Problem 10 - Algebra 1

Question

Simplify the expression if possible. $$\frac{15 x}{45}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x}{3}\).

Step 1: Identify the GCD of Numerator and Denominator

The first step is to identify the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 15 and 45 is 15.

Step 2: Divide Numerator and Denominator by GCD

Next, divide both the numerator and the denominator by the GCD. Thus, \(\frac{15 x}{45}\) simplifies to \(\frac{x}{3}\).

Step 3: Write the Simplified Expression

Write down the simplified expression, which is \(\frac{x}{3}\). This is the simplest form of the original expression.

Key Concepts

Greatest Common Divisor

When you're working with fractions, simplifying them can make calculations easier and clearer. The first step in simplifying a fraction like \(\frac{15x}{45}\) involves finding the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers exactly, without leaving a remainder. Finding the GCD requires some practice but can be made easier by listing out the factors:

• Factors of 15: 1, 3, 5, 15
• Factors of 45: 1, 3, 5, 9, 15, 45

Here, the largest common factor is 15, hence the GCD is 15. Knowing the GCD allows you to reduce the fraction by dividing both the numerator and denominator by this number. This step simplifies the fraction significantly.

Numerator and Denominator

In any fraction, such as \(\frac{15x}{45}\), there are two main parts: the numerator and the denominator. The numerator is the top number or expression, and the denominator is the bottom one. In our example, 15x is the numerator, while 45 is the denominator.Understanding these components is crucial when simplifying fractions. Once you've identified the GCD, which is 15 in this instance, you will need to divide both the numerator and the denominator by this GCD. By doing so, you obtain: \[\frac{15x}{45} = \frac{x}{3}\]The simplified fraction reflects a reduced form that is easier to interpret and work with. Always ensure you apply the same operation to both parts of the fraction, maintaining the equality of the expression.

Algebraic Fractions

Algebraic fractions, like \(\frac{15x}{45}\), involve variables and constants. Simplifying them requires the same basic principles as numerical fractions, but with a keener focus on any algebraic expressions involved.To simplify \(\frac{15x}{45}\), you follow the routine of finding the GCD and dividing both terms by it, resulting in \(\frac{x}{3}\). One key advantage of simplifying algebraic fractions is that it helps in solving equations more efficiently since it reduces complexity.Algebra itself might seem daunting at first, but by breaking it down into such manageable steps, it becomes more approachable. Always look to simplify your work when dealing with algebraic fractions to streamline further operations or solve equations seamlessly.