Problem 22
Question
The A.M. of a set of 50 numbers is 38 . If two numbers of the set, namely 55 and 45 are discarded, the A.M. of the remaining set of numbers is (A) \(38.5\) (B) \(37.5\) (C) \(36.5\) (D) 36
Step-by-Step Solution
Verified Answer
(B) 37.5
1Step 1: Understand the Problem
We are given a set of 50 numbers where the arithmetic mean (A.M.) is 38. Two numbers from this set, 55 and 45, are removed, and we need to find the new A.M. of the remaining 48 numbers.
2Step 2: Calculate the Original Sum
The arithmetic mean formula is \( \text{A.M.} = \frac{\text{Sum of all numbers}}{\text{Total number of values}} \). Given that the A.M. is 38 for 50 numbers, the total sum of these numbers is: \[50 \times 38 = 1900.\]
3Step 3: Subtract the Discarded Numbers
To find the sum of the remaining 48 numbers, subtract the two discarded numbers (55 and 45) from the original sum: \[1900 - 55 - 45 = 1800.\]
4Step 4: Calculate the New Arithmetic Mean
Now, divide the sum of the remaining numbers by the new total number of numbers to find the new A.M.: \[\frac{1800}{48} = 37.5.\]
5Step 5: Identify the Correct Answer
The new arithmetic mean of the numbers is 37.5. Therefore, the correct answer is (B) 37.5.
Key Concepts
Sum of NumbersDiscarded ValuesNew Set Calculation
Sum of Numbers
When dealing with the arithmetic mean (A.M.), the sum of numbers is a key component. The arithmetic mean is calculated using the formula \( \text{A.M.} = \frac{\text{Sum of all numbers}}{\text{Total number of values}} \). By rearranging this formula, you can find the sum if you know the A.M. and the total number of values. For example, if the A.M. of our 50-number set is 38, the sum of these numbers is 50 times 38, resulting in a total sum of 1900.
Understanding how to manipulate this basic formula can help you solve many problems involving averages. Knowing the sum allows you to reconstruct the effect of changes in the set, such as removing or adding numbers.
Remember, identifying the sum is often the first step in solving problems about averages or when comparing modified sets.
Understanding how to manipulate this basic formula can help you solve many problems involving averages. Knowing the sum allows you to reconstruct the effect of changes in the set, such as removing or adding numbers.
Remember, identifying the sum is often the first step in solving problems about averages or when comparing modified sets.
Discarded Values
Discarding values from a set means you are removing specific numbers. This affects both the sum and the total number count. In our exercise, the numbers 55 and 45 are discarded.
To adjust the sum of the set, subtract these discarded numbers from the original total. For instance, our original sum is 1900. Removing 55 and 45 changes the sum to 1900 - 55 - 45, which equals 1800.
This new sum will be used to calculate the new arithmetic mean with the reduced number of values.
The concept of discarded values is critical in understanding how averages change based on modifications to the data set. It is also useful in situations where you need to exclude outliers or erroneous data from a calculation.
To adjust the sum of the set, subtract these discarded numbers from the original total. For instance, our original sum is 1900. Removing 55 and 45 changes the sum to 1900 - 55 - 45, which equals 1800.
This new sum will be used to calculate the new arithmetic mean with the reduced number of values.
The concept of discarded values is critical in understanding how averages change based on modifications to the data set. It is also useful in situations where you need to exclude outliers or erroneous data from a calculation.
New Set Calculation
After understanding the changes caused by discarding values, calculating the new set is straightforward. With the updated sum and total number of values, you can find the new arithmetic mean.
For example, after removing two numbers from a 50-number set, we have 48 remaining numbers, and the sum is now 1800. The formula for arithmetic mean \( \text{A.M.} = \frac{\text{Sum}}{\text{Number of values}} \) is used again:
\( \frac{1800}{48} = 37.5 \). This gives us the new A.M.
This step shows the importance of recalculating when the set conditions change. Adjusting for discarded values is especially common in data analysis where certain data points may need to be excluded to ensure accuracy. The recalculated A.M. can reveal different insights from the data.
For example, after removing two numbers from a 50-number set, we have 48 remaining numbers, and the sum is now 1800. The formula for arithmetic mean \( \text{A.M.} = \frac{\text{Sum}}{\text{Number of values}} \) is used again:
\( \frac{1800}{48} = 37.5 \). This gives us the new A.M.
This step shows the importance of recalculating when the set conditions change. Adjusting for discarded values is especially common in data analysis where certain data points may need to be excluded to ensure accuracy. The recalculated A.M. can reveal different insights from the data.
Other exercises in this chapter
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