A particle goes A to M with a speed of 30 km/hr. and M to B with a speed of 50 km/hr. If MB = 5AM, the average speed in km/hr for the whole journey is –

Options

A. 45 km/hr
B. 40 km/hr
C. 36 km/hr
D. 34 km/hr

Answer

To solve the problem, we need to find the average speed of a particle that travels from point A to point M and then from point M to point B. The speeds for each segment and the relationship between the distances are given. Let's break down the solution step by step.

### Step 1: Define the distances
Let the distance from A to M be \( d \). According to the problem, the distance from M to B is given as \( MB = 5AM \). Therefore, we can express the distance from M to B as:
\[ MB = 5d \]

### Step 2: Calculate the total distance
The total distance traveled by the particle from A to B is the sum of the distances from A to M and from M to B:
\[ \text{Total Distance} = AM + MB = d + 5d = 6d \]

### Step 3: Calculate the time taken for each segment
We know that speed is equal to distance divided by time. Therefore, we can find the time taken for each segment:

- For the segment from A to M:
\[ \text{Speed} = 30 \text{ km/hr} \implies \text{Time} (T_1) = \frac{d}{30} \]

- For the segment from M to B:
\[ \text{Speed} = 50 \text{ km/hr} \implies \text{Time} (T_2) = \frac{MB}{50} = \frac{5d}{50} = \frac{d}{10} \]

### Step 4: Calculate the total time
Now, we can find the total time taken for the entire journey:
\[ \text{Total Time} = T_1 + T_2 = \frac{d}{30} + \frac{d}{10} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 10 is 30:
\[ \text{Total Time} = \frac{d}{30} + \frac{3d}{30} = \frac{4d}{30} = \frac{2d}{15} \]

### Step 5: Calculate the average speed
The average speed is defined as the total distance divided by the total time:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{6d}{\frac{2d}{15}} \]
To simplify this, we multiply by the reciprocal of the denominator:
\[ \text{Average Speed} = 6d \times \frac{15}{2d} = \frac{90}{2} = 45 \text{ km/hr} \]

### Final Answer
The average speed for the whole journey is:
\[ \boxed{45 \text{ km/hr}} \]