Problem 52 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. One-third of \(\frac{6 x}{y}\)

Step-by-Step Solution

Answer

\(\frac{2x}{y}\)

Step 1 - Understand the Problem

The problem asks to perform the operation involving one-third of the given fraction \(\frac{6x}{y}\).

Step 2 - Write the Expression

Write the mathematical expression for one-third of \(\frac{6x}{y}\). This can be represented as: \(\frac{1}{3} \times \frac{6x}{y}\).

Step 3 - Perform Multiplication

Multiply \(\frac{1}{3}\) by \(\frac{6x}{y}\). Use the property of multiplying fractions: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\). This gives: \(\frac{1 \times 6x}{3 \times y} = \frac{6x}{3y}\).

Step 4 - Simplify the Fraction

Simplify the fraction \(\frac{6x}{3y}\). Divide the numerator and the denominator by 3: \(\frac{6x \text{/} 3}{3y \text{/} 3} = \frac{2x}{y}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in algebra and many other areas of math. It makes fractions easier to work with and understand. The key idea is to reduce the fraction to its simplest form where the numerator and the denominator have no common factors other than 1.

For example, simplifying \(\frac{6x}{3y}\)\

Multiplying Fractions

Multiplying fractions is straightforward once you understand the rule: you multiply the numerators together and the denominators together.

In mathematical terms, if you have two fractions \(\frac{a}{b} \times \frac{c}{d}\), then their product is \(\frac{ac}{bd}\).

In our original exercise, we needed to find one-third of \(\frac{6x}{y}\), which can be written as \(\frac{1}{3} \times \frac{6x}{y}\). Using the rule for multiplying fractions: multiply the numerators (1 and 6x), and the denominators (3 and y)

This gives us \(\frac{1 \times 6x}{3 \times y} = \frac{6x}{3y}\). This straightforward application of the rule is very powerful and can be used in many types of fraction operations.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. In our example, the algebraic expression was \(\frac{6x}{y}\).

When dealing with algebraic expressions in fractions, the rules for fractions still apply but you also need to be mindful of the variables. Variables represent a number that can change or that we don't know yet.

In our case, 'x' and 'y' are variables. When you're asked to perform operations on algebraic fractions, always simplify the algebraic expression if possible, just as you would with numerical fractions.

Key tips for working with algebraic expressions in fractions:

• Perform operations as you would with numbers.
• Always simplify the final expression.
• Be careful with variables and ensure the expression remains meaningful.

Understanding these principles will make it easier for you to handle more complex algebraic problems.