Two cycles start from a house at interval of 10 minutes and travel with a speed of 6km/hr. With how much speed (km/hr.) should a woman coming from the opposite direction towards the house travel, to meet the cycles at an interval of 8 minutes?

Options

A. 1.7
B. 1.6
C. 1.5
D. 1.4

Answer

To solve the problem, we need to determine the speed at which a woman should travel to meet two cycles coming from the opposite direction at a specific interval. Here’s a step-by-step solution:

### Step 1: Determine the distance traveled by the cycles in 10 minutes
The speed of the cycles is given as 6 km/hr.

To find the distance traveled in 10 minutes, we convert 10 minutes into hours:
\[ 10 \text{ minutes} = \frac{10}{60} \text{ hours} = \frac{1}{6} \text{ hours} \]

Now, we can calculate the distance:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 6 \text{ km/hr} \times \frac{1}{6} \text{ hr} = 1 \text{ km} \]

### Step 2: Set up the equation for the woman's speed
Let the speed of the woman be \( x \) km/hr. When the woman travels towards the cycles, the effective speed at which they approach each other is the sum of their speeds:
\[ \text{Effective Speed} = x + 6 \text{ km/hr} \]

### Step 3: Determine the time taken to meet the cycles
The woman meets the cycles at an interval of 8 minutes. We convert 8 minutes into hours:
\[ 8 \text{ minutes} = \frac{8}{60} \text{ hours} = \frac{2}{15} \text{ hours} \]

### Step 4: Use the formula for distance
The distance between the woman and the cycles is 1 km (as calculated in Step 1). We can use the formula:
\[ \text{Distance} = \text{Effective Speed} \times \text{Time} \]
Substituting the known values:
\[ 1 \text{ km} = (x + 6) \text{ km/hr} \times \frac{2}{15} \text{ hr} \]

### Step 5: Solve for \( x \)
Now we can rearrange the equation:
\[ 1 = (x + 6) \times \frac{2}{15} \]
Multiplying both sides by 15:
\[ 15 = 2(x + 6) \]
Expanding the right side:
\[ 15 = 2x + 12 \]
Subtracting 12 from both sides:
\[ 3 = 2x \]
Dividing by 2:
\[ x = \frac{3}{2} = 1.5 \text{ km/hr} \]

### Conclusion
The speed at which the woman should travel to meet the cycles at an interval of 8 minutes is **1.5 km/hr**.