The average of 24 numbers is 43. If we subtract a number 'M' from each number, then the average becomes 38. What was the value of M?
5
Understanding the Average and the Effect of Subtraction
This problem involves calculating the value of a number 'M' that is subtracted from each number in a set, resulting in a change in the average of the set.
The average of a set of numbers is calculated by summing all the numbers and dividing by the count of numbers.
\text{Average} = \frac{\text{Sum of Numbers}}{\text{Number of Numbers}}
Analyzing the Given Information
- Initial number of values: 24
- Initial average: 43
- Action: A number 'M' is subtracted from each of the 24 numbers.
- New average: 38
Solving the Problem Using the Property of Averages
A key property of averages states that if a constant value is added to or subtracted from every number in a set, the average of the new set will be the original average plus or minus that constant value, respectively.
In this case, 'M' is subtracted from each number. Therefore, the new average is equal to the original average minus 'M'.
\text{New Average} = \text{Original Average} - \text{M}
We are given the original average (43) and the new average (38). We can substitute these values into the equation:
$$38 = 43 - M$$
Now, we need to solve for M. We can rearrange the equation to isolate M:
$$M = 43 - 38$$
Performing the subtraction:
$$M = 5$$
So, the value of M is 5.
Alternative Approach: Using the Sum
We can also solve this by calculating the sum of the numbers.
Step 1: Calculate the initial sum.
Using the formula: Sum = Average \(\times\) Number of Numbers
$$ \text{Initial Sum} = 43 \times 24 $$
$$ \text{Initial Sum} = 1032 $$
Step 2: Calculate the new sum.
The new average is 38, and there are still 24 numbers.
$$ \text{New Sum} = 38 \times 24 $$
$$ \text{New Sum} = 912 $$
Step 3: Relate the sums to M.
When M is subtracted from each of the 24 numbers, the total sum decreases by \(24 \times M\).
$$ \text{New Sum} = \text{Initial Sum} - (24 \times M) $$
Substitute the calculated sums:
$$ 912 = 1032 - (24 \times M) $$
Rearrange the equation to solve for \(24 \times M\):
$$ 24 \times M = 1032 - 912 $$
$$ 24 \times M = 120 $$
Now, solve for M:
$$ M = \frac{120}{24} $$
$$ M = 5 $$
Both methods confirm that the value of M is 5.
Revision Table: Key Concepts in Average
| Concept | Description | Formula/Property |
|---|---|---|
| Average (Mean) | The sum of all values divided by the number of values. | \( \text{Average} = \frac{\sum x}{n} \) |
| Effect of Adding/Subtracting a Constant | If a constant 'k' is added to (or subtracted from) each value, the new average is the old average plus (or minus) 'k'. | \( \text{New Avg} = \text{Old Avg} \pm k \) |
| Effect of Multiplying/Dividing by a Constant | If each value is multiplied (or divided) by a constant 'k' (k \(\neq\) 0), the new average is the old average multiplied (or divided) by 'k'. | \( \text{New Avg} = \text{Old Avg} \times k \) \( \text{New Avg} = \frac{\text{Old Avg}}{k} \) |
Additional Information: Understanding Changes in Data Sets
Understanding how operations on individual data points affect summary statistics like the average is crucial in data analysis and statistics.
- Consistent Operations: Adding or subtracting a constant from every number in a set shifts the entire data set uniformly along the number line. This uniform shift directly impacts the average by the same amount.
- Non-Consistent Operations: If different values are added or subtracted from different numbers in the set, the effect on the average is the total change in sum divided by the number of values.
- Impact on Other Statistics: While adding/subtracting a constant affects the mean (average), it does not change measures of dispersion like the range, variance, or standard deviation. Multiplying/dividing by a constant, however, affects both the mean and the dispersion measures.
This problem specifically used the property related to subtracting a constant, which simplifies the calculation significantly.
Important Questions from Average
The average of 28 numbers is 77. The average of first 14 numbers is 74 and the average of last 15 numbers is 84. If the 14 th number is excluded, then what is the average of remaining numbers? (correct to one decimal places)
24 students collected money for donation. The average contribution was Rs. 50. Later on, their teacher also contributed some money. Now the average contribution is Rs. 56. The teacher’s contribution is:
Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6 th number. If their average is 136, then the 6 th number is:
The average of five numbers is 30. If one number is excluded, then average becomes 31. What is the excluded number?
The average weight of 49 students in a class is 39 kg. Seven of them whose average weight is 40 kg leave the class and other seven students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?
