A,B and C have amounts in the ratio of 3:4:5. First B gives `1/4` th to A and `1/4` th to C then C gives `1/6` th to A. Find the final ratio of amount of A,B and C respectively

Options

A. `4:3:5`
B. `5:4:3`
C. `6:4:2`
D. `5:2:5`

Answer

To solve the problem step by step, we will first express the amounts of A, B, and C in terms of a variable based on their initial ratio, then we will calculate the changes in their amounts after the transactions, and finally, we will find the final ratio.

### Step 1: Define the initial amounts
Let the amounts of A, B, and C be represented as follows based on the ratio 3:4:5:
- Amount of A = 3x
- Amount of B = 4x
- Amount of C = 5x

### Step 2: Calculate the amount B gives to A and C
B gives `1/4` of his amount to A and `1/4` of his amount to C.
- Amount given by B to A = (1/4) * (4x) = x
- Amount given by B to C = (1/4) * (4x) = x

### Step 3: Update the amounts after B's transactions
After B gives x to A and x to C, the new amounts will be:
- Amount of A = 3x + x = 4x
- Amount of B = 4x - x - x = 2x
- Amount of C = 5x + x = 6x

### Step 4: Calculate the amount C gives to A
Now, C gives `1/6` of his amount to A.
- Amount given by C to A = (1/6) * (6x) = x

### Step 5: Update the amounts after C's transaction
After C gives x to A, the new amounts will be:
- Amount of A = 4x + x = 5x
- Amount of B = 2x (remains the same)
- Amount of C = 6x - x = 5x

### Step 6: Find the final ratio of A, B, and C
Now we have:
- Amount of A = 5x
- Amount of B = 2x
- Amount of C = 5x

The final ratio of A, B, and C is:
- A : B : C = 5x : 2x : 5x

### Step 7: Simplify the ratio
We can simplify the ratio by dividing each term by x:
- Final Ratio = 5 : 2 : 5

### Final Answer
The final ratio of amounts of A, B, and C is **5 : 2 : 5**.

---