Question

The average of \(A\) and \(B\) is 400 and the average of \(C\) and \(D\) is 600 the average of \(A, B, C\) and \(D\) is : (a) 500 (b) 450 (c) 525 (d) 625

Short answer

Answer: 500

Step-by-step solution

Average formula

The average of two or more numbers is the sum of the numbers divided by the quantity of the numbers. Using this formula, we can find the sum of each pair.

Calculate the sum of A and B

The average of \(A\) and \(B\) is 400, so the sum of the two numbers can be found with: \((A + B) / 2 = 400\) Multiplying both sides by 2, we get: \(A + B = 800\)

Calculate the sum of C and D

The average of \(C\) and \(D\) is 600, so the sum of the two numbers can be found with: \((C + D) / 2 = 600\) Multiplying both sides by 2, we get: \(C + D = 1200\)

Calculate the sum of A, B, C, and D

By adding the sums we found in steps 2 and 3, we can find the sum of \(A\), \(B\), \(C\), and \(D\): \(A + B + C + D = 800 + 1200 = 2000\)

Calculate the average of A, B, C, and D

To find the average of \(A\), \(B\), \(C\), and \(D\), we divide the total sum by the number of elements (which is 4): \(Average = (A + B + C + D) / 4 = 2000 / 4 = 500\) So, the average of \(A\), \(B\), \(C\), and \(D\) is 500, making the correct answer (a) 500.

Key concepts

Quantitative Aptitude

Quantitative aptitude is the ability to handle numbers and solve mathematical problems using basic arithmetic concepts. It's a skill that's especially valued in fields that require quick and accurate numerical analysis, such as finance, engineering, and technology. In educational contexts, questions on quantitative aptitude test a student’s ability to understand numerical data, analyze it, and apply standard mathematical procedures to find solutions.

Problems dealing with averages, percentages, ratios, and proportions frequently feature in quantitative aptitude assessments. Mastering these concepts can give students a solid foundation for problem-solving in a variety of real-world scenarios. When faced with an average calculation question, such as the one in our original exercise, it is critical to comprehend the concept of arithmetic mean and the process involved in finding it, which is a fundamental quantitative aptitude skill.

Arithmetic Mean

The arithmetic mean, often simply called the average, is a basic statistical measure used to describe the central tendency of a set of numbers. It is calculated by summing up all the values and then dividing by the number of values.

In the context of our exercise, the average of two numbers, such as \(A\) and \(B\), is found by adding them together and dividing by 2. When dealing with larger sets of numbers, the principle is the same—add up all the numbers and divide by the count of numbers. So, when finding the average of \(A, B, C,\) and \(D\), we add these four numbers and divide by 4. Understanding the arithmetic mean is not just about performing a calculation; it’s about grasping that the mean represents the point in a data set where all values balance each other out.

Problem-Solving

Problem-solving in mathematics, such as working through an exercise involving averages, involves a series of methodical steps. As illustrated by the exercise's step by step solution, the first step is understanding and applying the correct formula. Next, we use logical reasoning and manipulation of the equation to isolate the quantities we need.

Step-by-step problem-solving fosters a structured approach to tackling quantitative questions. This methodical process leads to a greater understanding of how each step affects the next, and it helps to uncover any errors along the way. When solving for the average, as in our textbook exercise, a systematic approach allows students to easily follow the logical sequence—calculate individual sums first, then derive the combined average. Problem-solving skills empower students to apply their arithmetic knowledge to a variety of situations, reinforcing their quantitative aptitude overall.