A boat covers (D + 80) km in downstream and 'D' km upstream in total `13(1)/(3)` hours Sum of speed of the boat in upstream and downstream is 48 km/hr and
speed of boat in still water is 300% more than speed of stream, Time taken by boat to cover (D + 30) km in upstream

Options

A. A)`12(1)/(3)` hours
B. B)`6(1)/(3)` hours
C. C)`8(1)/(3)` hours
D. D)`10(1)/(3)` hours

Answer

To solve the problem step by step, we will break down the information given and use it to find the required time taken by the boat to cover \(D + 30\) km upstream.

### Step 1: Understand the Given Information
We have:
- Distance downstream: \(D + 80\) km
- Distance upstream: \(D\) km
- Total time: \(13 \frac{1}{3}\) hours
- Sum of speeds in upstream and downstream: \(48\) km/hr
- Speed of the boat in still water is \(300\%\) more than the speed of the stream.

### Step 2: Convert Total Time to Improper Fraction
Convert \(13 \frac{1}{3}\) hours to an improper fraction:
\[ 13 \frac{1}{3} = \frac{40}{3} \text{ hours} \]

### Step 3: Set Up the Speed Equations
Let the speed of the stream be \(x\) km/hr. Then the speed of the boat in still water is \(4x\) km/hr (since it's \(300\%\) more than the speed of the stream).

From the information given:
\[ \text{Speed downstream} = 4x + x = 5x \text{ km/hr} \]
\[ \text{Speed upstream} = 4x - x = 3x \text{ km/hr} \]
The sum of these speeds is given as \(48\) km/hr:
\[ 5x + 3x = 48 \implies 8x = 48 \implies x = 6 \text{ km/hr} \]

### Step 4: Calculate the Speed of the Boat
Now, substituting \(x\) back to find the speed of the boat:
\[ \text{Speed of the boat} = 4x = 4 \times 6 = 24 \text{ km/hr} \]

### Step 5: Set Up the Time Equation
Using the formula for time, we can express the total time taken for downstream and upstream:
\[ \text{Time downstream} = \frac{D + 80}{5x} = \frac{D + 80}{30} \]
\[ \text{Time upstream} = \frac{D}{3x} = \frac{D}{18} \]
Setting up the equation for total time:
\[ \frac{D + 80}{30} + \frac{D}{18} = \frac{40}{3} \]

### Step 6: Find a Common Denominator and Solve for \(D\)
The common denominator for \(30\) and \(18\) is \(90\):
\[ \frac{3(D + 80)}{90} + \frac{5D}{90} = \frac{40}{3} \]
Multiplying through by \(90\):
\[ 3(D + 80) + 5D = 1200 \]
Expanding and combining like terms:
\[ 3D + 240 + 5D = 1200 \implies 8D + 240 = 1200 \implies 8D = 960 \implies D = 120 \text{ km} \]

### Step 7: Calculate Time for \(D + 30\) km in Upstream
Now we need to find the time taken to cover \(D + 30\) km upstream:
\[ D + 30 = 120 + 30 = 150 \text{ km} \]
Using the upstream speed:
\[ \text{Time} = \frac{150}{3x} = \frac{150}{18} = \frac{25}{3} \text{ hours} \]

### Step 8: Convert Time to Mixed Fraction
Convert \(\frac{25}{3}\) hours to a mixed fraction:
\[ \frac{25}{3} = 8 \frac{1}{3} \text{ hours} \]

### Final Answer
The time taken by the boat to cover \(D + 30\) km upstream is \(8 \frac{1}{3}\) hours.

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