The time taken by a person to cover 150km was 2.5 hours more than the time taken in return journey. If he returned at a speed of 10 km/hour more than the speed of going, what was the speed per hour in each direction?

Answer

To solve the problem step by step, we will define the variables and use the information given in the question.

### Step 1: Define the Variables
Let:
- \( x \) = speed of the person while going (in km/h).
- \( x + 10 \) = speed of the person while returning (in km/h).

### Step 2: Write the Time Equations
The distance covered in both directions is 150 km. The time taken to cover a distance is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So, the time taken to go is:
\[ \text{Time going} = \frac{150}{x} \]
And the time taken to return is:
\[ \text{Time returning} = \frac{150}{x + 10} \]

### Step 3: Set Up the Equation
According to the problem, the time taken to go is 2.5 hours more than the time taken to return:
\[ \frac{150}{x} = \frac{150}{x + 10} + 2.5 \]

### Step 4: Clear the Fractions
To eliminate the fractions, we can multiply through by \( x(x + 10) \):
\[ 150(x + 10) = 150x + 2.5x(x + 10) \]

### Step 5: Expand and Rearrange the Equation
Expanding both sides:
\[ 150x + 1500 = 150x + 2.5x^2 + 25x \]
Now, subtract \( 150x \) from both sides:
\[ 1500 = 2.5x^2 + 25x \]

### Step 6: Rearranging into Standard Quadratic Form
Rearranging gives us:
\[ 2.5x^2 + 25x - 1500 = 0 \]

### Step 7: Multiply to Eliminate Decimal
To eliminate the decimal, multiply the entire equation by 10:
\[ 25x^2 + 250x - 15000 = 0 \]

### Step 8: Simplify the Equation
Now, divide the entire equation by 25:
\[ x^2 + 10x - 600 = 0 \]

### Step 9: Factor the Quadratic Equation
We need to factor the quadratic equation:
\[ (x + 30)(x - 20) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 30 = 0 \) → \( x = -30 \) (not valid since speed cannot be negative)
2. \( x - 20 = 0 \) → \( x = 20 \)

### Step 11: Find the Speeds
Now we can find the speeds:
- Speed going = \( x = 20 \) km/h
- Speed returning = \( x + 10 = 30 \) km/h

### Final Answer
- Speed while going: **20 km/h**
- Speed while returning: **30 km/h**