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Chapter 2: Problem 14

Reduce each fraction to lowest terms. $$\frac{15 y}{25 y}$$

Short Answer

Expert verified
\(\frac{15y}{25y}\) reduces to \(\frac{3}{5}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of the numerator and the denominator. Here, the numerator is 15 and the denominator is 25. Factoring both, we find: - 15 can be factored into 3 脳 5. - 25 can be factored into 5 脳 5. The GCD of 15 and 25 is 5.
02

Simplify the Fraction

To reduce the fraction to its lowest terms, divide both the numerator and the denominator by their GCD. This gives: \[\frac{15y}{25y} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}\]The variables cancel out because they appear in both the numerator and the denominator as common terms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
The Greatest Common Divisor, often abbreviated as GCD, is a fundamental concept when working with fractions. It's the largest integer that can evenly divide two or more integers. In our case, with the fraction \( \frac{15y}{25y} \), the aim is to find the largest number that divides both 15 and 25 without leaving a remainder.

To figure out the GCD, we look at the factors of each number:
  • 15 breaks down into 3 and 5.
  • 25 breaks down into 5 and 5.
The common factor here is 5, which makes it the GCD. Knowing this helps simplify the fraction, making calculations easier and the numbers smaller. So remember, to find the GCD, factor each number and find the largest shared factor.
Simplification
Simplification involves reducing a fraction to its simplest form. By using the GCD, simplification becomes a straightforward process. For our fraction \( \frac{15y}{25y} \), we've established the GCD is 5. This means we need to divide both the numerator and the denominator by 5.

Here's how the simplification works:
  • Divide the top part (numerator) by 5, which changes 15 to 3.
  • Divide the bottom part (denominator) by 5, which changes 25 to 5.
This process results in \( \frac{3}{5} \). The goal is always to find the simplest version of a fraction to make it easier to work with in calculations. Simplifying reduces complexity and highlights the core value of the fraction.
Variables in Fractions
When dealing with fractions that include variables, like our example \( \frac{15y}{25y} \), it's essential to understand how they affect simplification. Variables represent unknown numbers, but they follow the same rules for multiplication and division.

In the fraction \( \frac{15y}{25y} \), the variable \( y \) appears in both the numerator and the denominator. Since they appear equally in both, they can be canceled out. This leaves only the numerical part to be simplified, aiding in reducing the fraction cleanly to \( \frac{3}{5} \).
  • If a variable is present in both parts of the fraction, treat it like a common factor.
  • Always check for common variables before simplifying the numerical part.
  • This helps avoid mistakes and ensures the fraction is correctly reduced.
Understanding this concept is crucial for dealing with variables and ensures accuracy in algebraic expressions.

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