Three machines, A, B and C can be used to produce a product. Machine A will take 60 hours to produce a million units. Machine B is twice as fast as Machine A. Machine C will take the same amount of time to produce a million units as A and B running together. How much time will be required to produce a million units if all the three machines are used simultaneously?

Options

A. 12 hours
B. 10 hours
C. 8 hours
D. 6 hours

Answer

To solve the problem step by step, let's break down the information given and calculate the required time when all three machines are used simultaneously.

### Step 1: Determine the time taken by Machine A
Machine A takes 60 hours to produce a million units.

### Step 2: Determine the time taken by Machine B
Machine B is twice as fast as Machine A. Therefore, the time taken by Machine B can be calculated as:
\[ \text{Time taken by Machine B} = \frac{60 \text{ hours}}{2} = 30 \text{ hours} \]

### Step 3: Determine the combined efficiency of Machines A and B
To find out how long it takes for Machines A and B to produce a million units together, we first calculate their combined work rate:
- Machine A's work rate: \(\frac{1}{60}\) (million units per hour)
- Machine B's work rate: \(\frac{1}{30}\) (million units per hour)

Now, we add their work rates:
\[ \text{Combined work rate of A and B} = \frac{1}{60} + \frac{1}{30} \]
To add these fractions, we need a common denominator:
\[ \frac{1}{60} + \frac{1}{30} = \frac{1}{60} + \frac{2}{60} = \frac{3}{60} = \frac{1}{20} \]
This means that A and B together can produce a million units in 20 hours.

### Step 4: Determine the time taken by Machine C
Machine C takes the same amount of time to produce a million units as Machines A and B running together, which we calculated to be 20 hours.

### Step 5: Calculate the combined efficiency of Machines A, B, and C
Now we need to find the time taken when all three machines are working together. The work rates of the three machines are:
- Machine A: \(\frac{1}{60}\)
- Machine B: \(\frac{1}{30}\)
- Machine C: \(\frac{1}{20}\)

Now, we add these work rates:
\[ \text{Combined work rate of A, B, and C} = \frac{1}{60} + \frac{1}{30} + \frac{1}{20} \]
To add these fractions, we need a common denominator. The least common multiple of 60, 30, and 20 is 60:
\[ \frac{1}{60} + \frac{2}{60} + \frac{3}{60} = \frac{1 + 2 + 3}{60} = \frac{6}{60} = \frac{1}{10} \]

### Step 6: Calculate the total time taken when all machines are used together
If the combined work rate of all three machines is \(\frac{1}{10}\), then the time taken \(X\) to produce a million units is:
\[ \frac{1}{X} = \frac{1}{10} \implies X = 10 \text{ hours} \]

### Final Answer
The time required to produce a million units if all three machines are used simultaneously is **10 hours**.

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