Problem 4 - A Complete Resource Book in Mathematics for JEE Main

Question

The mean weight of 9 items is 51 . If one more item is added to the series the mean becomes 16 . The value of the 10 th item is (A) 35 (B) 30 (C) 25 (D) 20

Step-by-Step Solution

Answer

The weight of the 10th item cannot logically be negative, implying possible misalignment in premise expectations.

Step 1: Calculate the Total Weight of 9 Items

The mean weight of 9 items is given as 51. The total weight of these 9 items can be found by multiplying the mean by the number of items. Therefore, the total weight is \( 51 \times 9 = 459 \).

Step 2: Calculate the New Total Weight with 10 Items

When the 10th item is added, the mean becomes 16. Thus, the total weight of all 10 items becomes \( 16 \times 10 = 160 \).

Step 3: Find the Weight of the 10th Item

The weight of the 10th item is the difference between the new total weight of the 10 items and the previous total weight of the 9 items. Therefore, the weight of the 10th item is \( 160 - 459 = -299 \). However, this indicates a mistake in interpretation as the mean should not reduce with more items added under typical conditions. Thus, the negative value indicates an inaccurate premise, illustrating either a miscalculation in expectations or an oversight in conditions.

Key Concepts

Weight Measurement

Understanding weight measurement is a foundational skill that is necessary for solving problems involving averages or means. In situations where we have multiple items or weights, each measured individually, it can be helpful to use statistical measures like the arithmetic mean to summarize data.
The arithmetic mean, or average, of a set of numbers is calculated by adding all the individual values together and dividing by the total number of values. This provides a single, central value that represents all the data points.
For example, if you have 9 items, each with their own weight, the arithmetic mean gives us an idea of what a typical weight might be. In our example, where the mean weight of 9 items is given as 51, this implies that if each item weighed the same, it would weigh 51 units. This is particularly useful for comparing different sets of data or identifying trends and patterns.

Total Weight Calculation

To understand the complete weight representation of a set of items or weights, calculating the total weight is essential. The total weight can be determined by multiplying the arithmetic mean by the number of items.
In the given problem, for 9 items with a mean weight of 51, the total weight calculation is straightforward:

• Multiply the mean (51) by the total number of items (9).
• This yields a total weight of 459 units for all 9 items combined.

When additional items are introduced, as in the problem when adding a 10th item, a new mean is calculated, which can affect this total.
For the new condition with a mean of 16 for 10 items, the total becomes 160 units.

• The change in total weight reflects the sum of values over both the original and new items added.

Understanding this basic concept allows students to track changes within a dataset, especially as they align with arithmetic mean and total adjustments.

Error Interpretation

Errors in calculations or logical deductions are potential pitfalls when dealing with data. This can occur, particularly in arithmetic problems involving averages and added quantities.
In our exercise, the weight of the 10th item being calculated as -299 is a prime example. Such a negative value signals an issue of interpretation rather than a mere computational mistake.
This may arise from a misunderstanding of how additional items typically impact a dataset. Normally, adding more items should either maintain or increase the mean, depending on the added weight.

• The negative outcome suggests incongruence with expected data trends.
• Potential missteps could include incorrect assumptions about item influence or miscalculating derived weights.

It's important to evaluate whether initial data, such as the mean values provided, aligns with logical and feasible responses. Identifying such issues helps solidify understanding, promoting critical thinking and the correct application of mathematical concepts.