To calculate the time Mathew would save by driving at a speed of \( k \) miles per hour instead of \( \frac{m}{X} \) miles per hour, we can follow these steps:
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Determine Mathew's original speed:
- Mathew drives \( m \) miles in \( X \) hours, so his speed is: \[ \text{original speed} = \frac{m}{X} \text{ miles per hour} \]
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Calculate the time taken at the original speed:
- The time taken to drive \( m \) miles at his original speed is \( X \) hours (as given).
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Calculate the time taken at the new speed \( k \) miles per hour:
- To find the time taken to drive \( m \) miles at the speed \( k \) miles per hour, we use the formula: \[ \text{time at } k = \frac{m}{k} \text{ hours} \]
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Calculate the time saved:
- The time saved by switching from the original speed to speed \( k \) is: \[ \text{time saved} = \text{original time} - \text{new time} \]
- Plugging in the expressions for the original and new times, we get: \[ \text{time saved} = X - \frac{m}{k} \]
Thus, the number of hours Mathew would save if he drove the same distance at \( k \) miles per hour is: \[ \text{time saved} = X - \frac{m}{k} \]