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A man made four trips of equal distances...

A man made four trips of equal distances. His speed in first trip was 60 km/hr and in each subsequent trip his speed was half of the previous trip. What is the average speed (in km/hr) of the man in these four trips?

A

16

B

30

C

`28.125`

D

`27.5`

Text Solution

AI Generated Solution

The correct Answer is:
To find the average speed of the man over the four trips, we will follow these steps: ### Step 1: Define the Distance for Each Trip Let the distance for each trip be \( D \) kilometers. ### Step 2: Determine the Speeds for Each Trip - The speed for the first trip is \( 60 \) km/hr. - The speed for the second trip is half of the first trip: \[ \text{Speed}_2 = \frac{60}{2} = 30 \text{ km/hr} \] - The speed for the third trip is half of the second trip: \[ \text{Speed}_3 = \frac{30}{2} = 15 \text{ km/hr} \] - The speed for the fourth trip is half of the third trip: \[ \text{Speed}_4 = \frac{15}{2} = 7.5 \text{ km/hr} \] ### Step 3: Calculate the Total Distance The total distance for the four trips is: \[ \text{Total Distance} = D + D + D + D = 4D \text{ kilometers} \] ### Step 4: Calculate the Time Taken for Each Trip Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \): - Time for the first trip: \[ T_1 = \frac{D}{60} \text{ hours} \] - Time for the second trip: \[ T_2 = \frac{D}{30} \text{ hours} \] - Time for the third trip: \[ T_3 = \frac{D}{15} \text{ hours} \] - Time for the fourth trip: \[ T_4 = \frac{D}{7.5} \text{ hours} = \frac{D \times 2}{15} = \frac{2D}{15} \text{ hours} \] ### Step 5: Calculate the Total Time Taken The total time taken for all four trips is: \[ \text{Total Time} = T_1 + T_2 + T_3 + T_4 = \frac{D}{60} + \frac{D}{30} + \frac{D}{15} + \frac{2D}{15} \] ### Step 6: Simplify the Total Time Expression To simplify, find a common denominator (which is 60): \[ \text{Total Time} = \frac{D}{60} + \frac{2D}{60} + \frac{4D}{60} + \frac{8D}{60} = \frac{(1 + 2 + 4 + 8)D}{60} = \frac{15D}{60} = \frac{D}{4} \text{ hours} \] ### Step 7: Calculate the Average Speed The average speed is given by the formula: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{4D}{\frac{D}{4}} = 4D \times \frac{4}{D} = 16 \text{ km/hr} \] ### Final Answer The average speed of the man in these four trips is \( 16 \) km/hr.
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