Problem 61 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.1) $$\frac{4}{3} \cdot x \text { when } x=\frac{1}{6}$$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{2}{9}\) when \(x = \frac{1}{6}\)

Step 1: Substitute the given value

Start by substituting \(x = \frac{1}{6}\) into the expression \(\frac{4}{3} \cdot x\). This gives us \(\frac{4}{3} \cdot \frac{1}{6}\)

Step 2: Multiply the fractions

Next, multiply the fractions together. Multiply the numerators (the top part of the fractions) together and the denominators (the bottom part) together. We get \((\frac{4}{1} \cdot \frac{1}{1}) / (\frac{3}{1} \cdot \frac{6}{1}) = \frac{4}{18}\)

Step 3: Simplify the Fraction

Finally, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which in this case is 2. This gives us \(\frac{4}{18}= \frac{2}{9}\)

Key Concepts

Fraction Multiplication

Fraction multiplication is a way of finding the product of two fractions. When multiplying fractions, the process is quite straightforward and follows a simple rule. You multiply the numerators (the top numbers) to get the new numerator and the denominators (the bottom numbers) to get the new denominator.

For example, when you have the fractions \( \frac{4}{3} \) and \( \frac{1}{6} \), you multiply the numerators together (\( 4 \times 1 \)) and the denominators together (\( 3 \times 6 \)). The result is \( \frac{4}{18} \).

This method makes it manageable to combine fractions without finding a common denominator first, unlike addition or subtraction of fractions.

Variable Substitution

Variable substitution is the process of replacing a variable in an algebraic expression with a given numeric value.

This tactic is particularly useful when you have an expression involving variables, and you want to find its value for specific numbers.

In the given exercise, the expression is \( \frac{4}{3} \cdot x \), and you need to evaluate it for \( x = \frac{1}{6} \). This substitution gives you a new expression \( \frac{4}{3} \cdot \frac{1}{6} \), which can be directly calculated.

It is an essential concept because it allows us to turn variable expressions into numerical calculations that can be evaluated easily.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its lowest terms. This means making the fraction as simple as possible by dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our exercise, after multiplying the fractions, you end up with \( \frac{4}{18} \). Both 4 and 18 can be divided by 2, which is the GCD. Thus, \( \frac{4}{18} \) simplifies to \( \frac{2}{9} \).

The simplified fraction represents the same value as the original, but in a more concise form. Simplifying is important in mathematics because it makes equations easier to understand and work with.