Two cycles start from a house with a speed of 20 km/hr. at interval of 15 minutes. With how must more speed (km//he.) the woman coming from the opposite directrion towards the house has to travel to meet the cycels at an intervel 10 minutes?

Options

A. 13
B. 11
C. 12
D. 14

Answer

To solve the problem step by step, we will break it down into manageable parts:

### Step 1: Determine the distance between the two cycles
- The speed of each cycle is given as 20 km/hr.
- The time interval between the two cycles is 15 minutes.
- We need to convert 15 minutes into hours:
\[ 15 \text{ minutes} = \frac{15}{60} \text{ hours} = \frac{1}{4} \text{ hours} \]
- Now, we can calculate the distance covered by the first cycle in this time:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 20 \text{ km/hr} \times \frac{1}{4} \text{ hr} = 5 \text{ km} \]
- Therefore, the distance between the two cycles when they start is 5 km.

### Step 2: Determine the time the woman has to meet the cycles
- The woman is supposed to meet the cycles at an interval of 10 minutes.
- We convert 10 minutes into hours:
\[ 10 \text{ minutes} = \frac{10}{60} \text{ hours} = \frac{1}{6} \text{ hours} \]

### Step 3: Set up the equation for relative speed
- Let the speed of the woman be \( x \) km/hr.
- The relative speed when the woman and the cycles are moving towards each other is the sum of their speeds:
\[ \text{Relative Speed} = 20 \text{ km/hr} + x \text{ km/hr} \]
- The distance they need to cover to meet is 5 km, and the time taken to meet is \( \frac{1}{6} \) hours. Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \):
\[ 5 \text{ km} = (20 + x) \text{ km/hr} \times \frac{1}{6} \text{ hr} \]

### Step 4: Solve for \( x \)
- Rearranging the equation gives:
\[ 5 = \frac{20 + x}{6} \]
- Multiplying both sides by 6:
\[ 30 = 20 + x \]
- Solving for \( x \):
\[ x = 30 - 20 = 10 \text{ km/hr} \]

### Step 5: Determine how much more speed is needed
- The woman needs to travel at a speed of 10 km/hr to meet the cycles.
- Since the cycles are traveling at 20 km/hr, we need to find how much more speed she needs compared to the cycles:
\[ \text{More speed required} = x - 20 = 10 - 20 = -10 \text{ km/hr} \]
- This means she needs to travel at 10 km/hr less than the cycles to meet them.

### Summary of the Solution
The woman must travel at a speed of 10 km/hr to meet the cycles at an interval of 10 minutes.