Question

Given that \(1 \mathrm{mi}=1760 \mathrm{yd}\), determine what conversion factor is appropriate to convert 1849 yd to miles; to convert \(2.781 \mathrm{mi}\) to yards.

Short answer

The appropriate conversion factors are as follows: To convert 1849 yards to miles, multiply by \(\frac{1 \text{ mile}}{1760 \text{ yards}}\). This gives approximately 1.051 miles. To convert 2.781 miles to yards, multiply by 1760 yards/mile. This gives approximately 4894.56 yards.

Step-by-step solution

Convert yards to miles

To find the conversion factor for converting yards to miles, we need to divide the given distance in yards by the number of yards in 1 mile (which is 1760 yards). The conversion factor will be: Conversion factor (yards to miles) = \(\frac{1 \text{ mile}}{1760 \text{ yards}}\) 2. Convert 1849 yards to miles

Apply the yards-to-miles conversion factor

Now, let's apply this conversion factor to the given distance of 1849 yards: Miles = 1849 yards × \(\frac{1 \text{ mile}}{1760 \text{ yards}}\) \(Miles = \frac{1849}{1760}\) miles 3. Find the conversion factor to convert miles to yards

Convert miles to yards

To find the conversion factor for converting miles to yards, we need to multiply the given distance in miles by the number of yards in 1 mile (which is 1760 yards). The conversion factor will be: Conversion factor (miles to yards) = 1760 yards/mile 4. Convert 2.781 miles to yards

Apply the miles-to-yards conversion factor

Now, let's apply this conversion factor to the given distance of 2.781 miles: Yards = 2.781 miles × 1760 yards/mile Yards = 2.781 × 1760 yards 5. Compute the final conversions

Perform the calculations

Now, calculate the final values for the given distances in their respective units: Miles = \(\frac{1849}{1760}\) miles ≈ 1.051 miles Yards = 2.781 × 1760 yards ≈ 4894.56 yards So, the appropriate conversion factor for 1849 yards is about 1.051 miles, and the appropriate conversion factor for 2.781 miles is about 4894.56 yards.

Key concepts

Yard to Mile Conversion

When dealing with different units of measurement, especially in distance, understanding how to convert from one unit to another is essential. In the context of converting yards to miles, it's important to recognize that this process relies on a specific relationship: 1 mile equals 1760 yards.

For instance, if a student is given a distance in yards and asked to convert it to miles, the student would divide the number of yards by 1760. This is based on the principle that dividing by a unit's equivalent in another unit yields a conversion. In our textbook example, converting 1849 yards to miles involves dividing by 1760, providing an approximation of 1.051 miles.

It's also beneficial to visualize this conversion: Imagine a mile-long stretch filled with yardsticks laid end to end; it would require 1760 of them to cover that mile. This imagery can help solidify the conceptual understanding of the unit conversion process.

Dimensional Analysis

Dimensional analysis, also known as the unit factor method or factor-label method, is a powerful tool used in chemistry and other sciences for converting between units. It involves using conversion factors to cancel out units until the desired unit is obtained.

To use dimensional analysis, you start with a given measurement and multiply by conversion factors that relate the units you're converting between. In each conversion factor, the numerator and denominator are equivalent amounts of different units, allowing one unit to cancel out while the other remains. In our previous yard to mile example, the conversion factor \( \frac{1 \text{ mile}}{1760 \text{ yards}} \) allowed us to cancel out 'yards' and end with our answer in 'miles'.

This method is not limited to simple conversions; it can be extended to complex calculations involving multiple steps and units. It demands careful attention to the units and an understanding of the relationships between different units in order to correctly apply the conversion factors.

Conversion Factors

Understanding conversion factors is crucial in the realm of unit conversions. A conversion factor is a ratio expressed as a fraction that denotes how many of one unit are equivalent to another unit. It bridges the gap between different measurement systems, allowing for a smooth transition from one unit to another.

In the context of our example, the conversion factors are \( \frac{1 \text{ mile}}{1760 \text{ yards}} \) and \( 1760 \text{ yards/mile} \). These values are exact and do not imply approximation; they're based on the definitions of the units involved. When using conversion factors to convert between units, ensure that the units you wish to cancel are diagonally opposite each other when you set up the conversion equation so that they effectively eliminate one another.

For students, remembering commonly used conversion factors can speed up the conversion process and avoid the need for redundant calculations. Making sure these factors are correctly incorporated into dimensional analysis will yield accurate and reliable results.