Question

Perform each of the following conversions, being sure to set up clearly the appropriate conversion factor in each case. a. \(2.23 \mathrm{m}\) to yards b. 46.2 yd to meters c. \(292 \mathrm{cm}\) to inches d. 881.2 in. to centimeters e. \(1043 \mathrm{km}\) to miles f. 445.5 mi to kilometers g. \(36.2 \mathrm{m}\) to kilometers h. 0.501 km to centimeters

Short answer

The short answers for each conversion are: a. \(2.23 \mathrm{m} = 2.43865 \mathrm{yd}\) b. \(46.2 \mathrm{yd} = 42.22688 \mathrm{m}\) c. \(292 \mathrm{cm} = 114.960692 \mathrm{in}\) d. \(881.2 \mathrm{in} = 2238.648 \mathrm{cm}\) e. \(1043 \mathrm{km} = 647.944153 \mathrm{mi}\) f. \(445.5 \mathrm{mi} = 716.68577 \mathrm{km}\) g. \(36.2 \mathrm{m} = 0.0362 \mathrm{km}\) h. \(0.501 \mathrm{km} = 50,100 \mathrm{cm}\)

Step-by-step solution

Recall the conversion factor between meters and yards

In order to convert meters to yards, we need to use the conversion factor: 1 meter = 1.09361 yards.

Apply the conversion factor

Multiply the given value in meters by the conversion factor: \(2.23 \mathrm{m} \times \frac{1.09361 \mathrm{yd}}{1\mathrm{m}} = 2.43865 \mathrm{yd}\) #b. Convert 46.2 yd to meters#

Recall the conversion factor between yards and meters

In order to convert yards to meters, we can use the inverse of the conversion factor we used before: 1 yard = 0.9144 meters.

Apply the conversion factor

Multiply the given value in yards by the conversion factor: \(46.2 \mathrm{yd} \times \frac{0.9144 \mathrm{m}}{1\mathrm{yd}} = 42.22688 \mathrm{m}\) #c. Convert 292 cm to inches#

Recall the conversion factor between centimeters and inches

In order to convert centimeters to inches, we need to use the conversion factor: 1 centimeter = 0.393701 inches.

Apply the conversion factor

Multiply the given value in centimeters by the conversion factor: \(292 \mathrm{cm} \times \frac{0.393701\mathrm{in}}{1\mathrm{cm}} = 114.960692\mathrm{in}\) #d. Convert 881.2 inches to centimeters#

Recall the conversion factor between inches and centimeters

In order to convert inches to centimeters, we can use the inverse of the conversion factor we used before: 1 inch = 2.54 centimeters.

Apply the conversion factor

Multiply the given value in inches by the conversion factor: \(881.2 \mathrm{in} \times \frac{2.54\mathrm{cm}}{1\mathrm{in}} = 2238.648\mathrm{cm}\) #e. Convert 1043 km to miles#

Recall the conversion factor between kilometers and miles

In order to convert kilometers to miles, we need to use the conversion factor: 1 kilometer = 0.621371 miles.

Apply the conversion factor

Multiply the given value in kilometers by the conversion factor: \(1043\mathrm{km} \times \frac{0.621371\mathrm{mi}}{1\mathrm{km}} = 647.944153\mathrm{mi}\) #f. Convert 445.5 miles to kilometers#

Recall the conversion factor between miles and kilometers

In order to convert miles to kilometers, we can use the inverse of the conversion factor we used before: 1 mile = 1.60934 kilometers.

Apply the conversion factor

Multiply the given value in miles by the conversion factor: \(445.5\mathrm{mi} \times \frac{1.60934\mathrm{km}}{1\mathrm{mi}} = 716.68577\mathrm{km}\) #g. Convert 36.2 m to kilometers#

Recall the conversion factor between meters and kilometers

In order to convert meters to kilometers, we need to use the conversion factor: 1 meter = 0.001 kilometers.

Apply the conversion factor

Multiply the given value in meters by the conversion factor: \(36.2\mathrm{m} \times \frac{0.001\mathrm{km}}{1\mathrm{m}} = 0.0362\mathrm{km}\) #h. Convert 0.501 km to centimeters#

Recall the conversion factor between kilometers and centimeters

In order to convert kilometers to centimeters, we need to use the conversion factor: 1 kilometer = 100,000 centimeters.

Apply the conversion factor

Multiply the given value in kilometers by the conversion factor: \(0.501\mathrm{km} \times \frac{100,000\mathrm{cm}}{1\mathrm{km}} = 50,100\mathrm{cm}\)

Key concepts

Conversion Factor

When converting from one unit to another, a conversion factor is a crucial tool. It's essentially a multiplier used to convert a quantity expressed in one unit into an equivalent expression in another. For example, when you want to convert meters to yards, you use the conversion factor 1 meter = 1.09361 yards.

This factor allows you to calculate the number of yards equivalent to meters by multiplying.

• Conversion factors can come in handy for various units, such as temperature, volume, and distance.
• It's important to set up the conversion factor so that the unit you want to get rid of cancels out, leaving the desired unit.
• Always double-check the accuracy of the conversion factor before using it, as precision is key in mathematical calculations.

Using conversion factors correctly requires practice, but once mastered, they are incredibly useful for solving a wide range of problems.

Metric System

The metric system is a standardized system used worldwide, with the exception of a few countries. Its main advantage is its simplicity and universal application. It's based on units of 10, making it easy to convert between different measurements.

Common units in the metric system include:

• "meter" for length,
• "liter" for volume,
• "gram" for mass.

When using the metric system, conversions are straightforward due to the decimal nature. For example, converting meters to kilometers simply involves multiplying by 0.001, as 1 kilometer equals 1000 meters.

This system's uniformity makes it ideal for scientific and technical fields that require precision and consistency. As you explore mathematical calculations, understanding the metric system becomes increasingly helpful.

Imperial System

The imperial system is a set of measurements commonly used in the United States. Unlike the metric system, the imperial system can be a bit challenging due to its lack of a base unit.

Some of the well-known units in the imperial system include:

• "inch" and "yard" for length,
• "gallon" for volume,
• "pound" for weight.

Conversions in the imperial system can be more complex, as they aren't based on factors of 10. For example, there are 12 inches in a foot and 3 feet in a yard.

This often means learning and remembering various conversion factors is necessary. Despite its complexity, being familiar with this system is essential for projects or exchanges involving regions where it is commonly used.

Mathematical Calculations

Accurate mathematical calculations are key in unit conversion. This involves applying conversion factors and ensuring the equations are set up correctly from the start.

To ensure accurate calculations, always:

• Write down the conversion factor and check units carefully,
• Align numbers correctly, especially when dealing with decimals,
• Recalculate to verify your result.

Also, note that in practical applications, rounding may be necessary, but you should always choose a level of precision appropriate for your needs.

For example, converting 2.23 meters to yards involves the multiplication of 2.23 by 1.09361, resulting in approximately 2.44 yards. By practicing carefully, you'll build confidence in mathematical conversions and reduce errors, whether in everyday tasks or advanced projects.