Problem 82 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{3}{16} \cdot \frac{4}{4} $$

Step-by-Step Solution

Answer

The final answer in simplest form is \( \frac{3}{16} \).

Step 1: Simplify the second fraction

\(\frac{4}{4}\) equals 1. Any number divided by itself will always result in 1. Therefore, the problem reduces to \( \frac{3}{16} \cdot 1 \).

Step 2: Multiply the fractions

Multiplying any number by one results in the original number remaining unchanged. Thus, \( \frac{3}{16} \cdot 1 \) equals \( \frac{3}{16} \).

Key Concepts

Fraction Multiplication

When multiplying fractions, the process is relatively straightforward: you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, consider the multiplication of two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\). The product of these fractions will be \(\frac{a \times c}{b \times d}\).

In our given exercise, we have \(\frac{3}{16}\) multiplied by \(\frac{4}{4}\). Following the rules of fraction multiplication, you would typically multiply the numerators and then the denominators. However, the second fraction simplifies to 1, as any number divided by itself is 1. This simplification turns our multiplication into \(\frac{3}{16} \cdot 1\), making the process even more straightforward.

Understanding this concept is crucial as it lays the foundation for more complex operations involving fractions. Let's remember that multiplication is just repeated addition, and when we multiply by a fraction, it's like finding a part of the whole – a skill that is useful in many real-world applications, such as adjusting recipes or calculating areas.

Simplest Form

A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In other words, the fraction is simplified to its lowest terms, and can’t be reduced any further. Simplifying a fraction may involve dividing both the numerator and the denominator by their greatest common factor (GCF).

In our exercise \(\frac{3}{16}\), the fraction is already in simplest form. The reason is that 3 and 16 have no common factors other than 1. Keeping fractions in their simplest form helps with clarity; it allows us to easily compare and work with them without dealing with unnecessary complexity. It also often leads to more accurate results in calculations and helps in identifying equivalent fractions. Therefore, being able to reduce fractions to their simplest form is an essential skill in mathematics.

Mathematical Operations

Mathematical operations are procedures or functions that take one or more values, perform a computation, and produce a result. The basic mathematical operations are addition, subtraction, multiplication, and division. When working with fractions, these operations adhere to specific rules that sometimes differ from those applied to whole numbers.

For instance, addition and subtraction with fractions require a common denominator, whereas multiplication, as shown in our exercise, does not. Each operation serves its purpose in various applications across different fields such as science, engineering, economics, and everyday life. Understanding these operations and the rules that govern them is not only fundamental to learning mathematics but also crucial in developing problem-solving skills.