Each of two trains - A&B of different length can cross a pole in 5 seconds and when they are moving in same direction, train - A crosses train B in `28(1)/(3)` sec. If sum of their length is 0.85 km, then find the ratio of their length?

Options

A. A) `7:11`
B. B)`10:7`
C. C)`11:13`
D. D)`10:13`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define the lengths of the trains
Let the length of train A be \( L_A \) and the length of train B be \( L_B \).

### Step 2: Calculate the lengths using the time taken to cross a pole
Since both trains can cross a pole in 5 seconds, we can express their lengths in terms of their speeds:
- Length of train A:
\[ L_A = 5 \times V_A \]
- Length of train B:
\[ L_B = 5 \times V_B \]

### Step 3: Use the information about crossing each other
When the two trains are moving in the same direction, train A crosses train B in \( 28 \frac{1}{3} \) seconds, which is equivalent to \( \frac{85}{3} \) seconds. The total distance covered while crossing each other is the sum of their lengths:
\[ L_A + L_B = (V_A - V_B) \times \frac{85}{3} \]

### Step 4: Substitute the sum of lengths
We know from the problem that:
\[ L_A + L_B = 0.85 \text{ km} = 850 \text{ m} \]
Thus, we can substitute this into our equation:
\[ 850 = (V_A - V_B) \times \frac{85}{3} \]

### Step 5: Solve for the difference in speeds
Rearranging the equation gives us:
\[ V_A - V_B = \frac{850 \times 3}{85} = 30 \text{ m/s} \]

### Step 6: Calculate the difference in lengths
From the lengths we defined earlier:
\[ L_A - L_B = 5V_A - 5V_B = 5(V_A - V_B) = 5 \times 30 = 150 \text{ m} \]

### Step 7: Set up the equations for lengths
Now we have two equations:
1. \( L_A + L_B = 850 \)
2. \( L_A - L_B = 150 \)

### Step 8: Solve the equations
Adding these two equations:
\[ (L_A + L_B) + (L_A - L_B) = 850 + 150 \]
\[ 2L_A = 1000 \implies L_A = \frac{1000}{2} = 500 \text{ m} \]

Now substituting \( L_A \) back into the first equation:
\[ 500 + L_B = 850 \implies L_B = 850 - 500 = 350 \text{ m} \]

### Step 9: Find the ratio of lengths
The ratio of the lengths of train A to train B is:
\[ \frac{L_A}{L_B} = \frac{500}{350} = \frac{10}{7} \]

### Final Answer
The ratio of the lengths of train A to train B is \( 10:7 \).

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