The average weight of 15 persons is increased by 1.2 kg when one of them whose weight is 51 kg is replaced by a new man. The weight of the new man is:
69 kg
This problem involves calculating the weight of a new person who replaces someone in a group, causing a change in the average weight of the group.
Understanding the Average Weight Concept
The average weight of a group of people is calculated by dividing the total weight of all individuals by the number of individuals in the group.
Average Weight $= \frac{\text{Total Weight}}{\text{Number of Persons}}$
If the average weight increases when one person is replaced by another, it means the weight of the new person is greater than the weight of the person who left. The increase in the total weight of the group is distributed among all members, causing the average to rise.
Analyzing the Given Information
- Initial number of persons: 15
- Weight of the person who left: 51 kg
- Increase in average weight: 1.2 kg
- We need to find the weight of the new man.
Calculating the Total Change in Weight
The increase in average weight is due to the difference between the new person's weight and the old person's weight. This difference adds to the total weight of the group. Since this increased total weight is then divided among all 15 persons to get the new average, the total increase in weight is the increase per person multiplied by the number of persons.
Increase in Total Weight = Increase in Average Weight $\times$ Number of Persons
Increase in Total Weight $= 1.2 \text{ kg/person} \times 15 \text{ persons}$
Increase in Total Weight $= 18 \text{ kg}$
Determining the Weight of the New Man
The increase in the total weight (18 kg) is the exact difference between the weight of the new man and the weight of the man who left.
Increase in Total Weight = Weight of New Man - Weight of Old Man
So, Weight of New Man = Weight of Old Man + Increase in Total Weight
Weight of New Man $= 51 \text{ kg} + 18 \text{ kg}$
Weight of New Man $= 69 \text{ kg}$
Therefore, the weight of the new man is 69 kg.
Verification
Let's assume the initial average weight was $A$ kg. The initial total weight was $15 \times A$.
When the 51 kg person leaves and the new man (let's call his weight $W$) joins, the new total weight is $15A - 51 + W$.
The new average weight is $(15A - 51 + W) / 15$.
We are told the new average is $A + 1.2$.
So, $\frac{15A - 51 + W}{15} = A + 1.2$
$15A - 51 + W = 15(A + 1.2)$
$15A - 51 + W = 15A + 15 \times 1.2$
$15A - 51 + W = 15A + 18$
Subtract $15A$ from both sides:
$-51 + W = 18$
$W = 18 + 51$
$W = 69$
This confirms our calculated weight of 69 kg for the new man.
Revision Table: Average Weight Concepts
| Concept | Formula/Explanation |
|---|---|
| Average | Sum of values / Number of values |
| Total Sum | Average $\times$ Number of values |
| Change in Average | Change in Total Sum / Number of values |
| Change in Total Sum (Replacement) | Weight of New Person - Weight of Old Person |
Additional Information: Solving Replacement Problems
Replacement problems in averages are common. The key is to understand that the change in the total sum is directly related to the difference between the value of the item or person leaving and the value of the item or person joining. This change in total sum is then distributed among all items/persons to affect the average.
- If the average increases, the new item/person has a higher value than the one replaced.
- If the average decreases, the new item/person has a lower value than the one replaced.
- The total change in sum is equal to the change in average multiplied by the total number of items/persons (which remains constant in a simple replacement).
Using the formula: Weight of New Person = Weight of Old Person + (Change in Average $\times$ Number of Persons) is a quick way to solve such problems.
If the average had decreased, the formula would be: Weight of New Person = Weight of Old Person - (Decrease in Average $\times$ Number of Persons).
Similar Questions
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Important Questions from Average
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