Problem 56 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{15 x}{75} $$

Step-by-Step Solution

Answer

\(\frac{x}{5}\)

Step 1: Identify Factors of Numerator and Denominator

First, identify the prime factors of the numerator and the denominator. The numerator consists of \(15x\) and the denominator is \(75\). The prime factors of \(15\) are \(3\) and \(5\), and for \(75\), the prime factors are \(3\) and \(5\), both repeated twice (i.e., \(3\) and \(5^2\)).

Step 2: Simplify the Numerical Parts

Now, the fraction \(15/75\) can be simplified by cancelling out the common factors. Both have a common prime factor of \(5\) which appears once in \(15\) and twice in \(75\), so one \(5\) can be cancelled out. Next, they also share a common factor of \(3\), which can also be cancelled out. This leaves us with \(1/5\).

Step 3: Include the Variable

Don’t forget to include the variable \(x\) from the numerator that was omitted during the numerical simplification. Ultimately, the simplification of \(\frac{15x}{75}\) results in \(\frac{x}{5}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime components. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For instance, in our exercise, we need to find the prime factors of the numbers in the fraction \(\frac{15}{75}\).

• 15 can be decomposed into prime numbers as \(3 \times 5\).
• 75 breaks down into the prime factors \(3 \times 5^2\) or written out, \(3 \times 5 \times 5\).

Prime factorization is crucial for simplifying fractions because it allows us to clearly see which numbers can be cancelled out in the numerator and the denominator. This paves the way for finding common factors.

Numerator and Denominator

A fraction is made up of two parts: the numerator and the denominator. The numerator is the top number, which represents the part of the whole we are considering. The denominator is the bottom number, representing the whole or the total number of parts.

• In the fraction \(\frac{15x}{75}\), \(15x\) is the numerator and represents 15 times the variable \(x\).
• The denominator, \(75\), represents the total or whole amount that is being divided into parts.

Understanding the roles of the numerator and denominator is vital. They help us visualize how a whole object or quantity can be divided, especially when simplifying expressions or comparing fractions.

Common Factors

Common factors are numbers that can divide both the numerator and the denominator without leaving a remainder. When simplifying fractions, we look for these common factors to reduce the fraction to its simplest form.

• For the fraction \(\frac{15x}{75}\), we found that both 15 and 75 have common factors of \(3\) and \(5\).
• By dividing both the numerator and the denominator by these common factors, the fraction can be simplified to its simplest form. In this case, dividing by \(3\) and \(5\) results in the numerical part becoming \(\frac{1}{5}\).

Finding and canceling out common factors simplifies calculations and makes it easier to understand and compare fractions or expressions.

Variables in Algebra

Variables represent unknown or changeable values in algebraic expressions. They are essential in expressing relationships or patterns in math. In our exercise,

The variable \(x\) appears in the numerator of the fraction \(\frac{15x}{75}\).

While simplifying, remember to keep the variable in the expression. It multiplied with numbers, not impacted by the common factors shared by 15 and 75.

In algebra, variables allow equations and expressions to be flexible and general. After simplifying, \(\frac{15x}{75}\) becomes \(\frac{x}{5}\), maintaining \(x\) as part of its result.