Analyzing Mark Averages: A Mathematical Puzzle

When dealing with group averages, we often encounter intriguing scenarios that challenge our understanding of arithmetic and averages. One such scenario involves determining the marks obtained in a specific paper when the averages of all papers, the first seven papers, and the last seven papers are known. Let's break down this problem step by step.

The Problem Statement

The average mark for 13 papers is 40. The average mark for the first 7 papers is 42, and the average mark for the last 7 papers is 35. What are the marks obtained in the 7th paper?

Calculating the Total Marks

To find the total marks of all 13 papers, we use the formula for the average:

[ text{Average} frac{text{Total Marks}}{text{Number of Papers}} ]

Therefore, the total marks for 13 papers are:

[ 40 times 13 520 ]

Breaking Down the Averages

Next, we calculate the total marks for the first 7 papers:

[ 42 times 7 294 ]

Similarly, we calculate the total marks for the last 7 papers:

[ 35 times 7 245 ]

Note that the 7th paper is included in both the first 7 papers and the last 7 papers. This overlap is crucial for solving the problem.

Setting Up the Equation

Let ( M_7 ) denote the marks obtained in the 7th paper. The total marks for the last 7 papers can be expressed as:

[ 245 294 - M_7 M_7 times 1 ]

This simplifies to:

[ 245 294 - M_7 ]

Solving for ( M_7 ):

[ M_7 294 - 245 49 ]

The marks obtained in the 7th paper are ( 49 ).

Solving for the 7th Paper Marks

The average mark for 13 papers is 40. Therefore, the total marks for all 13 papers are 520.

The total marks for the first 7 papers are 294.

The total marks for the last 7 papers are 245.

Considering the overlap, we can set up the equation: