A runner ran a particular distance in 30 minutes.
After some training they ran the same distance in 24 minutes.
What was the percentage increase in the runner’s average speed?


To calculate the percentage increase in the runner’s average speed, we can follow these steps:

1. **Find the initial and new average speeds**:
   - The runner's speed is the distance divided by the time.
   - Let \( d \) represent the distance run.
   - The initial time is 30 minutes, and the new time is 24 minutes.

   \[
   \text{Initial speed} = \frac{d}{30} \quad \text{(in units of distance per minute)}
   \]
   \[
   \text{New speed} = \frac{d}{24}
   \]

2. **Calculate the percentage increase**:
   The formula for percentage increase in speed is:
   
   \[
   \text{Percentage increase} = \left( \frac{\text{New speed} - \text{Initial speed}}{\text{Initial speed}} \right) \times 100
   \]

   Plug in the values:
   
   \[
   \text{Percentage increase} = \left( \frac{\frac{d}{24} - \frac{d}{30}}{\frac{d}{30}} \right) \times 100
   \]

   Simplify the expression:

   \[
   \text{Percentage increase} = \left( \frac{\frac{30d - 24d}{24 \times 30}}{\frac{d}{30}} \right) \times 100
   \]
   \[
   \text{Percentage increase} = \left( \frac{6d}{720} \times \frac{30}{d} \right) \times 100
   \]
   \[
   \text{Percentage increase} = \left( \frac{180}{720} \right) \times 100
   \]
   \[
   \text{Percentage increase} = 0.25 \times 100 = 25\%
   \]

Thus, the runner's average speed increased by **25%**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What formula would you use if the time intervals were in hours instead of minutes?
2. How would the result change if the runner reduced the time by a smaller percentage, say 10%?
3. What if the initial time was 40 minutes and the new time was 32 minutes? What would the percentage increase be?
4. How can the runner's speed be converted into different units, such as kilometers per hour?
5. How does this calculation change if the runner trains over a longer distance?

**Tip:** The percentage change formula can be applied to any comparison, not just speed, as long as you know the initial and new values.

A runner ran a particular distance in 30 minutes. After some training they ran the same distance in 24 minutes. What was the percentage increase in the runner’s average speed?

Learn how to calculate the percentage increase in a runner's speed after reducing their time from 30 minutes to 24 minutes over the same distance. This problem covers key algebraic concepts, including the formula for speed and percentage increase. Suitable for students in Grades 7-9.

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