This is a proportional reasoning problem where we use the scale of the map to determine real-world distances.

Step-by-Step Solution:

1. Determine the scale of the map: From the problem, $\frac{3}{4}$ inch on the map corresponds to 50 yards in reality. Thus, the scale is:

   $1 \, \text{inch on the map} = \frac{50}{\frac{3}{4}} \, \text{yards}$

   Simplify:

   $1 \, \text{inch on the map} = 50 \times \frac{4}{3} = \frac{200}{3} \, \text{yards}$

   $1 \, \text{inch on the map} \approx 66.67 \, \text{yards}$

2. Calculate the actual distance from home to school: The map shows 3 inches for the distance between Harper's current location and home. Using the scale:

   $\text{Distance from home} = 3 \, \text{inches} \times 66.67 \, \text{yards per inch}$

   $\text{Distance from home} = 200 \, \text{yards}$

Final Answer:

Harper is 200 yards from home.

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Would you like further clarification or details about the proportional reasoning?

Related Questions:

1. How can we convert this distance into other units (e.g., meters or miles)?
2. What if Harper were 1 inch from school on the map? What would the distance be in yards?
3. How can you use similar proportions in problems involving maps and scales?
4. What is the importance of simplifying ratios when solving these types of problems?
5. How would the answer change if the scale was in meters instead of yards?

Tip:

Always write the units next to numbers in proportional problems to avoid confusion and ensure accuracy in your calculations.