Problem 189 - Introductory Chemistry Atoms First

Question

Write two conversion factors that express the relationship between: (a) Grams and kilograms, using 1 and 1000 (b) Kilograms and grams, using 1 and \(0.001\) (c) Yards and feet (d) Meters and centimeters, using 1 and 100 (e) Meters and centimeters, using 1 and \(0.01\)

Step-by-Step Solution

Answer

(a) Grams and kilograms: \( \frac{1000 \text{ g}}{1 \text{ kg}} \) and \( \frac{1 \text{ kg}}{1000 \text{ g}}\) (b) Kilograms and grams: \( \frac{1 \text{ g}}{0.001 \text{ kg}} \) and \( \frac{0.001 \text{ kg}}{1 \text{ g}} \) (c) Yards and feet: \( \frac{1 \text{ yd}}{3 \text{ ft}} \) and \( \frac{3 \text{ ft}}{1 \text{ yd}} \) (d) Meters and centimeters: \( \frac{1 \text{ m}}{100 \text{ cm}} \) and \( \frac{100 \text{ cm}}{1 \text{ m}}\) (e) Meters and centimeters: \( \frac{1 \text{ m}}{0.01 \text{ cm}} \) and \( \frac{0.01 \text{ cm}}{1 \text{ m}} \)

Step 1: Conversion factor 1 from grams to kilograms

For every 1000 grams, we have 1 kilogram, since 1 kilogram is equal to 1000 grams: \( \frac{1000 \text{ g}}{ 1 \text{ kg}} \)

Step 2: Conversion factor 2 from kilograms to grams

For each kilogram, there are 1000 grams: \( \frac{1 \text{ kg}}{1000 \text{ g}}\) (b) Kilograms and grams, using 1 and \(0.001\)

Step 3: Conversion factor 3 from grams to kilograms

For each gram, there is \(0.001\) kilogram: \( \frac{1 \text{ g}}{0.001 \text{ kg}} \)

Step 4: Conversion factor 4 from kilograms to grams

For each kilogram, there are 1000 grams, and since \(0.001\) is the reciprocal of 1000: \( \frac{0.001 \text{ kg}}{1 \text{ g}} \) (c) Yards and feet

Step 5: Conversion factor 5 from yards to feet

For every yard, there are 3 feet: \( \frac{1 \text{ yd}}{3 \text{ ft}} \)

Step 6: Conversion factor 6 from feet to yards

For each foot, there is \(\frac{1}{3}\) of a yard: \( \frac{3 \text{ ft}}{1 \text{ yd}} \) (d) Meters and centimeters, using 1 and 100

Step 7: Conversion factor 7 from meters to centimeters

For every meter, there are 100 centimeters: \( \frac{1 \text{ m}}{100 \text{ cm}} \)

Step 8: Conversion factor 8 from centimeters to meters

For each centimeter, there are \(0.01\) meters: \( \frac{100 \text{ cm}}{1 \text{ m}}\) (e) Meters and centimeters, using 1 and \(0.01\)

Step 9: Conversion factor 9 from meters to centimeters

For every meter, there are 100 centimeters: \( \frac{1 \text{ m}}{0.01 \text{ cm}} \)

Step 10: Conversion factor 10 from centimeters to meters

For each centimeter, we have \(0.01\) meters: \( \frac{0.01 \text{ cm}}{1 \text{ m}} \)

Key Concepts

Grams to Kilograms

When it comes to converting grams to kilograms, knowing the relationship between these two units is essential. Kilograms are the metric unit for measuring mass, commonly used around the world. They are bigger than grams. Therefore, when converting grams to kilograms, we must understand that:

• 1 kilogram is equal to 1000 grams.
• This means for every 1000 grams, you have 1 kilogram.

To convert grams to kilograms, you divide the number of grams by 1000. This is because the conversion factor from grams to kilograms is 1/1000, or as a multiplier, it is 0.001.Remember:

• The formula for conversion reads: \( \frac{1 \text{ g}}{0.001 \text{ kg}} \)
• This conversion factor makes unit conversion straightforward.
• For reverse conversion, multiply kilograms by 1000 to find the equivalent grams.

This is helpful in everyday scenarios like cooking or weighing small items.

Yards to Feet

The conversion between yards and feet is commonly used in the United States, especially in construction and sports fields.

• 1 yard is equal to 3 feet.
• This means for every yard, you have 3 feet.

To convert yards to feet, multiply the number of yards by 3.This is simple:

• If you have a length in yards and want to know it in feet, just apply: \( \frac{1 \text{ yd}}{3 \text{ ft}} \).
• For reverse conversion, divide feet by 3 to get the equivalent yards.

Having this conversion knowledge is especially practical for doing quick calculations without the need for a calculator, ensuring measurements are precise and accurate during important projects.

Meters to Centimeters

Meters and centimeters are both metric units of length commonly used worldwide.

• 1 meter is equal to 100 centimeters.
• This means for every meter, there are 100 centimeters.

To convert meters to centimeters, multiply the number of meters by 100. It is such a simple conversion because:

• Each meter is 100 times larger than a centimeter.
• The formula for conversion is straightforward: \( \frac{1 \text{ m}}{100 \text{ cm}} \).

Reverse conversion from centimeters to meters can be done by dividing by 100.This clearance in conversion helps in architectural designs, textile measurements, and many educational tasks. Understanding and performing these conversions correctly is a fundamental skill in science, mathematics, and daily life.