Problem 12 - Algebra 1

Question

Simplify the expression. $$\frac{4 x}{3} \cdot \frac{1}{x}$$

Step-by-Step Solution

Answer

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

Step 1: Identify common factors

Inspect the fractions and notice that both fractions have a common factor, 'x'.

Step 2: Cancel out common factors

Multiply the fractions. The 'x' in the numerator of the first fraction \(\frac{4x}{3}\) and 'x' in the denominator \(\frac{1}{x}\) cancel out.

Step 3: Calculate the resulting expression

The resulting simplified expression is \(\frac{4}{3}\).

Key Concepts

Simplification

Simplification in algebra is the process of reducing an expression to its simplest form. This makes it easier to analyze and solve. When simplifying, we often look to eliminate unnecessary terms or factors. For example, in the expression \(\frac{4x}{3} \cdot \frac{1}{x}\), we focus on simplifying the multiplication by cancelling out common terms. Simplification helps in both arithmetic and algebraic expressions to make them more straightforward and easier to work with. The key is to keep the expression equivalent to the original one but in a reduced form.

Fractions

Fractions are a way of representing a part of a whole. They consist of a numerator and a denominator. In the expression \(\frac{4x}{3} \cdot \frac{1}{x}\), there are two fractions being multiplied. The first fraction, \(\frac{4x}{3}\), has '4x' as the numerator and '3' as the denominator. The second fraction, \(\frac{1}{x}\), has '1' as the numerator and 'x' as the denominator.

• Numerator: The top part of the fraction, indicating how many parts are considered.
• Denominator: The bottom part of the fraction, showing the total number of equal parts.

Working with fractions requires a good understanding of how these two parts interact. Properly managing numerators and denominators is crucial during operations such as addition, subtraction, multiplication, and division of fractions.

Multiplication

When we multiply fractions, we multiply the numerators together and the denominators together. It follows the rule: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \).

In our case, with the expression \( \frac{4x}{3} \cdot \frac{1}{x} \), this principle applies directly. So we multiply:

• Numerator: \(4x \times 1 = 4x\)
• Denominator: \(3 \times x = 3x\)

However, before completing the multiplication, it is wise to first simplify by cancelling common factors whenever possible. This can significantly simplify the calculation and the final expression.

Common Factors

Identifying common factors in expressions is crucial for simplifying them. A common factor is a number or variable that divides two terms exactly. In the multiplication of the fractions \( \frac{4x}{3} \) and \( \frac{1}{x} \), both have 'x' appearing once in the numerator and once in the denominator.

• Common factors can be cancelled out immediately if they appear in both the numerator and denominator at least once.
• Cancelling 'x' in \( \frac{4x}{3} \cdot \frac{1}{x} \) simplifies the problem as it leaves us with \( \frac{4}{3} \).

By removing the 'x', the expression is now easier to handle and reduces potential errors in further computations. Always look for opportunities to simplify by cancelling common factors in algebraic expressions.