A person travelled 2560 km partly by aeroplane, partly by ship and rest by car. The time taken by aeroplane, ship and car are in the ratio 1: 4:5 and the ratio of average speed by all three means is 20:1 : 8. If average speed in whole trip is 64 km/hr then find the speed of ship, time taken by ship and distance travelled by ship.

Options

A. 10 km/hr, 16 hr, 160 km
B. 20 km/hr, 12 hr, 150 km
C. 30 km/hr, 18 hr, 140 km
D. 23 km/hr, 17 hr, 120 km

Answer

To solve the problem step by step, we will follow the given ratios and the average speed to find the speed of the ship, the time taken by the ship, and the distance traveled by the ship.

### Step 1: Understand the Ratios
We are given:
- The time ratios for aeroplane, ship, and car as \(1:4:5\).
- The speed ratios for aeroplane, ship, and car as \(20:1:8\).
- The total distance traveled is \(2560\) km.
- The average speed for the whole trip is \(64\) km/hr.

### Step 2: Calculate Total Time Taken for the Journey
Using the average speed formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
We can rearrange this to find the total time:
\[ \text{Total Time} = \frac{\text{Total Distance}}{\text{Average Speed}} = \frac{2560 \text{ km}}{64 \text{ km/hr}} = 40 \text{ hours} \]

### Step 3: Determine Time Taken by Each Mode of Transport
Let the time taken by the aeroplane be \(x\). Then, according to the ratios:
- Time taken by aeroplane = \(x\)
- Time taken by ship = \(4x\)
- Time taken by car = \(5x\)

The total time can be expressed as:
\[ x + 4x + 5x = 40 \]
\[ 10x = 40 \implies x = 4 \text{ hours} \]

Thus, the time taken by each mode is:
- Time taken by aeroplane = \(4\) hours
- Time taken by ship = \(4 \times 4 = 16\) hours
- Time taken by car = \(5 \times 4 = 20\) hours

### Step 4: Calculate the Speeds of Each Mode of Transport
Let the speeds of aeroplane, ship, and car be \(20k\), \(k\), and \(8k\) respectively (based on the speed ratio \(20:1:8\)).

### Step 5: Use the Total Distance to Find \(k\)
The distances traveled by each mode can be calculated as:
- Distance by aeroplane = Speed × Time = \(20k \times 4 = 80k\)
- Distance by ship = Speed × Time = \(k \times 16 = 16k\)
- Distance by car = Speed × Time = \(8k \times 20 = 160k\)

Adding these distances gives us the total distance:
\[ 80k + 16k + 160k = 2560 \]
\[ 256k = 2560 \implies k = 10 \]

### Step 6: Calculate the Speed of the Ship
Now substituting \(k\) back to find the speed of the ship:
\[ \text{Speed of ship} = k = 10 \text{ km/hr} \]

### Step 7: Calculate the Distance Traveled by the Ship
Using the time taken by the ship:
\[ \text{Distance by ship} = \text{Speed} \times \text{Time} = 10 \text{ km/hr} \times 16 \text{ hours} = 160 \text{ km} \]

### Summary of Results
- Speed of ship = \(10\) km/hr
- Time taken by ship = \(16\) hours
- Distance traveled by ship = \(160\) km