Eight persons with an average weight of 49kg were in a lift. Two more persons Anita and Kamal boarded the lift and the average weight of the person in the lift increased by 2.8kg. What is the average weight of Anita and Kamal?
(d) 63 kg
Calculating Average Weight of Persons in a Lift
This problem involves calculating the average weight of two new persons who join a group, causing the overall average weight to change. We will first find the total weight of the initial group, then the total weight of the group after the new persons join, and finally use these totals to find the combined weight and average weight of the new persons.
Understanding the Initial State
Initially, there were 8 persons in the lift. The average weight of these 8 persons was 49 kg.
The formula for average is: $\text{Average} = \frac{\text{Total Sum}}{\text{Number of Items}}$
From this, we can find the total weight of the initial 8 persons:
$\text{Total Weight} = \text{Average Weight} \times \text{Number of Persons}$
Initial Total Weight: $8 \text{ persons} \times 49 \text{ kg/person} = 392 \text{ kg}$.
Analyzing the State After New Persons Board
Two more persons, Anita and Kamal, boarded the lift. This means the total number of persons in the lift is now:
New Number of Persons = Initial Persons + New Persons = $8 + 2 = 10$ persons.
The average weight of the persons in the lift increased by 2.8 kg. The new average weight is:
New Average Weight = Original Average Weight + Increase in Average Weight
New Average Weight = $49 \text{ kg} + 2.8 \text{ kg} = 51.8 \text{ kg}$.
Now, we can find the total weight of all 10 persons in the lift:
New Total Weight = New Average Weight $\times$ New Number of Persons
New Total Weight = $51.8 \text{ kg/person} \times 10 \text{ persons} = 518 \text{ kg}$.
Calculating the Combined Weight of Anita and Kamal
The difference between the new total weight and the initial total weight is the combined weight of the two new persons, Anita and Kamal.
Combined Weight of Anita and Kamal = New Total Weight - Initial Total Weight
Combined Weight of Anita and Kamal = $518 \text{ kg} - 392 \text{ kg} = 126 \text{ kg}$.
Determining the Average Weight of Anita and Kamal
We know the combined weight of Anita and Kamal is 126 kg, and there are 2 of them. To find their average weight, we divide their combined weight by the number of persons (which is 2).
Average Weight of Anita and Kamal = $\frac{\text{Combined Weight of Anita and Kamal}}{\text{Number of New Persons}}$
Average Weight of Anita and Kamal = $\frac{126 \text{ kg}}{2 \text{ persons}} = 63 \text{ kg}$.
So, the average weight of Anita and Kamal is 63 kg.
Summary of Calculations
| Description | Calculation | Result |
|---|---|---|
| Initial Number of Persons | 8 | |
| Initial Average Weight | 49 kg | |
| Initial Total Weight | $8 \times 49$ kg | 392 kg |
| New Number of Persons | $8 + 2$ | 10 |
| New Average Weight | $49 + 2.8$ kg | 51.8 kg |
| New Total Weight | $10 \times 51.8$ kg | 518 kg |
| Combined Weight of Anita & Kamal | $518 - 392$ kg | 126 kg |
| Average Weight of Anita & Kamal | $\frac{126}{2}$ kg | 63 kg |
Conclusion on Average Weight
Based on our calculations, the average weight of Anita and Kamal is 63 kg.
Revision Table: Key Concepts in Average Calculation
| Concept | Definition | Formula |
|---|---|---|
| Average (Mean) | A measure of central tendency; the sum of a set of values divided by the number of values. | $\frac{\text{Sum of all values}}{\text{Number of values}}$ |
| Total Sum | The sum of all values in a set. | Average $\times$ Number of values |
| Change in Average | The difference between a new average and an old average, usually due to adding or removing items. | New Average - Old Average |
Additional Information: Working with Averages
Understanding how averages change when new items are added or removed is a common type of problem in quantitative aptitude.
- When items are added: If the added items have a value greater than the current average, the average will increase. If their value is less than the current average, the average will decrease. If their value is exactly equal to the current average, the average will remain the same.
- In this problem, the average increased (from 49 kg to 51.8 kg). This indicates that the average weight of the two new persons (Anita and Kamal) must be greater than the original average of 49 kg. Our calculated average of 63 kg is indeed greater than 49 kg, which aligns with this concept.
- When solving problems involving changes in average, always calculate the total sum before and after the change. This is often the most straightforward way to find the sum or average of the items causing the change.
- Remember that the formula for average can be rearranged to find the total sum if you know the average and the number of items: Total Sum = Average $\times$ Number of Items.
Important Questions from Average
The average salary of employees of a factory is ₹15,000. The average salary of 250 of the employees is ₹16,000 and that of the remaining employees is ₹13,750. Total number of employees in the factory is:
Find the mean of prime numbers between 1 and 30.
If mode and mean of a data are 18 and 21 respectively, then median of the data is
The average of the numbers 5, 3, 9, 11, 29 and (p+1) is 12. The average increases by 2 when the numbers (q – 5) and (q +11) are also included. Find the value of q.
In a particular week, the average earning per day of a plumber from Monday to Wednesday remained ₹580 and from Thursday to Saturday it was ₹612. If the average earning for the whole week was ₹675, then how much did he earn on Sunday?
