B is thrice as efficient as A and C is 0.75 times as efficient as A. What is the ratio of number of days taken by A, B and C when they work individually?
A. `4:1:3`
B. `3:1:4`
C. `2:3:1`
D. `5:1:2`

Options

A. C
B. A
C. D
D. B

Answer

To solve the problem, we need to find the ratio of the number of days taken by A, B, and C when they work individually based on their efficiencies.

### Step 1: Define the efficiencies
Let's assume the efficiency of A is 100 units.

- Since B is thrice as efficient as A, the efficiency of B = 3 × 100 = 300 units.
- C is 0.75 times as efficient as A, so the efficiency of C = 0.75 × 100 = 75 units.

### Step 2: List the efficiencies
Now we have:
- Efficiency of A = 100
- Efficiency of B = 300
- Efficiency of C = 75

### Step 3: Find the number of days taken by each
The number of days taken to complete a work is inversely proportional to efficiency. Therefore, we can express the number of days taken by A, B, and C as follows:

- Days taken by A = Total work / Efficiency of A = 1 / 100
- Days taken by B = Total work / Efficiency of B = 1 / 300
- Days taken by C = Total work / Efficiency of C = 1 / 75

### Step 4: Calculate the ratio of days taken
To find the ratio of days taken by A, B, and C, we need to take the reciprocal of their efficiencies:

- Days taken by A = 1 / 100
- Days taken by B = 1 / 300
- Days taken by C = 1 / 75

Now, we can express the ratio of days taken as:
- Ratio = Days taken by A : Days taken by B : Days taken by C
- Ratio = (1/100) : (1/300) : (1/75)

### Step 5: Simplify the ratio
To simplify this ratio, we can find a common denominator. The least common multiple (LCM) of 100, 300, and 75 is 600.

Now we convert each term:
- Days taken by A = 600 / 100 = 6
- Days taken by B = 600 / 300 = 2
- Days taken by C = 600 / 75 = 8

Thus, the ratio of days taken by A, B, and C is:
- Ratio = 6 : 2 : 8

### Step 6: Further simplify the ratio
Now, we can simplify the ratio by dividing each term by 2:
- Ratio = 3 : 1 : 4

### Final Answer
The ratio of the number of days taken by A, B, and C when they work individually is **3:1:4**.