Problem 66 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{x+3}{\frac{1}{3}}$$

Step-by-Step Solution

Answer

3x + 9

Step 1: Understand the Expression

The expression is \( \frac{x+3}{\frac{1}{3}} \). Here, you need to divide the numerator \( x + 3 \) by the denominator \( \frac{1}{3} \).

Step 2: Simplify the Division

To simplify \( \frac{x+3}{\frac{1}{3}} \), multiply the numerator by the reciprocal of the denominator. The reciprocal of \( \frac{1}{3} \) is 3.

Step 3: Perform the Multiplication

Multiply \( x + 3 \) by 3: \( (x + 3) \times 3 \).

Step 4: Distribute the Multiplication

Distribute the 3 to both terms inside the parentheses: \( 3 \times x + 3 \times 3 = 3x + 9 \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator (the top part) and a denominator (the bottom part). For example, in the fraction \( \frac{1}{3} \), 1 is the numerator, and 3 is the denominator. Fractions are used in various arithmetic operations like addition, subtraction, multiplication, and division. Understanding how to work with fractions is crucial in algebra and other areas of math. In the given exercise, the fraction \( \frac{1}{3} \) is in the denominator, which requires us to find its reciprocal (or its 'flipped' form) to solve the problem.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form. This involves combining like terms or performing arithmetic operations. In the provided exercise, we started with the expression \( \frac{x+3}{\frac{1}{3}} \). To simplify, we need to perform division. Since dividing by a fraction is the same as multiplying by its reciprocal, we change the division into multiplication. We then multiply the numerator \( x + 3 \) by the reciprocal of the denominator \( \frac{1}{3} \), which is 3. This simplifies to the expression \( (x + 3) \times 3 \).

Distributive Property

The distributive property is a key algebraic principle that allows you to multiply a single term by each term inside parentheses. It states that \( a(b + c) = ab + ac \). We use this property to simplify expressions involving multiplication over addition or subtraction. In the exercise, after rewriting the division into multiplication, we have \( (x + 3) \times 3 \). Applying the distributive property, we multiply 3 by each term inside the parentheses: \( 3 \times x \) and \( 3 \times 3 \). This simplifies the expression to \( 3x + 9 \). This step is essential for breaking down and simplifying complex algebraic expressions.