To solve the problem step by step, we need to first determine how much A, B, and C were supposed to receive based on the intended ratio and then how much they actually received based on the mistaken ratio. Finally, we will calculate how much A gained from the error.
### Step 1: Determine the intended distribution
The intended ratio for A, B, and C is 3:4:5.
First, we calculate the total parts in the intended ratio:
- Total parts = 3 + 4 + 5 = 12
Next, we find the value of one part:
- Value of one part = Total amount / Total parts = 2820 / 12 = 235
Now, we calculate the amount each person should receive:
- Amount for A = 3 parts = 3 * 235 = 705
- Amount for B = 4 parts = 4 * 235 = 940
- Amount for C = 5 parts = 5 * 235 = 1175
### Step 2: Determine the mistaken distribution
The mistaken ratio for A, B, and C is given as \( \frac{1}{3} : \frac{1}{4} : \frac{1}{5} \).
To convert these fractions into a common ratio, we find the least common multiple (LCM) of the denominators (3, 4, and 5):
- LCM(3, 4, 5) = 60
Now, we convert each fraction:
- \( \frac{1}{3} = \frac{20}{60} \)
- \( \frac{1}{4} = \frac{15}{60} \)
- \( \frac{1}{5} = \frac{12}{60} \)
Thus, the mistaken ratio is 20:15:12.
Next, we calculate the total parts in the mistaken ratio:
- Total parts = 20 + 15 + 12 = 47
Now, we find the value of one part in the mistaken distribution:
- Value of one part = Total amount / Total parts = 2820 / 47 ≈ 60
Now, we calculate the amount each person actually received:
- Amount for A = 20 parts = 20 * 60 = 1200
- Amount for B = 15 parts = 15 * 60 = 900
- Amount for C = 12 parts = 12 * 60 = 720
### Step 3: Calculate A's gain
Now, we can find out how much A gained by the error:
- Gain for A = Amount received in mistaken distribution - Amount intended = 1200 - 705 = 495
Thus, A's gain by the error is **Rs. 495**.
### Summary of the Solution:
- Intended amount for A: Rs. 705
- Mistaken amount for A: Rs. 1200
- Gain for A: Rs. 495
