A jogger desires to run a certain course in `1/4` less time than he usually takes. But what per cent must be incresase his average running speed to accomplish the goal ?

Options

A. `50%`
B. `20%`
C. `25%`
D. `33(1)/(3)` %

Answer

To solve the problem of how much the jogger must increase his average running speed to run a course in \( \frac{1}{4} \) less time, we can follow these steps:

### Step 1: Understand the relationship between speed, distance, and time
The basic formula that relates speed (S), distance (D), and time (T) is:
\[ D = S \times T \]
This means that distance is equal to speed multiplied by time.

### Step 2: Define the variables
Let:
- \( T \) = original time taken to run the course
- \( S \) = original speed
- \( D \) = distance of the course

From the formula, we can express the distance as:
\[ D = S \times T \]

### Step 3: Calculate the new time
The jogger wants to run the course in \( \frac{1}{4} \) less time. Therefore, the new time \( T' \) is:
\[ T' = T - \frac{1}{4}T = \frac{3}{4}T \]

### Step 4: Calculate the new speed
To cover the same distance \( D \) in the new time \( T' \), the new speed \( S' \) must be:
\[ D = S' \times T' \]
Substituting for \( T' \):
\[ D = S' \times \frac{3}{4}T \]

### Step 5: Set the equations equal to each other
Since the distance \( D \) remains constant, we can set the two equations for distance equal to each other:
\[ S \times T = S' \times \frac{3}{4}T \]

### Step 6: Solve for the new speed
Dividing both sides by \( T \) (assuming \( T \neq 0 \)):
\[ S = S' \times \frac{3}{4} \]
Rearranging gives:
\[ S' = \frac{4}{3}S \]

### Step 7: Calculate the increase in speed
The increase in speed is:
\[ \text{Increase} = S' - S = \frac{4}{3}S - S = \left(\frac{4}{3} - 1\right)S = \frac{1}{3}S \]

### Step 8: Calculate the percentage increase
To find the percentage increase in speed:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{S}\right) \times 100 = \left(\frac{\frac{1}{3}S}{S}\right) \times 100 = \frac{1}{3} \times 100 = 33.33\% \]

### Final Answer
The jogger must increase his average running speed by **33.33%** to accomplish his goal.

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