Question

Given that \(1 \mathrm{mi}=1760\) 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 \(\frac{1 \;\text{mile}}{1760\;\text{yards}}\) for converting yards to miles and \(\frac{1760\;\text{yards}}{1 \;\text{mile}}\) for converting miles to yards. Using these factors, 1849 yards is approximately 1.0506 miles, and 2.781 miles is approximately 4891.36 yards.

Step-by-step solution

Conversion factor for yards to miles

Since we are given that 1 mile is equal to 1760 yards, we can write this conversion factor as a fraction. The conversion factor will be the fraction with miles in the numerator and yards in the denominator, as shown below: Conversion factor (yards to miles) = \(\frac{1 \;\text{mile}}{1760\;\text{yards}}\)

Conversion factor for miles to yards

To determine the conversion factor for converting miles to yards, we simply find the reciprocal of the conversion factor for yards to miles, as shown below: Conversion factor (miles to yards) = \(\frac{1}{\frac{1 \;\text{mile}}{1760\;\text{yards}}}\) Now, we can find the reciprocal by flipping the numerator and the denominator: Conversion factor (miles to yards) = \(\frac{1760\;\text{yards}}{1 \;\text{mile}}\)

Converting 1849 yards to miles

Now that we have the conversion factor for yards to miles, we can convert 1849 yards to miles by multiplying the value in yards by the conversion factor: Miles = \(1849 \;\text{yards} \times \frac{1 \;\text{mile}}{1760\;\text{yards}}\) The yards will cancel out and we are left with: Miles = \(\frac{1849}{1760}\;\text{miles}\) ≈ 1.0506 miles

Converting 2.781 miles to yards

Similarly, we can convert 2.781 miles to yards by multiplying the value in miles by the conversion factor (miles to yards): Yards = \(2.781\;\text{miles} \times \frac{1760\;\text{yards}}{1 \;\text{mile}}\) The miles will cancel out and we are left with: Yards = \(2.781 \times 1760\;\text{yards}\) ≈ 4891.36 yards So, the appropriate conversion factors are \(\frac{1 \;\text{mile}}{1760\;\text{yards}}\) for converting yards to miles and \(\frac{1760\;\text{yards}}{1 \;\text{mile}}\) for converting miles to yards, and the converted values are approximately 1.0506 miles and 4891.36 yards, respectively.

Key concepts

Yards to Miles Conversion

Converting yards to miles involves understanding the relationship between these two units of measurement. A yard is a smaller unit compared to a mile. Therefore, when we convert a distance measured in yards to miles, the resulting number will be smaller.
The conversion factor is the key to this conversion process. Since it is given that one mile equals 1760 yards, the conversion factor from yards to miles is \( \frac{1 \;\text{mile}}{1760\;\text{yards}} \). This fraction helps to translate any quantity from yards to its equivalent in miles.
To perform the conversion, you need to multiply the number of yards by the conversion factor. This multiplication makes the unit "yards" cancel out, leaving you with the unit "miles." For example, if we want to convert 1849 yards to miles, we use the formula:

• Miles = 1849 yards \( \times \frac{1 \;\text{mile}}{1760\;\text{yards}} \)

This calculation gives a result of approximately 1.0506 miles.

Miles to Yards Conversion

Just like how we convert yards to miles, converting miles to yards uses the reciprocal of the yards to miles conversion factor. This process involves moving from a larger unit (miles) to a smaller unit (yards), resulting in a larger numerical value.
The conversion factor here is \( \frac{1760\;\text{yards}}{1 \;\text{mile}} \). This means that for every mile, you have 1760 yards.
When converting from miles to yards, you multiply the number of miles by this conversion factor. The unit "miles" cancels out, leaving you with "yards." For instance, when converting 2.781 miles to yards:

• Yards = 2.781 miles \( \times \frac{1760\;\text{yards}}{1 \;\text{mile}} \)

This results in approximately 4891.36 yards, effectively translating from the larger unit of miles to the smaller unit, yards.

Conversion Factor

The concept of a conversion factor is crucial in unit conversion tasks. It represents a ratio that is used to convert measurements from one unit to another.
A conversion factor for unit conversion is derived from the equivalence between two units. For yards and miles, it is based on the fact that 1 mile equals 1760 yards. This relationship allows the expression of conversion factors such as \( \frac{1 \;\text{mile}}{1760\;\text{yards}} \) and \( \frac{1760\;\text{yards}}{1 \;\text{mile}} \).
The beauty of the conversion factor is its ability to "cancel out" units during multiplication. When multiplying a measurement by an appropriate conversion factor, the units we are converting from (e.g., yards or miles) will cancel, leaving only the units we are converting to (e.g., miles or yards).

• Ensures consistent and accurate conversion.
• Makes it easy to switch between units of different scales.
• Adaptable to any two units given their relationship.

Understanding and applying conversion factors is essential for accurately navigating through different unit systems, ensuring clarity and precision in calculations.