 450 years

Find the total age of 19 people including Dronacharya

Use the average formula again to find the total age of 19 people, including Dronacharya. We are given that the average age of 19 people is 26 years: Total age = Average * (Number of people) Total age = 26 * 19 Calculate the total age: Total age = 494 years

Calculate the age of Dronacharya

Now that we have both total ages, we can find Dronacharya's age by subtracting the total age of the 18 students from the total age of the 19 people (including Dronacharya): Dronacharya's age = (Total age of 19 people) - (Total age of 18 students) Dronacharya's age = 494 - 450 Calculate Dronacharya's age: Dronacharya's age = 44 years The correct answer is (b) 44.

Key concepts

Understanding the Arithmetic Mean

The arithmetic mean, often referred to as the average, is a fundamental concept in mathematics. It is used to find the central value in a data set. To calculate it, you sum up all the numbers in your data set and then divide by the number of entries in the set.
For example, if you have a set of ages like [20, 22, 24], the average age would be calculated as follows:

• Add all the ages: 20 + 22 + 24 = 66
• Divide by the number of ages: 66 ÷ 3 = 22

So, the arithmetic mean or average age in this example is 22.
In our exercise, we initially calculated the average age of 18 students as 25. This helped us find the total cumulative age of these students by rearranging the mean formula: Total age = Average × Number of people. It's a practical application of the arithmetic mean, allowing us to back-track from a known average to a sum total.

Problem-Solving Techniques in Mathematics

Mathematics is all about solving problems, and being systematic helps. In this problem, we handled a multi-step process to solve for an unknown—Dronacharya's age. The key is to carefully understand given information and systematically apply formulas.
Here's how we approached the problem:

• Identified the known quantities: average ages and number of persons, both with and without Dronacharya.
• Used established formulas to calculate unknowns: total age calculations for both scenarios.
• Substracted the smaller total age from the larger one to isolate Dronacharya's age.

Always begin with what you know, translate the problem into mathematical formulas, and solve step-by-step. This problem-solving technique is useful in age-related arithmetic problems and beyond.

Solving Age-Related Problems

Age-related problems often involve averages since they require understanding cumulative and relative ages of individuals in a group. In the given problem, we dealt with initially knowing the average of a group and needing to find the age of an additional person.
Such problems typically follow these steps:

• Determine the total age of known groups using the average given.
• Recalculate when a new member (or information) is introduced that changes the average.
• Find differences to single out unknowns (individual ages, extra members, etc.).

These problems highlight the importance of arithmetic operations and reasoning. The step-by-step process helps in simplifying what might initially seem complex, ultimately leading one to the solution. Understanding this process can aid any age-related word problem, enhancing problem-solving skills.