Question

The average of \(a, b\) and \(c\) is 79 and the average of \(a\) and \(c\) is also 79 . Then the value of \(b\) is : (a) 0 (b) 79 (c) \(-79\) (d) none of these

Short answer

Answer: Based on the analysis and solution steps above, the value of \(b\) is 79.

Step-by-step solution

Write down the average equations for given information

According to the problem, we have two average values: 1. The average of \(a, b\) and \(c\) is 79 2. The average of \(a\) and \(c\) is also 79 Let's write down the respective equations: 1. \(\frac{a + b + c}{3} = 79\) 2. \(\frac{a + c}{2} = 79\)

Solve the first average equation

Solving the first equation for the sum of \(a, b\) and \(c\), we have: \(a + b + c = 79 \cdot 3\) \(a + b + c = 237\) (1)

Solve the second average equation

Solving the second equation for the sum of \(a\) and \(c\), we have: \(a + c = 79 \cdot 2\) \(a + c = 158\) (2)

Solve for \(b\) using the two equations

Subtract equation (2) from equation (1), \((a + b + c) - (a + c) = 237 - 158\) \(b = 79\) The value of \(b\) is 79 which corresponds to option (b).

Key concepts

Arithmetic Mean

When dealing with averages, we are often referencing the concept of the arithmetic mean. It is the most commonly used measure of central tendency, widely known simply as 'the average'. The arithmetic mean is calculated by adding up a set of numbers and then dividing the sum by the count of those numbers. For instance, if we wish to find the average of three numbers, say 5, 7, and 9, we add them up to get 21 and then divide by 3, the number of values, resulting in an average of 7.

In our exercise, the concept of arithmetic mean is foundational. It's used to determine the average value of different variables. This application of the mean helps to simplify complex problems, making them more manageable algebraically. By understanding that the average can represent a common value that all numbers can relate to, it becomes easier to decipher relationships between those numbers, particularly when some of the numbers are unknown variables.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. These expressions can include numbers, variables (letters used to represent unknown values), and arithmetic operations. Algebra plays a crucial role in solving problems that involve unknown quantities.

In the context of our exercise, algebraic equations are derived from the given average problems. We form two separate equations from the given average values and use them to solve for the unknown variable, which is 'b' in this case. This process often involves manipulation such as adding, subtracting, or substituting equations. For instance, we subtract one equation from another to isolate and hence find the value of 'b'. These methods are part of a larger set of tools in algebra that allow us to handle and solve various equations efficiently.

Quantitative Aptitude

Quantitative aptitude refers to the ability to handle numerical and mathematical tasks. It is a skill that is often assessed in academic and professional environments. Understanding mathematical concepts like the arithmetic mean and being able to manipulate algebraic equations are vital components of this aptitude.

In the given problem, quantitative aptitude comes into play when students interpret the word problem, translate it into mathematical equations, and then solve them accurately. Working through the exercise requires the ability to reason quantitatively and solve problems using mathematical concepts. Enhancing quantitative aptitude involves the practice of such problems, along with a deep understanding of the underlying mathematical principles. Real-world applications of quantitative aptitude range from budgeting and investing to science and engineering fields, making it an indispensable part of academia and various careers.