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

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 (Option C).

Step 1: Understand the Problem

The problem gives us the mean weight of a group of seven students and the weights of six students. We need to find the weight of the seventh student.

Step 2: Recall the Formula for Mean

Mean is calculated as the sum of all values divided by the number of values. In this case, the mean weight is given as 55 kg for 7 students.

Step 3: Set Up the Equation

Let the weight of the seventh student be \( x \). Then, the equation for the mean weight is given by:\[\frac{52 + 58 + 55 + 53 + 56 + 54 + x}{7} = 55\]

Step 4: Simplify the Equation

First, calculate the sum of the given weights:\[ 52 + 58 + 55 + 53 + 56 + 54 = 328 \]Now, substitute back into the equation:\[\frac{328 + x}{7} = 55\]

Step 5: Solve for the Seventh Student's Weight

Multiply both sides of the equation by 7 to eliminate the fraction:\[ 328 + x = 385 \]Subtract 328 from both sides to find \( x \):\[ x = 385 - 328 = 57 \]

Step 6: Verify the Solution

Sum up all the weights, including the seventh student:\[ 52 + 58 + 55 + 53 + 56 + 54 + 57 = 385 \]Calculate the mean:\[ \frac{385}{7} = 55 \]This matches the given mean, confirming that the solution is correct.

Key Concepts

Equation Setup

To find the weight of the seventh student, we first need to set up an equation that incorporates all the knowns and unknowns.
The problem provides the weights of six students, and we will let the weight of the seventh student be represented as a variable, say \( x \).
When dealing with mean calculations, the setup involves forming an equation using the formula for the arithmetic mean.
In this case, the equation can be written as follows:

• The total weight of all students is needed, which equals the weight of the first six students plus the weight of the seventh student \( (x) \).
• Divide this sum by the total number of students, which is 7, to equate it to the mean weight given (55 kg).

Thus, the setup of the equation using these data goes as:\[\frac{52 + 58 + 55 + 53 + 56 + 54 + x}{7} = 55\]Understanding how to set up this equation is essential. It provides the foundation needed to solve the problem.

Problem Solving Steps

Once the equation is established, solving it becomes straightforward if approached step by step.

• Step 1: Calculate the sum of the known weights: \(52 + 58 + 55 + 53 + 56 + 54 = 328\). This step simplifies the equation for easier solving.
• Step 2: Substitute the sum back into the equation, forming \(\frac{328 + x}{7} = 55\).
• Step 3: To handle the fraction, multiply the entire equation by 7 across both sides:\[328 + x = 385\]
• Step 4: Isolate \(x\) by subtracting 328 from both sides of the equation,leading to \(x = 385 - 328 = 57\).

Taking it one small leap at a time ensures you stay grounded in the method.
Paying close attention to each little arithmetic operation is key in many math problems.

Arithmetic Mean

The arithmetic mean is a very common measure of central tendency in statistics.
It's also known as the average.
Calculating the arithmetic mean is simple; you sum up all values in the dataset and then divide by the number of values.
In classroom settings, calculating the mean helps in determining at a glance what the average of a group is.Here, the problem involves the mean weight of students:

• The sum of student weights is needed.
• Then you simply count the students (or data points) involved.
• Finally, divide the total weight by this count to find the mean: \( \frac{385}{7} = 55\).

Remember, a key property of the arithmetic mean is that all deviations from the mean sum to zero.
That means, mathematically, it balances all values in the dataset equally.