A man travels from Point P to Qat 90 km/hr and from Qto R at 60 km/hr. The total distance between P to R is 200 km. If his average speed is 75 km/hr then find the distance between P and Q.

Options

A. 80 km
B. 120 km
C. 100 km
D. 150 km

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the Variables
Let the distance from Point P to Q be \( x \) km. Therefore, the distance from Q to R will be \( 200 - x \) km since the total distance from P to R is 200 km.

**Hint:** Define your variables clearly to represent the unknown distances.

### Step 2: Calculate Time Taken for Each Segment
The time taken to travel from P to Q at a speed of 90 km/hr is given by:
\[ \text{Time from P to Q} = \frac{x}{90} \]
The time taken to travel from Q to R at a speed of 60 km/hr is given by:
\[ \text{Time from Q to R} = \frac{200 - x}{60} \]

**Hint:** Use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to find the time for each segment.

### Step 3: Set Up the Average Speed Equation
The average speed for the entire journey is given as 75 km/hr. The formula for average speed is:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Thus, we can set up the equation:
\[ 75 = \frac{200}{\frac{x}{90} + \frac{200 - x}{60}} \]

**Hint:** Remember that average speed is total distance divided by total time.

### Step 4: Simplify the Equation
First, calculate the total time:
\[ \text{Total Time} = \frac{x}{90} + \frac{200 - x}{60} \]
To combine these fractions, find a common denominator (which is 180):
\[ \text{Total Time} = \frac{2x}{180} + \frac{3(200 - x)}{180} = \frac{2x + 600 - 3x}{180} = \frac{600 - x}{180} \]
Now substitute this back into the average speed equation:
\[ 75 = \frac{200}{\frac{600 - x}{180}} \]

**Hint:** Finding a common denominator helps in simplifying the fractions.

### Step 5: Cross Multiply to Solve for \( x \)
Cross multiplying gives:
\[ 75 \cdot \frac{600 - x}{180} = 200 \]
This simplifies to:
\[ 75(600 - x) = 200 \cdot 180 \]
Calculating the right side:
\[ 75(600 - x) = 36000 \]

**Hint:** Cross-multiplication is a powerful tool to eliminate fractions.

### Step 6: Expand and Solve for \( x \)
Expanding the left side:
\[ 45000 - 75x = 36000 \]
Now, isolate \( x \):
\[ 45000 - 36000 = 75x \]
\[ 9000 = 75x \]
\[ x = \frac{9000}{75} = 120 \]

**Hint:** Isolate the variable to find its value.

### Conclusion
The distance between Point P and Q is \( 120 \) km.

**Final Answer:** The distance between P and Q is **120 km**.