Question

Convert \(2.255 \times 10^{10} \mathrm{~g}\) to each unit. (a) \(\mathrm{kg}\) (b) \(\mathrm{Mg}\) (c) \(\mathrm{mg}\) (d) metric tons (1 metric ton \(=1000 \mathrm{~kg}\) )

Short answer

2.255 \(\times\) 10^{10} \mathrm{~g}\) equals (a) 2.255 \(\times\) 10^{7} \mathrm{~kg}\), (b) 2.255 \(\times\) 10^{4} \mathrm{~Mg}\), (c) 2.255 \(\times\) 10^{13} \mathrm{~mg}\), and (d) 2.255 \(\times\) 10^{4} metric tons.

Step-by-step solution

Convert to kilograms (kg)

To convert grams (g) to kilograms (kg), divide the number of grams by 1000, because there are 1000 grams in a kilogram. So, to convert 2.255 \(\times\) 10^{10} g to kg, divide it by 1000: \[2.255 \times 10^{10} \text{ g} \div 1000 = 2.255 \times 10^{7} \text{ kg}\]

Convert to megagrams (Mg)

A megagram (Mg) is equivalent to a metric ton, which is 1000 kilograms. To convert kilograms to megagrams, you divide the number of kilograms by 1000. So, to convert 2.255 \(\times\) 10^{7} kg to Mg: \[2.255 \times 10^{7} \text{ kg} \div 1000 = 2.255 \times 10^{4} \text{ Mg}\]

Convert to milligrams (mg)

To convert grams to milligrams, multiply the number of grams by 1000, because there are 1000 milligrams in a gram. To convert 2.255 \(\times\) 10^{10} g to mg: \[2.255 \times 10^{10} \text{ g} \times 1000 = 2.255 \times 10^{13} \text{ mg}\]

Convert to metric tons

Since one metric ton equals 1000 kilograms, to convert kilograms to metric tons, you divide the number of kilograms by 1000. To convert 2.255 \(\times\) 10^{7} kg to metric tons: \[2.255 \times 10^{7} \text{ kg} \div 1000 = 2.255 \times 10^{4} \text{ metric tons}\]

Key concepts

Converting Grams to Kilograms

Understanding how to convert grams (g) to kilograms (kg) is essential when dealing with mass in the metric system. Since 1 kilogram equals 1000 grams, converting is as simple as dividing the number of grams by 1000.

For example, let's take the exercise where you have to convert 2.255 \times 10^{10} g to kilograms. By following the conversion rule, you divide the given number by 1000:
\[2.255 \times 10^{10} \text{ g} \div 1000 = 2.255 \times 10^{7} \text{ kg}\]

This process is practical because it merely requires one to move the decimal point three places to the left.

Converting Kilograms to Megagrams

A megagram (Mg), which is the same as a metric ton, is not as commonly used as kilograms or grams, but it is the standard unit for larger masses. Just like converting grams to kilograms, converting kilograms to megagrams (or metric tons) involves dividing by 1000, because there are 1000 kilograms in a megagram.

To convert from the exercise's 2.255 \times 10^{7} kg to megagrams, you divide by 1000:
\[2.255 \times 10^{7} \text{ kg} \div 1000 = 2.255 \times 10^{4} \text{ Mg}\]

Remember, when working with conversions, keep track of your units to avoid confusion!

Converting Grams to Milligrams

When you need to express mass in smaller units, milligrams (mg) come into play. To convert from grams to milligrams, you multiply by 1000 because there are 1000 milligrams in a gram. It's essentially the opposite process of converting grams to kilograms.

Using the same exercise, converting 2.255 \times 10^{10} g to milligrams is straightforward:
\[2.255 \times 10^{10} \text{ g} \times 1000 = 2.255 \times 10^{13} \text{ mg}\]

This conversion is handy in fields like medicine and chemistry where precise, smaller measurements are crucial.

Understanding Metric Tons

The metric ton is a unit of mass equal to 1,000 kilograms or one megagram. It's widely used in commerce and everyday language to express heavy weights like the mass of vehicles or produced materials.

In our exercise, converting kilograms to metric tons follows the same method as converting to megagrams, since they are equivalent:
\[2.255 \times 10^{7} \text{ kg} \div 1000 = 2.255 \times 10^{4} \text{ metric tons}\]

An interesting fact is that in some countries, 'ton' could mean either a short ton (907.185 kg) or a long ton (1016.05 kg), but a 'metric ton' is universally recognized as 1000 kg.