Problem 27 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{24 x y}{40 y}$$

Step-by-Step Solution

Answer

The fraction \(\frac{24xy}{40y}\) simplifies to \(\frac{3x}{5}\).

Step 1: Identify the GCD

First, identify the greatest common divisor (GCD) of the numerator and the denominator. The numerator is \(24xy\) and the denominator is \(40y\). The numerical GCD of 24 and 40 is 8.

Step 2: Factorize Numerator and Denominator

Write the numerator and the denominator in terms of their prime factors and common variables. The numerator \(24xy\) becomes \(8 imes 3 imes x imes y\) and the denominator \(40y\) becomes \(8 imes 5 imes y\).

Step 3: Cancel Common Factors

Cancel the common factors from the numerator and the denominator. Both have \(8\) and \(y\) as common factors. After cancelling, we have \(3x\) in the numerator and \(5\) in the denominator.

Step 4: Write Simplified Fraction

With the common factors cancelled, the fraction \(\frac{24xy}{40y}\) simplifies to \(\frac{3x}{5}\).

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is an important concept when it comes to simplifying fractions. What do we mean by the greatest common divisor? It's the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. This helps in reducing fractions to their simplest form.

Think of it as finding common ground between two opponents. In our case, the numerator and the denominator are the opponents, and we are trying to find a compromise that brings them together by reducing the fraction.

To find the GCD, you can either list the divisors of each number and choose the largest one they share, or you can use the prime factorization method to help identify it. This can sound tricky, but with practice, it becomes a lot easier!

Prime Factorization

Prime factorization is like breaking a number down into a list of prime numbers that multiply together to give the original number. A prime number is a number greater than one that has no divisors other than one and itself.

For example, when you look at the number 24, you can break it down into its prime components: it’s made up of the prime numbers 2 and 3. Written as prime factors, the number 24 is:

• 24 = 2 \( \times \) 2 \( \times \) 2 \( \times \) 3

Similarly, 40 breaks down into:

• 40 = 2 \( \times \) 2 \( \times \) 2 \( \times \) 5

Prime factorization is especially helpful when you’re simplifying algebraic fractions, as it enables you to visualize any common factors easily. Once you identify common factors, you can efficiently reduce the fraction to its simplest form.

Simplifying Algebraic Fractions

Now, let's put all of this knowledge together to simplify algebraic fractions. Simplifying an algebraic fraction entails reducing a fraction that contains variables, so that it’s as simple as possible. It’s a bit like cleaning up your room so that it’s easier to find things!

Firstly, factor the numerator and denominator to identify any common factors. These factors often include both numbers and variables. Once you've identified and cancelled these, what remains is the fraction in its simplest form.

Consider the example fraction \( \frac{24xy}{40y} \). After finding the GCD, which includes the number 8 and variable y, and cancelling these out from both the numerator and denominator, you are left with the simplified fraction \( \frac{3x}{5} \). Remember, cancelling is like a mini subtraction where you're taking away the common parts to leave behind only what’s necessary.

By practicing, you become quicker and more accurate at simplifying these expressions, making it a powerful tool in solving algebraic equations and simplifying answers.