Problem 75 - Chemistry Matter and Change

Question

When converting from meters to centimeters, how do you decide which values to place in the numerator and denominator of the conversion factor?

Step-by-Step Solution

Answer

When converting from meters to centimeters, you must use the relationship \(1 m = 100 cm\). To create a conversion factor, write the ratio \(\frac{100\, cm}{1\, m}\), with meters in the denominator and centimeters in the numerator. Multiply the given value in meters by this conversion factor and simplify to find the equivalent value in centimeters: \(Value_{m} * \frac{100\, cm}{1\, m} = Value_{cm}\). The meters units will cancel out, leaving you with the value in centimeters.

Step 1: Understand the conversion relationship

The relationship between meters and centimeters is that 1 meter = 100 centimeters. This means that there are 100 centimeters in 1 meter. Keep in mind that this ratio should be used when converting between these two units.

Step 2: Identify the starting value and desired unit

Suppose you have a certain length in meters that you want to convert to centimeters. Write down the value in meters. Identify the desired unit: in this case, centimeters.

Step 3: Write the conversion factor ratio

Create a conversion factor using the relationship between meters and centimeters. Write down the conversion ratio: \(\frac{100 \, cm}{1\, m}\)

Step 4: Arrange the conversion factor

Arrange the conversion ratio in such a way that meters are in the denominator and centimeters are in the numerator. This is important since we want the meters to cancel each other out while performing the conversion. So, the conversion factor is: \(\frac{100 \, cm}{1\, m}\).

Step 5: Multiply the value by the conversion factor

Now, multiply the given value in meters by the conversion factor to find the equivalent value in centimeters: \(Value_{m} * \frac{100 \, cm}{1\, m} = Value_{cm}\).

Step 6: Simplify and cancel the units

The meters unit in the numerator will cancel out the meters unit in the denominator. Perform the multiplication and you will be left with the value in centimeters.

Key Concepts

Conversion Factor

A conversion factor acts like a fancy translator between two different types of units. Imagine you're turning a sentence from English into another language using a dictionary. That's similar to how a conversion factor helps translate one measurement into another. In our exercise, we want to convert meters into centimeters. Here's how it works:

• First, we understand that 1 meter is equivalent to 100 centimeters.
• With this knowledge, we can set up a fraction or ratio known as the conversion factor to make the conversion easy.

This fraction \( \frac{100 \, cm}{1 \, m} \) is essential because it tells you exactly how many centimeters make up one meter. What's neat is that by multiplying your number of meters by this conversion factor, you end up with centimeters, almost like magic! This technique ensures that you stick to the correct relationship, helping you avoid mistakes when you want to switch from one unit to another.

Length Measurement

Length is one of the most fundamental ways we measure the world around us. It helps us know how far, long, tall, or wide something is. For length measurement, there are different units that can be used depending on the context - like how tall a tree is or how long a car can be. The basic units of length that many people start learning with are meters and centimeters. Understanding these units can help you communicate more accurately:

• A meter is a unit often used for measuring larger spaces like rooms.
• Centimeters are smaller and more precise, making them useful for measuring smaller objects like a book's width or a pencil's length.

When you measure something, you choose the best unit based on the size of what you're observing. Understanding which units work best for different tasks makes measurements useful and meaningful in everyday life.

Metric System

The metric system is a brilliant way of simplifying how we measure things. It's used all around the world, which helps people communicate scientific and everyday measurements clearly, even if they speak different languages. The beauty of the metric system is in its simplicity:

• It is based on the number 10, making conversions straightforward.
• This system uses a series of base units like meters for length, grams for weight, and liters for volume.

Working with the metric system involves using prefixes, such as 'centi' or 'milli,' that indicate multiplication by a fraction of the number 10. For example:

• 'Centi-' means one hundredth, so one centimeter is one hundredth of a meter.
• 'Milli-' implies one thousandth, such as a millimeter being one thousandth of a meter.

This orderly sequence of units makes calculations more predictable and reduces errors, especially when converting units.