Problem 73 - Algebra A Combined Function

Question

Perform each indicated operation and simplify. $$ \frac{3}{5}\left(\frac{5}{3} x\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( x \).

Step 1: Identify the Operation

We are given the task of performing the operation which is the multiplication of two fractions, specifically \( \frac{3}{5} \times \frac{5}{3}x \). Notice that \( x \) is the variable involved and the operation involves both numeric and variable multiplication.

Step 2: Multiply the Fractions

Multiply the numeric parts of the fractions: \( \frac{3}{5} \times \frac{5}{3} \). Use the rule for multiplying fractions: multiply the numerators and the denominators. Here, \( 3 \times 5 = 15 \) for the numerator, and \( 5 \times 3 = 15 \) for the denominator, making the fraction \( \frac{15}{15} \).

Step 3: Simplify the Fraction

The fraction \( \frac{15}{15} \) simplifies to 1 because any number divided by itself is 1.

Step 4: Apply to the Expression

Now that we know \( \frac{15}{15} = 1 \), substitute back into the expression: \( 1 \times x = x \). Therefore, the original expression simplifies to \( x \).

Key Concepts

Multiplication of Fractions

When multiplying fractions, the process is straightforward: multiply the numerators together and the denominators together. For example, if you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), the product is \( \frac{a \times c}{b \times d} \).

This creates a new fraction which may then require simplification.
In the exercise example, both fractions are \( \frac{3}{5} \) and \( \frac{5}{3} \). When multiplied, the numerators 3 and 5 multiply to give 15, as do the denominators.
This results in \( \frac{15}{15} \), showing us that multiplication flipped the original sequence, simplifying our work ahead.

Simplifying Expressions

Simplifying in algebra is crucial for making expressions easier to understand and work with.
Once you have performed the multiplication of fractions, simplification follows.

• Identify the greatest common factor of the numerator and denominator,
• and divide both by it to reduce the fraction to its simplest form.

In the example \( \frac{15}{15} \), both the numerator and denominator are the same number. Thus, reducing the fraction is straightforward: divide both by 15 to get the simplest form, which is 1.
With a simplified expression, any remaining calculations with the variable become noticeably easier.

Variables in Algebra

Variables in algebra are symbols that stand in for unknown or changeable numbers, commonly represented by letters like \( x \), \( y \), or \( z \). Understanding how they function within expressions can drastically improve your algebraic skills.

In our exercise, the variable \( x \) was part of the fraction \( \frac{5}{3}x \). Multiplying by a fraction containing a variable is no different from multiplying numeric fractions.
The variable travels along with the fraction throughout the operation.

• Combine fractions first,
• before reattaching the variable to the simplified result.

Therefore, after the operation and simplification, 1 times \( x \) simply leaves \( x \) on its own,making continuity of algebraic expressions with variables straightforward and clearer.