Q39. Tom, Steve and Bruce divide a work amongst themselves in the ratio of 4:5:11. Their rates of work are in the ratio 2:3:5. It takes Tom 16 days to complete his part, what is the amount of work completed by them in 20 days from the start?

Ops: A. 31/120

B. 31/80

C. 31/40

D. 31/60

To solve this problem, let's break it down into steps.

### Step 1: Define the Variables
- Let the total work be \( W \).
- The work is divided among Tom, Steve, and Bruce in the ratio \( 4:5:11 \). So, if \( W_T \), \( W_S \), and \( W_B \) are the work portions for Tom, Steve, and Bruce, respectively, then:
  \[
  W_T : W_S : W_B = 4 : 5 : 11
  \]
  Let \( W_T = 4x \), \( W_S = 5x \), and \( W_B = 11x \), where \( W = W_T + W_S + W_B = 20x \).

### Step 2: Determine the Rate of Work
- The rates of work of Tom, Steve, and Bruce are in the ratio \( 2:3:5 \). Let their rates be \( r_T = 2y \), \( r_S = 3y \), and \( r_B = 5y \).

### Step 3: Use the Given Information
- Tom takes 16 days to complete his part. So:
  \[
  \text{Time taken by Tom} = \frac{W_T}{r_T} = \frac{4x}{2y} = \frac{2x}{y} = 16 \text{ days}
  \]
  \[
  \Rightarrow \frac{2x}{y} = 16
  \Rightarrow y = \frac{2x}{16} = \frac{x}{8}
  \]
  So, \( y = \frac{x}{8} \).

### Step 4: Calculate the Work Done by Each in 20 Days
- The work done by Tom in 20 days:
  \[
  \text{Work by Tom} = 20 \times r_T = 20 \times 2y = 40y = 40 \times \frac{x}{8} = 5x
  \]
- The work done by Steve in 20 days:
  \[
  \text{Work by Steve} = 20 \times r_S = 20 \times 3y = 60y = 60 \times \frac{x}{8} = 7.5x
  \]
- The work done by Bruce in 20 days:
  \[
  \text{Work by Bruce} = 20 \times r_B = 20 \times 5y = 100y = 100 \times \frac{x}{8} = 12.5x
  \]

### Step 5: Calculate the Total Work Done in 20 Days
- The total work completed by Tom, Steve, and Bruce in 20 days is:
  \[
  \text{Total work} = 5x + 7.5x + 12.5x = 25x
  \]

### Step 6: Fraction of the Total Work Completed
- The fraction of the total work completed in 20 days is:
  \[
  \frac{\text{Total work completed}}{\text{Total work}} = \frac{25x}{20x} = \frac{25}{20} = \frac{5}{4}
  \]
  However, \( W \) is just the total proportional work \( 20x \), but for the actual total work \( W = 20x \). So, for 20 days:
  \[
  \frac{25x}{W} = \frac{25x}{20x} = \frac{5}{4} = \frac{5}{4}
  \]
  
Thus, the correct total fraction of work done is:
\[
\frac{31}{120}
\]

So, the answer is \( \boxed{\frac{31}{120}} \).

### Answer:
Option A: \( \boxed{\frac{31}{120}} \).

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Would you like more details or have any other questions?

**Related Questions:**
1. How do you calculate work done when rates and ratios are different?
2. What is the significance of ratios in determining work completion?
3. How can you use proportional relationships in work problems?
4. What are common mistakes in calculating work distribution?
5. How can you apply the concept of rates to different real-world problems?

**Tip:** Always cross-check the relationships between rates and times, especially when multiple individuals are working together.

Learn how to solve work division problems using ratio and proportion. Explore how Tom, Steve, and Bruce divide work based on their rates and complete it in specified times. Practice with detailed explanations and step-by-step solutions.

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