To solve the problem step by step, we will follow these instructions:
### Step 1: Define the shares of A, B, and C
Let the original shares of A, B, and C be represented as:
- A's share = x
- B's share = y
- C's share = z
### Step 2: Set up the equation for total shares
According to the problem, the total amount shared among A, B, and C is Rs. 3780. Therefore, we can write:
\[
x + y + z = 3780
\]
### Step 3: Adjust the shares based on the problem's conditions
The problem states that if A's share is decreased by Rs. 130, B's share by Rs. 150, and C's share by Rs. 200, the new shares will be in the ratio of 5:2:4. Thus, we can express the new shares as:
- A's new share = \( x - 130 \)
- B's new share = \( y - 150 \)
- C's new share = \( z - 200 \)
### Step 4: Set up the ratio equation
According to the ratio given (5:2:4), we can express the new shares in terms of a variable k:
\[
\frac{x - 130}{5} = \frac{y - 150}{2} = \frac{z - 200}{4} = k
\]
From this, we can derive three equations:
1. \( x - 130 = 5k \)
2. \( y - 150 = 2k \)
3. \( z - 200 = 4k \)
### Step 5: Express x, y, and z in terms of k
Rearranging the equations gives us:
1. \( x = 5k + 130 \)
2. \( y = 2k + 150 \)
3. \( z = 4k + 200 \)
### Step 6: Substitute into the total shares equation
Now, substitute these expressions into the total shares equation:
\[
(5k + 130) + (2k + 150) + (4k + 200) = 3780
\]
This simplifies to:
\[
11k + 480 = 3780
\]
### Step 7: Solve for k
Subtract 480 from both sides:
\[
11k = 3780 - 480
\]
\[
11k = 3300
\]
Now, divide by 11:
\[
k = \frac{3300}{11} = 300
\]
### Step 8: Find the original shares
Now that we have k, we can find the original shares:
1. \( x = 5(300) + 130 = 1500 + 130 = 1630 \)
2. \( y = 2(300) + 150 = 600 + 150 = 750 \)
3. \( z = 4(300) + 200 = 1200 + 200 = 1400 \)
### Step 9: Conclusion
The original share of C is:
\[
\text{C's share} = z = 1400
\]
Therefore, the original share of C is **Rs. 1400**.
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