Answered

Ishi walked a total of $2 \frac{1}{2}$ miles on a treadmill. He walked at a constant rate of 4 miles per hour. Which expression shows how long, in minutes, Ishi walked on the treadmill?


A$2.5 \text { miles } \times 4 \frac{\text { miles }}{\text { hour }} \times \frac{60 \text { minutes }}{1 \text { hour }}$


B$2.5 \text { miles } \div 4 \frac{\text { miles }}{\text { hour }} \times \frac{60 \text { minutes }}{1 \text { hour }}$


C$2.5 \text { miles } \div 4 \frac{\text { miles }}{\text { hour }} \div \frac{60 \text { minutes }}{1 \text { hour }}$


D$2.5 \text { miles } \times 4 \frac{\text { miles }}{\text { hour }} \div \frac{60 \text { minutes }}{1 \text { hour }}$

Verified Answer :

  • Define the distance as \(2.5\) miles and speed as \(4\) miles per hour.
  • Use the formula for time: \(\text{Time (hours)} = \text{Distance} \div \text{Speed}\).
  • Convert hours to minutes by multiplying the time in hours by 60.
  • The correct expression for the time in minutes is \(\boxed{2.5 \text{ miles} \div 4 \frac{\text{miles}}{\text{hour}} \times \frac{60 \text{ minutes}}{1 \text{ hour}}}\).

Explanation

  1. Analyzing the Relationship Between Distance, Speed, and Time

To determine how long Ishi walked on the treadmill, we first analyze the given information. Ishi walked a total distance of \(D = 2 \frac{1}{2} = 2.5\) miles at a constant speed of \(S = 4\) miles per hour. The relationship between distance, speed, and time is given by the formula:
\[ \text{Time (in hours)} = \frac{\text{Distance}}{\text{Speed}} \]
Substituting the given values, the time Ishi walked in hours is:
\[ T_h = \frac{2.5 \text{ miles}}{4 \text{ miles/hour}} = 2.5 \div 4 \text{ hours} \]

  1. Converting Time from Hours to Minutes

The question asks for the time in minutes. Since there are 60 minutes in 1 hour, we must convert the time from hours to minutes by multiplying the time in hours by 60 minutes per hour:
\[ \text{Time (in minutes)} = \text{Time (in hours)} \times 60 \frac{\text{minutes}}{\text{hour}} \]
Substituting the expression we found for the time in hours:
\[ T_m = (2.5 \div 4) \times 60 \]
This shows that to get the time in minutes, we take the distance in miles, divide it by the speed in miles per hour, and multiply the result by 60.

  1. Selecting the Correct Expression

Comparing our derived expression \(2.5 \div 4 \times 60\) with the provided choices:

  1. \(2.5 \times 4 \times 60\)
  2. \(2.5 \div 4 \times 60\)
  3. \(2.5 \div 4 \div 60\)
  4. \(2.5 \times 4 \div 60\)

The expression that matches our result is the second option: \(2.5 \text{ miles} \div 4 \frac{\text{miles}}{\text{hour}} \times \frac{60 \text{ minutes}}{1 \text{ hour}}\).

  1. Conclusion

Ishi's walk duration in minutes is calculated by first dividing his total distance by his speed to find the time in hours, and then multiplying by 60 to convert it to minutes. Therefore, the correct expression is \(\boxed{2.5 \text{ miles} \div 4 \frac{\text{miles}}{\text{hour}} \times \frac{60 \text{ minutes}}{1 \text{ hour}}}\).

Examples

Understanding unit conversions is a fundamental skill used in many everyday situations. For example, if you know the distance to a park in kilometers and your average walking speed in kilometers per hour, you can calculate the duration of your walk in minutes to better plan your afternoon. Similarly, when following a recipe that gives ingredient amounts in ounces but requires grams, you use conversion factors to ensure your measurements are accurate. Mastering these relationships helps in everything from traveling and cooking to managing budgets and time.

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