Problem 16 - Fundamentals of Mathematics

Question

\(\frac{1}{4}\) of 3 is what number?

Step-by-Step Solution

Answer

The answer is \(\frac{3}{4}\).

Step 1: Understand the Problem

The problem is asking for the result when one-quarter (\(\frac{1}{4}\)) of the number 3 is calculated. This means multiplying 3 by \(\frac{1}{4}\) to find a portion of it.

Step 2: Set Up the Calculation

To find \(\frac{1}{4}\) of 3, we will set up the multiplication expression. This is expressed mathematically as: \(3 \times \frac{1}{4}\).

Step 3: Perform the Multiplication

Multiply 3 by \(\frac{1}{4}\). This involves multiplying the number by the numerator of the fraction and dividing by the denominator:\[ 3 \times \frac{1}{4} = \frac{3}{4} \].

Step 4: Simplify the Result

The fraction \(\frac{3}{4}\) is already in its simplest form. No further simplification is needed.

Key Concepts

Fractions

Fractions are an essential part of mathematics, helping us describe parts of a whole. They are written as two numbers separated by a line, like this: \(\frac{a}{b}\). The top number, \(a\), is called the numerator and tells us how many parts are being considered. The bottom number, \(b\), is the denominator, showing into how many equal parts the whole is divided.

Fractions can represent more than just parts of a whole; they can indicate division. For example, \(\frac{1}{4}\) can mean one part out of four or be understood as dividing 1 by 4. Knowing how to work with fractions is crucial, as they appear frequently in different areas of math, from geometry to algebra.

To get comfortable with fractions, always remember that they tell you about division; the numerator is divided by the denominator. This perspective helps tremendously when performing operations like addition, subtraction, multiplication, or division with fractions.

Basic Arithmetic

Basic arithmetic consists of the fundamental operations of math: addition, subtraction, multiplication, and division. In arithmetic involving fractions, multiplication allows you to find a part of a whole number by combining the value of the whole with a specific fraction.

For example, to find \(\frac{1}{4}\) of 3, you set up a multiplication problem: \(3 \times \frac{1}{4}\). This operation determines how much one-fourth of three is. The arithmetic rules simplify the process greatly by providing a clear path to find the answer.

Multiplying a whole number by a fraction entails:

• Multiplying the whole number (in this case, 3) by the numerator.
• Dividing the result by the denominator, which represents the number of divisions in the whole.

This process helps convert the multiplication expression into an understandable fraction or whole number answer.

Simplifying Fractions

Simplifying fractions is the process of making them easier to understand or work with. A fraction is simplified when the numerator and the denominator cannot be divided evenly by any number other than one.

In our exercise, the fraction \(\frac{3}{4}\) resulted from multiplying 3 by \(\frac{1}{4}\). Since there are no common factors in the numbers 3 and 4, \(\frac{3}{4}\) is already in its simplest form. Though simplifying might seem unnecessary in this case, it's an important skill when dealing with more complex fractions.

To simplify effectively:

• Factor both the numerator and the denominator completely.
• Divide both by the greatest common factor (GCF) to find the simplest form of the fraction.

While beginners often skip simplification when fractions are simple, practicing it prepares you for handling intricate fractional expressions later on.