Question

There were five sections in MAT paper. The average score of Pooja in first 3 sections was 83 and the average in the last 3 sections was 97 and the average of all the sections (i.e, whole paper) was 92 , then her score in the third section was: (a) 85 (b) 92 (c) 88 (d) none of these

Short answer

Answer: (d) None of these. (Her score in the third section was 80.)

Step-by-step solution

Find the total scores in different sections

Since we know the average scores, we can find the total scores by multiplying the average by the number of sections. Let's denote her scores in the 5 sections as S1, S2, S3, S4, and S5. Total score in first 3 sections = (Average in first 3 sections) * 3 = 83 * 3 = 249 Total score in last 3 sections = (Average in last 3 sections) * 3 = 97 * 3 = 291 Total score in the whole paper = (Average in the whole paper) * 5 = 92 * 5 = 460

Create the equation

We now have the following relationships involving her scores in the different sections: S1 + S2 + S3 = 249 S3 + S4 + S5 = 291 S1 + S2 + S3 + S4 + S5 = 460

Solve for S3 (score in the third section)

We first need to eliminate S1, S2, S4, and S5 from the equations. We can do this by subtracting the first equation from the third equation and then subtracting the result from the second equation: (S1 + S2 + S3 + S4 + S5) - (S1 + S2 + S3) = 460 - 249 => S4 + S5 = 211 Now, subtracting this result from the second equation: (S3 + S4 + S5) - (S4 + S5) = 291 - 211 => S3 = 80 Finally, her score in the third section is 80, which is not included in the options (a), (b), or (c). Therefore, the correct answer is (d) none of these.

Key concepts

Mathematical Problem Solving

Mathematical problem solving involves a logical sequence of steps to arrive at a solution. It often begins with understanding the problem by carefully examining given data, which in this case are the average scores across several sections. Once the problem is understood, a strategy for solving it is developed. For Pooja's score problem, using average scores to find the total scores for each section is crucial. This involves basic arithmetic operations and algebra to simplify complex relationships. Solving the problem requires optimizing the use of equations derived from known averages to find unknown values—specifically, her score in the third section.

• Identify what is known: Section averages and overall average.
• Strategize how to efficiently use this information to solve for the unknown.
• Implement the strategy and solve by performing necessary operations like subtraction.

Arithmetic Mean

The concept of arithmetic mean, or average, is central in determining the total scores in various sections of Pooja's MAT paper. The arithmetic mean is calculated by taking the sum of a set of values and dividing it by the number of values. In Pooja's case:

• Average score for first 3 sections is 83, calculated as: \[\text{Total score} = 83 \times 3 = 249\]
• Average score for last 3 sections is 97, resulting in: \[\text{Total score} = 97 \times 3 = 291\]
• The overall average is 92 for all 5 sections, giving: \[\text{Total score} = 92 \times 5 = 460\]

Understanding arithmetic mean helps in breaking down the total score needed from given averages, making it easier to manage and solve larger problems.

Equation Formulation

When dealing with problems that require finding unknown values, formulating equations is a powerful approach. Equations represent mathematical relationships between known and unknown quantities. Here:

• We form equations based on section averages to determine total scores: \[S1 + S2 + S3 = 249,\]\[S3 + S4 + S5 = 291\]
• By also considering the overall total: \[S1 + S2 + S3 + S4 + S5 = 460\]

The next step is to manipulate these equations to eliminate unnecessary variables, focusing instead on what's needed—in this case, solving for \(S3\). This process involves logical steps such as subtraction to isolate the variable of interest, helping us conclude that \(S3 = 80\), which is crucial for resolving Pooja's scoring issue.Formulating and solving these equations not only resolves the problem but reinforces the importance of equations in mathematical problem solving.