Problem 73 - Prealgebra

Question

Reduce to lowest terms. $$\frac{75 x^{2} y^{3}}{100 x y^{2}}$$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{3xy}{4}\).

Step 1: Factor Numerator and Denominator

First, examine the numerator and the denominator of the fraction: Numerator: \(75x^2y^3\)- Factor \(75\) into prime factors: \(75 = 3 \times 5^2\).- Therefore, numerator in factored form: \(3 \times 5^2 \times x^2 \times y^3\).Denominator: \(100xy^2\)- Factor \(100\) into prime factors: \(100 = 2^2 \times 5^2\).- Therefore, denominator in factored form: \(2^2 \times 5^2 \times x \times y^2\).

Step 2: Cancel Common Factors

Look for common factors between the numerator and the denominator:- Both numerator and denominator have \(5^2\), which can be cancelled.- Both have \(x\), so cancel one \(x\) from each.- Both have \(y^2\), so cancel \(y^2\) from each.After cancellation:\[\frac{3 \times x \times y}{2^2} = \frac{3xy}{4}\]

Step 3: Final Step: Simplify the Fraction

Since all common factors have been cancelled, we are left with:\[\frac{3xy}{4}\]This fraction is in its simplest form as there are no common factors in the numerator and the denominator.

Key Concepts

Prime Factorization

Prime factorization is like breaking down a number into its most basic building blocks, known as prime numbers. Let’s dive into what primes and prime factorization mean.

• Prime Numbers: A prime number is a number greater than 1 that has no divisors other than 1 and itself. For example, 2, 3, 5, 7, etc.
• Factorization Process: To factor a number means to break it down into numbers that, when multiplied together, give the original number. With prime factorization, the trick is to use only prime numbers in this breakdown.

For example, in the original exercise, we used prime factorization to separate 75 into 3 and two 5's (written as \(3 \times 5^2\)). Similarly, 100 became \(2^2 \times 5^2\). This step is crucial as it makes spotting common factors between the numerator and denominator much simpler.

Common Factors

Simply put, common factors are numbers or variables that appear in both the numerator and the denominator of a fraction and can therefore be cancelled out.
Here’s how to identify and utilize them:

• Identify Common Numbers: After using prime factorization to break elements into their parts, check what's common in both the numerator and denominator. In our exercise, \(5^2\), \(x^1\), and \(y^2\) appeared in both.
• Cancel Them Out: Once identified, you can cancel these common factors, because essentially, you’re dividing them by themselves resulting in the value of 1.

This canceling step is what simplifies the fraction. It turns a complicated looking fraction into something more manageable.

Simplest Form

Bringing a fraction to its simplest form means reducing it to the point where no common factors exist between the numerator and the denominator. Let’s explore how:

• The End Goal: The main objective is to ensure the numerator and the denominator have no shared factors other than 1. This means every number and variable in the numerator and denominator are completely unique to each part.
• Why Simplify?: A simplified fraction is easier to understand and use in further calculations. Mathematicians love simplicity, and this process eliminates unnecessary complexity.

In our step-by-step solution, we ended with \(\frac{3xy}{4}\), which is in its simplest form as no further common factors can be found. Maintaining simplicity is key for clarity and accuracy going forward.