Question

The mean weight per bag in a group of 6 bags of rice is \(119\text{ kg}\). The individual weights of 5 of them are \(115\text{ kg}\), \(109\text{ kg}\), \(129\text{ kg}\), \(117\text{ kg}\) and \(114\text{ kg}\). What is weight of the other bag of the group?

• A. \(129\text{ kg}\)
• B. \(130\text{ kg}\)
• C. \(131\text{ kg}\)
• D. \(132\text{ kg}\)

Hint:

To simplify the addition, you can use a deviation method.
Assume a baseline of 119. The deviations of the 5 bags from 119 are:
\(-4, -10, +10, -2, -5\).
The sum of these deviations is \(-11\).
For the average to remain 119, the deviation of the 6th bag must be \(+11\).
Thus, the 6th bag is \(119 + 11 = 130\text{ kg}\). This method is faster and less prone to arithmetic errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The arithmetic mean is defined as the sum of all observations divided by the total number of observations.
By rearranging this definition, we can find the sum of all values if the mean and the sample size are known.

Step 2: Key Formula or Approach:
The formula for the arithmetic mean of \(N\) observations is:
\[ \bar{X} = \frac{\sum_{i=1}^{N} X_i}{N} \]
This can be rewritten to find the total sum of the observations:
\[ \sum_{i=1}^{N} X_i = N \times \bar{X} \]

Step 3: Detailed Explanation:
We are given:
- Total number of bags, \(N = 6\)
- Mean weight of the 6 bags, \(\bar{X} = 119\text{ kg}\)
Using our formula, we find the total combined weight of all 6 bags:
\[ \text{Total Weight} = 6 \times 119 = 714\text{ kg} \]
We are given the individual weights of 5 of these bags:
- \(X_1 = 115\text{ kg}\)
- \(X_2 = 109\text{ kg}\)
- \(X_3 = 129\text{ kg}\)
- \(X_4 = 117\text{ kg}\)
- \(X_5 = 114\text{ kg}\)
Let us calculate the sum of the weights of these 5 known bags:
\[ \text{Sum of } 5 \text{ bags} = 115 + 109 + 129 + 117 + 114 \]
\[ \text{Sum of } 5 \text{ bags} = 584\text{ kg} \]
Let \(X_6\) represent the weight of the remaining 6th bag.
We write the sum of all 6 bags as:
\[ \text{Sum of } 5 \text{ bags} + X_6 = \text{Total Weight} \]
\[ 584 + X_6 = 714 \]
Solving for \(X_6\):
\[ X_6 = 714 - 584 \]
\[ X_6 = 130\text{ kg} \]
The weight of the other bag of the group is \(130\text{ kg}\), which corresponds to the second option.

Step 4: Final Answer:
Therefore, the correct option is (B).