`(1)/(10)` of a rod is coloured red, `(1)/(20)` orange,`(1)/(30)` yellow, `(1)/(40)` green `(1)/(50)`, blue , `(1)/(60)` black and the rest is violet. If the length of the violet portion of the rod is 12.08 metres, then the length of the rod is

Options

A. 16 m
B. 18 m
C. 20 m
D. 30 m

Answer

To solve the problem, we need to find the total length of the rod based on the given colored portions and the length of the violet portion. Let's break it down step by step.

### Step 1: Define the Length of the Rod
Let the total length of the rod be \( L \) meters.

### Step 2: Calculate the Colored Portions
The colored portions of the rod are as follows:
- Red: \( \frac{1}{10}L \)
- Orange: \( \frac{1}{20}L \)
- Yellow: \( \frac{1}{30}L \)
- Green: \( \frac{1}{40}L \)
- Blue: \( \frac{1}{50}L \)
- Black: \( \frac{1}{60}L \)

### Step 3: Find the Total Fraction of Colored Portions
To find the total fraction of the rod that is colored, we need to add all these fractions together. First, we need to find a common denominator for these fractions. The least common multiple (LCM) of the denominators (10, 20, 30, 40, 50, 60) is 600.

Now, we convert each fraction:
- Red: \( \frac{1}{10} = \frac{60}{600} \)
- Orange: \( \frac{1}{20} = \frac{30}{600} \)
- Yellow: \( \frac{1}{30} = \frac{20}{600} \)
- Green: \( \frac{1}{40} = \frac{15}{600} \)
- Blue: \( \frac{1}{50} = \frac{12}{600} \)
- Black: \( \frac{1}{60} = \frac{10}{600} \)

Adding these fractions together:
\[ \frac{60 + 30 + 20 + 15 + 12 + 10}{600} = \frac{147}{600} \]

### Step 4: Calculate the Fraction of the Violet Portion
The fraction of the rod that is violet is the remainder after subtracting the colored portions from 1:
\[ \text{Violet Portion} = 1 - \frac{147}{600} = \frac{600 - 147}{600} = \frac{453}{600} \]

### Step 5: Relate Violet Portion to Length
We know that the length of the violet portion is given as 12.08 meters. Therefore, we can set up the equation:
\[ \frac{453}{600}L = 12.08 \]

### Step 6: Solve for \( L \)
To find \( L \), we rearrange the equation:
\[ L = 12.08 \times \frac{600}{453} \]

Calculating this:
\[ L = 12.08 \times \frac{600}{453} \approx 16 \text{ meters} \]

### Conclusion
The total length of the rod is approximately **16 meters**.

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