Problem 28 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{50 x y}{75 x}$$

Step-by-Step Solution

Answer

The fraction reduces to \(\frac{2y}{3}\).

Step 1: Identify Common Factors

First, identify the common factors in both the numerator and the denominator. The numerator is \(50xy\), and the denominator is \(75x\). Notice that both have the factor \(x\).

Step 2: Factor Numbers

Look at the numerical part: 50 and 75. Find the greatest common divisor (GCD) of 50 and 75. Both can be divided by 25, which is the GCD.

Step 3: Cancel Common Factors

Cancel out the common factor, \(x\), and the GCD, 25, from both the numerator and the denominator. So, \(\frac{50xy}{75x}\) can be reduced to \(\frac{(50 \div 25)(xy \div x)}{(75 \div 25)(x \div x)}\), which simplifies to \(\frac{2y}{3}\).

Step 4: Simplify Fraction

After canceling, you are left with \(\frac{2y}{3}\). Check to make sure there are no other common factors in the new numerator and denominator.

Key Concepts

Greatest Common Divisor (GCD)

Understanding the concept of the Greatest Common Divisor (GCD) is fundamental when simplifying fractions. The GCD is the largest number that divides two or more numbers without leaving a remainder. To simplify fractions, identifying the GCD helps in reducing the numerical portion effectively.

The GCD can be found through various methods:

• Listing factors: List all factors of each number and identify the greatest one that is common.
• Prime factorization: Break down each number into prime factors and multiply the common prime factors.
• Euclidean algorithm: A more complex method, but very efficient for large numbers.

By using the GCD, you ensure the fraction is reduced to its simplest form, where the numerator and denominator have no other common divisors other than 1.

Numerators and Denominators

A fraction consists of a numerator and a denominator. The numerator is the top part of the fraction, representing how many parts you have. The denominator is the bottom part, indicating into how many parts the whole is divided. In the fraction \( \frac{50xy}{75x} \), 50xy is the numerator, and 75x is the denominator.

When reducing a fraction, both the numerator and the denominator should be treated similarly. Look for:

• Common variables: Variables appearing in both should be simplified wherever possible.
• Numerical common factors: Use the GCD to reduce the numbers ensuring the fraction is in its simplest form.

Proper management of both numerators and denominators leads to a neat and simplified result.

Reducing Fractions

Simplifying or reducing fractions means to make the fraction as simple as possible. You do this by dividing the top and bottom by their greatest common factor. This retains the value of the fraction but makes it easier to read and work with.

The steps are straightforward:

• Identify the GCD of the numerator and the denominator.
• Divide both numerator and denominator by their GCD.
• Simplify any remaining terms, like canceling out equal variables.

Reducing fractions not only simplifies calculations but also helps in understanding the underlying relationships between numerators and denominators more clearly.

In the example given, reducing \( \frac{50xy}{75x} \) involves recognizing 25 as the GCD and cancelling out factor x, resulting in \( \frac{2y}{3} \), which is the simplest form.