Problem 84 - Algebra 1: Concepts and Skills

Question

Simplify. \(\frac{25}{15}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{25}{15}\) is \(\frac{5}{3}\).

Step 1: Identifying the GCD

The first step is to find the greatest common divisor (GCD) or the highest common factor (HCF) of the numbers 25 and 15. The number 5 is the largest number which divides both 25 and 15 without any remainder.

Step 2: Simplifying the Fraction

After identifying the GCD, the numerator and the denominator of the fraction are divided by the GCD. In this case, they are both divided by 5. Thus, the fraction \(\frac{25}{15}\) becomes \(\frac{25/5}{15/5}\) which simplifies to \(\frac{5}{3}\).

Key Concepts

Greatest Common Divisor (GCD)

Understanding the Greatest Common Divisor (GCD), also known as the greatest common factor (GCF), is crucial when attempting to simplify fractions. It's the largest number that can divide two or more integers without leaving a remainder. To find it, consider the factors of each number, and then compare these factors to find the highest number they share.

For example, the factors of 25 are 1, 5, and 25, while the factors of 15 are 1, 3, 5, and 15. The shared factors are 1 and 5, but since we’re looking for the greatest, the GCD is 5. Calculating the GCD is the first step in fraction reduction, providing a pathway to simplify numerical expressions efficiently.

Simplifying Numerical Expressions

The technique of simplifying numerical expressions helps clarify complex mathematical problems by reducing them to their simplest form. When dealing with fractions, this process involves using the GCD to divide both the numerator (top number) and the denominator (bottom number).

To illustrate, with the numerical expression \(\frac{25}{15}\), we divide both 25 and 15 by the GCD of 5. Simplification of numerical expressions allows us to convert the fraction to \(\frac{25/5}{15/5}\), or \(\frac{5}{3}\). Simplification is especially important in algebra and calculus, where it enables students to more easily solve for variables and integration problems.

Fraction Reduction

Fraction reduction is a specific instance of simplifying numerical expressions, aiming to scale down fractions to their simplest form. This not only makes calculations easier but also helps in comparing the values of various fractions.

As previously demonstrated, when we reduced \(\frac{25}{15}\) by dividing both terms by their GCD of 5, we achieved the reduction to \(\frac{5}{3}\). This process is universal for all fractions, irrespective of their complexity. Reduced fractions are vital in all forms of mathematical calculations — be it in measuring ingredients for a recipe or solving an integral in calculus, the concept of fraction reduction proves to be an essential mathematical skill.