The weight of 9 members out of 22 are measured consecutively and their average weight is calculate after each member is weighted. If the average weight increases by 1 kg each time, how much heavier is the 9th person than the first one ?

Options

A. 16 kg
B. 18 kg
C. 20 kg
D. 44 kg

Answer

To solve the problem, we will denote the weight of the first member as \( W_1 \) and the weight of the 9th member as \( W_9 \). We know that the average weight increases by 1 kg each time a new member is weighed.

### Step-by-step solution:

1. **Understanding the Average Weight Calculation:**
The average weight after weighing the first member is simply \( W_1 \).

2. **Calculating the Average After Each Member:**
- After weighing the first member, the average is \( W_1 \).
- After weighing the second member, the average becomes \( \frac{W_1 + W_2}{2} \).
- After weighing the third member, the average is \( \frac{W_1 + W_2 + W_3}{3} \).
- Continuing this way, after weighing the 9th member, the average is \( \frac{W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 + W_9}{9} \).

3. **Setting Up the Average Weight Increases:**
According to the problem, the average weight increases by 1 kg after each member is weighed. Therefore, we can express the average weights as follows:
- After the first member: \( W_1 \)
- After the second member: \( W_1 + 1 \)
- After the third member: \( W_1 + 2 \)
- ...
- After the ninth member: \( W_1 + 8 \)

4. **Setting Up the Equation for the 9th Member:**
The average after the 9th member can also be expressed as:
\[ \frac{W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 + W_9}{9} = W_1 + 8 \]

5. **Multiplying Both Sides by 9:**
\[ W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 + W_9 = 9(W_1 + 8) \]
\[ W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 + W_9 = 9W_1 + 72 \]

6. **Rearranging the Equation:**
Now, we can rearrange the equation to find the total weight of the 9 members:
\[ W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 + W_9 = 8W_1 + 72 \]

7. **Finding the Weight of the 9th Member:**
We know that the average weight after the 8th member is \( W_1 + 7 \):
\[ \frac{W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8}{8} = W_1 + 7 \]
Multiplying by 8:
\[ W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 = 8W_1 + 56 \]
Now, substituting this back into the previous equation:
\[ 8W_1 + 56 + W_9 = 9W_1 + 72 \]
Simplifying gives:
\[ W_9 = W_1 + 16 \]

8. **Calculating the Difference:**
The difference between the weight of the 9th member and the first member is:
\[ W_9 - W_1 = (W_1 + 16) - W_1 = 16 \text{ kg} \]

### Final Answer:
The 9th person is 16 kg heavier than the first person.