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Let's analyze the problem step-by-step.

### Problem Statement:
The question states:
1. A boat travels \( x \) km downstream and \( (x - 40) \) km upstream.
2. The time taken to travel \( x \) km downstream is the same as the time taken to travel \( (x - 40) \) km upstream.
3. The downstream and upstream speeds of the boat are in the ratio of 5:4.
4. We need to find the value of \( x \).

### Solution:
Let's denote:
- \( V_d \) as the downstream speed.
- \( V_u \) as the upstream speed.
- The downstream and upstream speeds are in the ratio of \( 5 : 4 \). Thus, we can write:
  \[
  \frac{V_d}{V_u} = \frac{5}{4}
  \]
  So, let \( V_d = 5k \) and \( V_u = 4k \), where \( k \) is some constant.

#### Step 1: Time Equations
Since time taken is distance divided by speed, the time taken to go downstream is:
\[
\text{Time}_{\text{downstream}} = \frac{x}{V_d} = \frac{x}{5k}
\]
The time taken to go upstream is:
\[
\text{Time}_{\text{upstream}} = \frac{x - 40}{V_u} = \frac{x - 40}{4k}
\]

According to the problem, these times are equal:
\[
\frac{x}{5k} = \frac{x - 40}{4k}
\]

#### Step 2: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[
4k \cdot x = 5k \cdot (x - 40)
\]
\[
4x = 5x - 200
\]
Bringing like terms together:
\[
4x - 5x = -200
\]
\[
-x = -200
\]
\[
x = 200
\]

### Answer:
The value of \( x \) is **200**.

### Verification:
1. **Downstream Distance:** \( x = 200 \) km
2. **Upstream Distance:** \( x - 40 = 160 \) km
3. **Downstream Speed:** \( V_d = 5k \)
4. **Upstream Speed:** \( V_u = 4k \)

The times taken for downstream and upstream would indeed match, confirming \( x = 200 \) as correct.

Thus, the answer is:
**(d) 200**

Would you like a further breakdown or have any other questions?

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Here are 5 related questions for additional practice:
1. How would the solution change if the downstream and upstream speed ratio were 3:2?
2. If the downstream distance was different from the upstream distance, how would you set up the equation?
3. How does changing the downstream speed affect the time comparison?
4. If both distances were equal, would the problem still hold true?
5. What if the time taken for downstream travel was less than upstream travel?

**Tip:** Always check ratios given in word problems as they often give hints for variable substitution.

The time taken by a boat to go 'x' km downstream is equal to that time taken by the boat to go (x - 40) km upstream. If the ratio between the downstream and upstream speed of boat is 5:4, respectively, then find the value of x.
Options:
a) 240
b) 160
c) 180
d) 200
e) 150

Solve a boat and stream problem where the downstream and upstream speeds of a boat are in a 5:4 ratio. Given that the time taken downstream equals the time taken upstream for different distances, learn how to calculate the distance using algebra and proportion.

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