Problem 23 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The mean weight per student in a group of seven students is \(55 \mathrm{~kg}\) If the individual weights of 6 students are \(52,58,55,53,56\) and 54 ; then weight of the seventh student is (A) \(55 \mathrm{~kg}\) (B) \(60 \mathrm{~kg}\) (C) \(57 \mathrm{~kg}\) (D) \(50 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The weight of the seventh student is 57 kg.

Step 1: Understand the Mean Formula

The mean (average) of a set of numbers is calculated by dividing the sum of all the numbers by the total count of numbers. Here, the mean weight given is 55 kg for 7 students.

Step 2: Set Up the Equation

Let's denote the weight of the seventh student as \( x \). The formula for the mean gives us the equation: \( \frac{52 + 58 + 55 + 53 + 56 + 54 + x}{7} = 55 \).

Step 3: Calculate the Sum of Given Weights

Add up the weights of the 6 known students: \( 52 + 58 + 55 + 53 + 56 + 54 = 328 \).

Step 4: Plug Values into the Equation

Substitute the sum into the mean equation: \( \frac{328 + x}{7} = 55 \).

Step 5: Solve for the Unknown Weight

To find \( x \), multiply both sides by 7: \( 328 + x = 385 \). Then solve for \( x \) by subtracting 328 from both sides: \( x = 385 - 328 = 57 \).

Step 6: Verify the Calculation

Double-check the calculation to ensure no errors were made: 328 (sum of six weights) + 57 (weight of the seventh student) equals 385. Dividing 385 by 7 gives 55, confirming the mean is correct.

Key Concepts

Average Weight Calculation

Calculating the average weight, also known as the mean, is a common task in mathematics, particularly useful in statistics. To find the average, you add up all the weights and divide by the number of weights. In the given exercise, the group consists of seven students. Their average or mean weight is given as 55 kg.
However, only the weights of six students are known: 52, 58, 55, 53, 56, and 54 kg. To find the weight of the seventh student, we need to understand that the average is essentially evenly distributing the total weight among all students.

• Sum up the given weights.
• Include the unknown weight as a variable.
• Use the mean formula to set up an equation.

This straightforward approach can be used in any scenario where you need to find missing data, especially when averages are involved.

Solving Equations

Solving equations is at the heart of algebra and is crucial for finding unknown quantities in a problem. In this exercise, we denote the unknown weight of the seventh student as \( x \).
We formulate the problem as an equation: \( \frac{52 + 58 + 55 + 53 + 56 + 54 + x}{7} = 55 \). This equation represents the mean calculation with the unknown weight part of the sum.
Step-by-step, we solve for \( x \):

• Add the known weights: 52 + 58 + 55 + 53 + 56 + 54 = 328 kg.
• Substitute into the equation: \( \frac{328 + x}{7} = 55 \).
• Multiply through by 7, simplifying to: \( 328 + x = 385 \).
• Isolate \( x \) by subtracting 328 from both sides, yielding \( x = 57 \).

This method of solving equations by isolating the variable is a fundamental technique in mathematics.

Mathematics Problem Solving

Mathematical problem solving involves a systematic process of understanding the problem, developing a plan (such as setting up an equation), carrying out the plan, and then looking back at the solution to verify its correctness. This process enhances our logical reasoning skills and is applicable across various mathematical topics.
In tackling this exercise, firstly, we identified all elements involved: the average weight, the known weights, and the unknown weight. We set up an equation to embody our understanding of the problem in mathematical terms.
Following the steps undergoes the process:

• Understand the problem and the given data.
• Translate this information into a mathematical equation.
• Solve the equation using algebraic methods to find the unknown.
• Lastly, verify our solution to ensure accuracy by checking if the average equation holds true.

This systematic approach ensures a thorough understanding and skillful tackling of similar mathematical problems.