Problem 59 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{4}{7} \cdot \frac{7}{4}$$

Step-by-Step Solution

Answer

The product is 1.

Step 1: Identify the fractions to multiply

In this problem, we are given two fractions: \(\frac{4}{7}\) and \(\frac{7}{4}\). Our task is to multiply these fractions and simplify the result.

Step 2: Multiply the numerators

To find the product of the fractions, first multiply the numerators. Here, the numerators are 4 and 7. So, \(4 \times 7 = 28\).

Step 3: Multiply the denominators

Next, multiply the denominators. Here, the denominators are 7 and 4. So, \(7 \times 4 = 28\).

Step 4: Form the new fraction

Combine the products of the numerators and denominators to form the new fraction: \(\frac{28}{28}\).

Step 5: Simplify the fraction

The fraction \(\frac{28}{28}\) can be simplified. Since both the numerator and the denominator are the same, the fraction simplifies to 1.

Key Concepts

Fraction Simplification

When working with fractions, simplification is an essential part of the calculation process. It means expressing a fraction in its simplest form. This often involves reducing it so that the numerator and denominator have no common factors other than 1. When multiplying fractions, simplification helps you find the most condensed version of the resultant fraction.

In our example, by multiplying \( \frac{4}{7} \times \frac{7}{4} \), we get \( \frac{28}{28} \). Since both the numerator and denominator are equal, they have a common factor of 28. Thus, we simplify the fraction to 1:

• Check if the numerator and denominator are identical.
• Divide both by their greatest common factor.
• Write the result as a whole number if possible.

This makes it easier to understand and compare fractions in their most basic form.

Numerator and Denominator

Fractions are made up of two parts: the numerator and the denominator. The numerator is the top number, showing how many parts we have or are considering, while the denominator is the bottom number, indicating the total number of equal parts the whole is divided into.

In our example, the fractions \( \frac{4}{7} \) and \( \frac{7}{4} \) both have their own numerators and denominators. When multiplying fractions, you multiply the numerators together and the denominators together. This step-by-step process forms the new fraction before simplification:

• Multiply numerators to find the new numerator: \( 4 \times 7 = 28 \).
• Multiply denominators to find the new denominator: \( 7 \times 4 = 28 \).

Understanding these terms is crucial. This helps in recognizing how and why fractions combine the way they do.

Elementary Mathematics

Elementary mathematics involves the basics of arithmetic including addition, subtraction, multiplication, and division, along with a fundamental understanding of numbers and their relationships. Working with fractions is an important aspect of elementary mathematics, as it lays the groundwork for more advanced topics.

Multiplying fractions exemplifies the importance of capturing basic arithmetic skills. It teaches:

• The step-by-step process of performing operations on fractions.
• The value of simplification for clearer, correct answers.
• Conceptual understanding of part-whole relationships.

Fractions, combining the conceptual views of division and multiplication, are essential in everyday situations, helping to build a foundation for meaningful number recognition and manipulation. By mastering these skills early on, students are better prepared for more complex mathematical concepts and problems.