Problem 39 - Fundamentals of Mathematics

Question

Use a unit fraction to convert 4 yd to feet.

Step-by-Step Solution

Answer

4 yards is equal to 12 feet.

Step 1: Understand the Conversion Relationship

First, we need to understand the relationship between yards and feet. We know that 1 yard is equal to 3 feet.

Step 2: Set Up the Conversion Fraction

To convert 4 yards to feet, we will use the conversion relationship as a unit fraction. Therefore, we set up the fraction \( \frac{3 \, \text{feet}}{1 \, \text{yard}} \).

Step 3: Apply the Conversion Fraction

Multiply the number of yards by the conversion fraction: \( 4 \, \text{yards} \times \frac{3 \, \text{feet}}{1 \, \text{yard}} \). Yards will cancel out, leaving us with feet.

Step 4: Perform the Calculation

Calculate the multiplication: \( 4 \times 3 = 12 \). So, 4 yards is equal to 12 feet.

Key Concepts

Understanding Unit Fractions

In the realm of unit conversion, a unit fraction plays a pivotal role. But what exactly is it? A unit fraction is a fraction where the numerator is one unit of the desired measurement, and the denominator is one unit of the original measurement.

By using unit fractions, you transform measurements without altering their quantity. It's like using a mathematical switch that changes units seamlessly. In conversion tasks, such as converting yards to feet, the unit fraction keeps track of which unit is being converted to which other unit.

For example, when converting from yards to feet, you will use the unit fraction \( \frac{3 \, \text{feet}}{1 \, \text{yard}} \). Notice here, 3 feet is equivalent to 1 yard.

This fraction allows you to maintain the accuracy of the conversion, ensuring your final answer reflects the true measurement in the new unit.

The Conversion Relationship

Understanding the conversion relationship is crucial for successful unit conversions. The conversion relationship is simply the known equivalence between two different units of measure.

For example:

• 1 yard is equivalent to 3 feet

This relationship tells us how many of one unit fits within another, acting as the foundation for creating the unit fraction.

In unit conversions, recognizing this relationship is the first step. Once you've established it, you can confidently set up your conversion fraction. Knowing the conversion relationship ensures that you're not arbitrarily switching numbers but making a mathematically correct transformation of units.

When you know that 1 yard equals 3 feet, you can then form your conversion fraction appropriately. This relationship also helps in problem-solving across various fields, such as physics, chemistry, and everyday life scenarios.

The Role of Multiplication in Unit Conversion

Multiplication is a key operation in the process of unit conversion. It works together with the unit fraction to change units seamlessly.

When we multiply by a unit fraction, we're not just multiplying numbers. We're also transforming the units of measurement.

Consider our practical example: converting 4 yards to feet. We set up our multiplication as:

• \(4 \, \text{yards} \times \frac{3 \, \text{feet}}{1 \, \text{yard}}\)

At this point, performing multiplication cancels out any unwanted unit. Here, the unit yard cancels out, leaving feet as our desired outcome.

This illustrates how multiplication isn't just about getting bigger numbers, it's about transforming what's being measured. Thus, multiplication with a unit fraction gives us the answer in the units we need, maintaining both the quantity and the measurement integrity of the original amount. Think of it as changing the costume of a character without altering the character itself.