A person takes 20 minutes more to cover a certain distance by decreasing his speed by 20%. What is the time taken to cover the distance at his original speed ?

Options

A. A) 1hr
B. B) 1 hr 20 min
C. C) 1 hr 10 min
D. D) 50 min

Answer

To solve the problem step by step, let's denote the variables clearly and break down the calculations.

### Step 1: Define the Variables
Let:
- \( d \) = distance (in km)
- \( k \) = original speed (in km/h)
- \( T_0 \) = time taken to cover the distance at original speed (in hours)
- \( T_d \) = time taken to cover the distance at decreased speed (in hours)

### Step 2: Express the Time Taken at Original Speed
The time taken to cover the distance at original speed is given by:
\[ T_0 = \frac{d}{k} \]

### Step 3: Calculate the Decreased Speed
The speed after a 20% decrease is:
\[ \text{Decreased speed} = k - 0.2k = 0.8k \]

### Step 4: Express the Time Taken at Decreased Speed
The time taken to cover the distance at the decreased speed is:
\[ T_d = \frac{d}{0.8k} \]

### Step 5: Relate the Two Times
According to the problem, the time taken at the decreased speed is 20 minutes more than the time taken at the original speed. We can express this as:
\[ T_d = T_0 + \frac{20}{60} \quad \text{(since 20 minutes = } \frac{1}{3} \text{ hours)} \]
Substituting the expressions for \( T_0 \) and \( T_d \):
\[ \frac{d}{0.8k} = \frac{d}{k} + \frac{1}{3} \]

### Step 6: Simplify the Equation
To eliminate \( d \), we can multiply through by \( 0.8k \):
\[ d = 0.8 \cdot d + \frac{0.8k}{3} \]
Rearranging gives:
\[ d - 0.8d = \frac{0.8k}{3} \]
\[ 0.2d = \frac{0.8k}{3} \]
\[ d = \frac{0.8k}{3 \cdot 0.2} = \frac{0.8k}{0.6} = \frac{4k}{3} \]

### Step 7: Substitute Back to Find \( T_0 \)
Now we can substitute \( d \) back into the equation for \( T_0 \):
\[ T_0 = \frac{d}{k} = \frac{\frac{4k}{3}}{k} = \frac{4}{3} \text{ hours} \]

### Step 8: Convert Hours to Minutes
To convert \( T_0 \) into minutes:
\[ T_0 = \frac{4}{3} \times 60 = 80 \text{ minutes} \]

### Conclusion
The time taken to cover the distance at the original speed is **80 minutes**.

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