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Chapter 8: Problem 1

Reduce each of the following fractions to lowest terms. $$ \frac{30 y}{35 y} $$

Short Answer

Expert verified
Question: Reduce the fraction \(\frac{30 y}{35 y}\) to its lowest terms. Answer: \(\frac{6}{7}\)

Step by step solution

01

Determine the GCD of the coefficients

Find the greatest common divisor of 30 and 35. One way to do this is to list the factors of each number and find the largest factor they have in common. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 35: 1, 5, 7, 35 The GCD of 30 and 35 is 5.
02

Simplify the coefficients with the GCD

Divide both coefficients by the GCD: Numerator: \(30 \div 5 = 6\) Denominator: \(35 \div 5 = 7\) So the simplified coefficients are 6 and 7.
03

Cancel the common variable

Since both the numerator and the denominator have the same variable, y, we can cancel it out: $$ \frac{6y}{7y} $$ After canceling y, we are left with: $$ \frac{6}{7} $$
04

Write the final simplified fraction

The simplified fraction is: $$ \frac{6}{7} $$

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Key Concepts

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

Greatest Common Divisor
Understanding the Greatest Common Divisor (GCD) is crucial when simplifying algebraic fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. It's like finding the biggest shared piece you can evenly break two numbers into. Simplifying a fraction starts with identifying the GCD of its numerator and denominator. Once found, both parts of the fraction can be divided by this number to reduce the fraction to its lowest terms. Think of it as peeling away the common layers until you're left with the simplest form possible.
Factorization
Next up, let's talk about factorization. Factorization is like breaking down a number into its building blocks. These blocks are the number's factors鈥攖he whole numbers that multiply together to give the original number. For example, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30 because you can multiply pairs of these numbers to get 30. When simplifying fractions, factorization helps you see which numbers (or variables) can be divided out. It's a bit like taking apart a Lego structure to see how you can reassemble it in a more streamlined form.
Algebraic Expression Simplification
Simplifying an algebraic expression means reducing it to its most basic form without changing its value. This process, much like simplifying numerical fractions, often involves finding common factors, but with variables included. For instance, with the fraction \(\frac{30y}{35y}\), you identify common numerical factors and also notice that the variable 'y' appears in both the numerator and the denominator, hinting at further simplification. It's like distilling a complicated recipe down to its essential ingredients, ensuring you keep the flavor without the clutter.
Variable Cancellation
When a variable appears in both the numerator and the denominator of a fraction, you can often 'cancel' it out, which is another form of simplification. This variable cancellation process is based on the property that a number (or variable) divided by itself equals one. For instance, in the algebraic fraction \(\frac{6y}{7y}\), the variable 'y' can be cancelled from both sides of the fraction. Think of it as if you owed someone \(5 and at the same time, they owed you \)5; the debts cancel each other out and you're left with a clean slate.

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Most popular questions from this chapter

For the following problems, perform the divisions. $$ \frac{20 y^{2}+15 y-4}{4 y+3} $$

For the following problems, perform the indicated operations. $$ \frac{7 a}{a+6}+\frac{5 a}{a-8} $$

For the following problems, perform the divisions. $$ \frac{4 x^{3}+4 x^{2}-3 x-2}{2 x-1} $$

For the following problems, perform the indicated operations. $$ \frac{x+6}{x-1} \cdot \frac{x+7}{x+6} $$

For the following problems, solve the rational equations. $$ \frac{4 y-5}{4}+\frac{8 y+1}{6}=\frac{-69}{12} $$
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