Problem 58 - Fundamentals of Mathematics

Question

State the numerator and denominator and write in words each of the fractions appearing in the statements for the following 10 problems. The fraction \(\frac{4}{3}\) is used in finding the volume of a sphere.

Step-by-Step Solution

Answer

The fraction \(\frac{4}{3}\) is 'four-thirds'.

Step 1: Identify the Fraction

The given fraction in the statement is \(\frac{4}{3}\). This fraction is used in the context of finding the volume of a sphere.

Step 2: Identify the Numerator

The numerator in the fraction \(\frac{4}{3}\) is 4.

Step 3: Identify the Denominator

The denominator in the fraction \(\frac{4}{3}\) is 3.

Step 4: Write Numerator in Words

The numerator, 4, is written in words as 'four'.

Step 5: Write Denominator in Words

The denominator, 3, is written in words as 'three'.

Step 6: Write the Fraction in Words

The fraction \(\frac{4}{3}\), written in words, is 'four-thirds.'

Key Concepts

Understanding the Numerator

When you look at a fraction like \( \frac{4}{3} \), the numerator is the top number. In this case, it is "4." The numerator tells you how many parts you actually have. It's as if you have a pizza divided into three slices, and you take four of those slices, perhaps combining them with slices from another pizza.

• The numerator represents a portion or part of a whole.
• It is always positioned above the fraction bar.
• You can visualize it as the number of items counted.

For our fraction \( \frac{4}{3} \), writing the numerator as a word would simply be: "four." This is straightforward but essential for two crucial reasons: it numerically describes part of a value and is critical in operations like addition and subtraction of fractions.

Exploring the Denominator

The denominator in a fraction like \( \frac{4}{3} \) is found at the bottom, here it is "3." This number is significant because it shows into how many equal parts the whole is divided. Consider our pizza again: if you divide one pizza into three slices, each slice represents one-third of the pizza.

• The denominator represents the total, or whole, number of equal parts.
• It indicates the size or value of each fractional part.
• It provides the framework to understand how large each "piece" is.

For \( \frac{4}{3} \), the denominator written in words is: "three." By understanding the role of the denominator, you can better comprehend the size of each piece or portion indicated by the fraction.

Calculating the Volume of a Sphere

In the realm of geometry, the formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). This equation is used to find how much space is inside a spherical object.
Here's how each part contributes:

• \( V \) represents the volume you're solving for.
• \( \frac{4}{3} \) is a critical fraction and contributes to the calculation of the volume.
• \( \pi \) is a constant (approximately 3.14159) that helps measure the curvature of the sphere.
• \( r \) stands for the radius of the sphere. It is the distance from the center of the sphere to any point on its surface.
• The cubed sign \( r^3 \) implies that the radius is multiplied by itself twice to form a volume component.

Understanding the fraction \( \frac{4}{3} \) is vital for this formula. It ensures that when you plug in the radius, you'll calculate the correct volume, scaled perfectly to the dimensions of a sphere. With a clear grasp of each component, solving for the volume can be a straightforward process.