A person takes 1 h more than that of his friend to reach a party. The distance travelled by the person is 20 km more than that of his friend. Also given that the speed of the persons is 16 km/h , whereas that of his friend is 15 km/h , find the distance travelled by his friend to reach the party

Options

A. 60 km
B. 50 km
C. 80 km
D. 30 km

Answer

To solve the problem step by step, let's break down the information given and set up the equations accordingly.

### Step 1: Define Variables
Let the distance travelled by the friend be \( x \) km.
Then, the distance travelled by the person will be \( x + 20 \) km (since he travels 20 km more than his friend).

**Hint:** Define variables for unknowns to make equations easier to set up.

### Step 2: Write the Time Equations
The time taken by the friend can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For the friend:
\[ \text{Time}_{\text{friend}} = \frac{x}{15} \quad \text{(since speed of friend is 15 km/h)} \]
For the person:
\[ \text{Time}_{\text{person}} = \frac{x + 20}{16} \quad \text{(since speed of person is 16 km/h)} \]

**Hint:** Use the formula for time to relate distance and speed.

### Step 3: Set Up the Equation
According to the problem, the person takes 1 hour more than his friend to reach the party. Thus, we can set up the equation:
\[ \text{Time}_{\text{person}} = \text{Time}_{\text{friend}} + 1 \]
Substituting the time equations:
\[ \frac{x + 20}{16} = \frac{x}{15} + 1 \]

**Hint:** Set up an equation based on the relationship between the times.

### Step 4: Clear the Fractions
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators, which is 240:
\[ 240 \left(\frac{x + 20}{16}\right) = 240 \left(\frac{x}{15} + 1\right) \]
This simplifies to:
\[ 15(x + 20) = 16x + 240 \]

**Hint:** Multiplying by the LCM helps to simplify calculations by removing fractions.

### Step 5: Expand and Rearrange
Expanding the left side:
\[ 15x + 300 = 16x + 240 \]
Now, rearranging the equation:
\[ 15x + 300 - 240 = 16x \]
This simplifies to:
\[ 15x + 60 = 16x \]
Subtracting \( 15x \) from both sides gives:
\[ 60 = x \]

**Hint:** Rearranging helps isolate the variable.

### Step 6: Conclusion
Thus, the distance travelled by the friend is \( x = 60 \) km.

**Final Answer:** The distance travelled by his friend to reach the party is **60 km**.

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