Question

The mean weight of a group of 9 students is \(19 \mathrm{~kg}\). If a boy of weight \(29 \mathrm{~kg}\) is joined in the group, then find the mean weight of 10 students. The following are the steps involved in solving the above problem. Arrange them in sequential order. (a) The mean weight of 10 students \(=\frac{200}{10} \mathrm{~kg}\) (b) The total weight of 9 students \(=9 \times 19 \mathrm{~kg}=171 \mathrm{~kg}\) (c) The total weight of 10 students \(=(171+29) \mathrm{kg}=200 \mathrm{~kg}\) (d) \(\therefore\) The mean weight \(=20 \mathrm{~kg}\) (1) bcad (2) bdac (3) bdca (4) bcda

Short answer

Answer: The correct sequence of steps is (4) bcda.

Step-by-step solution

Determine the total weight of 9 students

To find the total weight of the initial group of 9 students, we multiply the mean weight by the number of students: Total weight = 9 * 19 kg = 171 kg.

Find the total weight of 10 students

To find the total weight of 10 students after the boy joins the group, we simply add his weight to the existing total weight: Total weight of 10 students = 171 kg + 29 kg = 200 kg.

Calculate the mean weight of 10 students

To find the mean weight of the group after the boy joins, we divide the total weight by the number of students (10): Mean weight = 200 kg / 10 = 20 kg.

Write down the correct sequence of steps

Based on the step-by-step solution, the correct sequence is (b) -> (c) -> (a) -> (d). Hence, the answer is \((4)~\text{bcda}\).

Key concepts

Arithmetic Mean

The arithmetic mean, often referred to as the average, is a key concept in mathematics. It is used to summarize a group of numbers by distributing the total equally among all the numbers. To find the arithmetic mean, you simply add up all the numbers in your set, and then divide that sum by the number of elements in the set.
For example, if you have the weights of 10 students, average them by adding all the weights together. Then, divide by 10, the total number of students. Mathematically, if the total weight is represented as \( W \) and the number of students is \( n \) , then the arithmetic mean \( \bar{x} \) is given by the formula: \[ \bar{x} = \frac{W}{n} \]
In the given problem, once we calculated the total weight of all 10 students, we divided it by 10 to find their mean weight.

Sequential Reasoning

Sequential reasoning involves following a sequence of steps to reach a logical conclusion. It's like assembling a puzzle where each step brings you closer to solving a problem. In mathematics, this means organizing information and procedures in the correct order.
For the problem of finding the mean weight after adding a new student, it's crucial to follow the steps correctly:

• First, calculate the total weight of the initial group.
• Add the weight of the new student.
• Then compute the new mean weight.

Each step builds upon the previous one to ensure an accurate result. Inconsistencies or errors in the sequence can lead to incorrect conclusions. Utilizing sequential reasoning helps us identify the right order, such as in this problem where the sequence is crucial to getting the right mean.

Mathematics Problem Solving

Mathematics problem solving requires a clear understanding of the concepts and a strategic approach to apply them. This involves identifying the problem, understanding the mathematical principles involved, and executing a plan to find a solution.
In this exercise, we had to understand the concept of mean and apply it to find the total weight first before calculating the new mean after a change in group composition. Our strategy involved breaking down the problem:

• First, find the total weight of the old group.
• Second, address the change caused by adding a new weight.
• Last, calculate the new mean.

By following the correct steps, we efficiently solved the problem. Problem-solving in mathematics is often about order, clarity, and execution.

Total Weight

The total weight refers to summing up individual weights in a group to get a cumulative figure. Calculating the total weight is an essential step in finding the mean or average weight of a group.
In the given problem, we initially calculated the total weight of the 9 students by multiplying the mean weight by the number of students: \( 9 \times 19 = 171 \mathrm{~kg} \).
Adding the weight of the new student (29 kg) resulted in the new total weight: \( 171 + 29 = 200 \mathrm{~kg} \).
Understanding the total weight helps us progress seamlessly to find an updated mean when there are changes in group composition. This is an indispensable part of comprehending how averages and total sums work together in arithmetic operations.