Mastering Averages of Combined Groups for Competitive Exams

Welcome to the module on Averages of Combined Groups! This is a crucial concept in quantitative aptitude, frequently tested in competitive exams like the CAT. Understanding how to calculate the average when different groups are combined is key to solving many problems efficiently.

The Core Concept: Weighted Average

When we combine groups with different averages, we can't simply average the averages. Instead, we use the concept of a weighted average. Each group's average contributes to the overall average based on the 'weight' or size of that group. The formula for the average of two combined groups is:

The average of combined groups is a weighted average.

To find the average of combined groups, you multiply each group's average by its size (number of elements), sum these products, and then divide by the total number of elements across all groups.

Let Group 1 have $n_1$ elements with an average $A_1$ , and Group 2 have $n_2$ elements with an average $A_2$ . The sum of elements in Group 1 is $S_1 = n_1 \times A_1$ , and the sum of elements in Group 2 is $S_2 = n_2 \times A_2$ . When combined, the total sum is $S_{total} = S_1 + S_2 = (n_1 \times A_1) + (n_2 \times A_2)$ . The total number of elements is $N_{total} = n_1 + n_2$ . Therefore, the combined average ( $A_{combined}$ ) is: $A_{combined} = \frac{S_{total}}{N_{total}} = \frac{(n_1 \times A_1) + (n_2 \times A_2)}{n_1 + n_2}$ This principle extends to more than two groups.

What is the fundamental principle used to calculate the average of combined groups?

Weighted Average.

Illustrative Example

Consider a class with two sections. Section A has 30 students with an average score of 70. Section B has 20 students with an average score of 80. What is the average score of the entire class?

To solve this, we apply the weighted average formula. The 'weights' are the number of students in each section.

Group 1 (Section A): $n_1 = 30$ , $A_1 = 70$ Group 2 (Section B): $n_2 = 20$ , $A_2 = 80$

Sum of scores in Section A = $30 \times 70 = 2100$ Sum of scores in Section B = $20 \times 80 = 1600$

Total sum of scores = $2100 + 1600 = 3700$ Total number of students = $30 + 20 = 50$

Combined Average = $\frac{3700}{50} = 74$

This visualizes how the higher average of Section B pulls the overall average closer to 80 than to 70, reflecting its 'weight' in the combined group.

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In the example, which section's average had a greater influence on the combined average, and why?

Section A's average had a greater influence because it had more students (30) compared to Section B (20).

Variations and Problem-Solving Strategies

Problems involving combined averages can be presented in various ways. You might be given the combined average and asked to find the average of one group, or the number of elements in a group. Always identify the knowns (number of elements and averages of individual groups) and the unknown. Rearranging the weighted average formula is often necessary.

A common shortcut: If the number of elements in two groups are in a ratio $x:y$ , their averages will influence the combined average in the inverse ratio $y:x$ relative to the combined average. For example, if Group A has 30 students and Group B has 20 (ratio 3:2), the combined average will be closer to Group A's average if Group A's average is lower, and closer to Group B's average if Group B's average is lower.

Practice Scenarios

Practice problems often involve mixing quantities of different prices, combining mixtures with different concentrations, or analyzing performance data across different teams or periods. The underlying principle of weighted average remains constant.

What is a common scenario where the concept of combined averages is applied in real-world problems?

Mixing different quantities of items with different prices or concentrations.

Learning Resources

Weighted Average Explained - Byjus(blog)

This article provides a clear explanation of the weighted average concept with examples relevant to competitive exams.

Average of Combined Groups - Indiabix(documentation)

A concise explanation and practice problems specifically on the average of combined groups, a staple for exam preparation.

CAT Quantitative Aptitude: Averages - Unacademy(blog)

This resource offers a comprehensive overview of averages, including combined groups, tailored for CAT aspirants.

Understanding Weighted Averages - Khan Academy(tutorial)

Khan Academy offers a foundational understanding of weighted means, which is directly applicable to combined averages.

Average Questions for Competitive Exams - Gradeup (now Byju's Exam Prep)(blog)

A collection of practice questions on averages, including those involving combined groups, with explanations.

Quantitative Aptitude for CAT - Average Concepts(blog)

This blog post breaks down various average concepts, including the crucial topic of combined averages, with practical examples.

Weighted Average - Math is Fun(documentation)

A user-friendly explanation of weighted averages with interactive elements and clear examples.

CAT Quant: Averages - Practice Problems with Solutions(blog)

Provides practice problems and solutions for averages, with a focus on exam-relevant question types.

The Concept of Average in CAT - YouTube(video)

A video tutorial explaining the fundamental concepts of averages, likely covering combined groups for CAT preparation.

Average of Combined Groups - Practice Questions(documentation)

A dedicated set of practice questions and answers specifically for the 'Average of Combined Groups' topic.