Question

\(\frac{3 x}{27}=\frac{2}{9}\)

Short answer

The solution is \( x = 2 \).

Step-by-step solution

Simplify the Denominators

Both sides of the equation have fractions with denominators that can be simplified. On the left, simplify \( \frac{3x}{27} \) by dividing both the numerator and denominator by 3, which gives \( \frac{x}{9} \). The equation then becomes \( \frac{x}{9} =\frac{2}{9} \).

Compare Numerators

Since the denominators are the same, we can set the numerators equal to each other to solve for \( x \). So, \( x = 2 \).

Key concepts

Understanding Numerators

In fractions, the numerator is the number located above the fraction line. It tells us how many parts of a whole we have. In the exercise equation \( \frac{3x}{27} = \frac{2}{9} \), the numerators are \( 3x \) and \( 2 \).
Let's break down why numerators matter in solving equations.

• The numerator represents the number of equal parts being considered. If we think of a pie cut into sections, the numerator shows how many of those pie pieces are being discussed.
• In our equation, the numerator \( 3x \) suggests that the term \( x \) is multiplied by 3. This affects how we approach solving for \( x \) as we need to ensure balance while keeping track of these segments.
• After simplification of the fraction \( \frac{3x}{27} \) to \( \frac{x}{9} \), solving gets easier since the numerators can now be directly compared (\( x \) and \( 2 \)).

When the denominators match, as they do post-simplification (both are 9), we focus solely on the numerators to solve the problem. In our case, we equate \( x \) to 2 to maintain the integrity of the original equation.

The Role of Denominators

Denominators, found below the fraction line, indicate the total number of equal parts into which something is divided. In the equation \( \frac{3x}{27} = \frac{2}{9} \), the denominators are initially 27 and 9.
Understanding denominators helps us visualize the scale or the size of each fraction:

• The denominator shows how many parts make up a whole. The larger the denominator, the smaller each part becomes.
• In our math exercise, simplifying \( \frac{3x}{27} \) by dividing both the numerator and the denominator by 3 results in \( \frac{x}{9} \). This process aims to make the fractions easier to compare or solve by matching their denominators.
• With both fractions reduced to a common denominator (9), we can focus on solving the equation by dealing only with the numerators. This common ground formed by equal denominators negates the need to manage fraction size complexity, as comparing fractions becomes straightforward.

The trick with denominators in equations is simplifying when possible, as it leads to a clearer path to find what the equation's terms equate to.

The Art of Simplification

Simplification in math is akin to cleaning a messy room; it removes clutter so that the solution is clear and direct. In the equation provided, simplification makes the math simpler and less error-prone.
Here's how simplification aids solving the equation:

• By breaking down \( \frac{3x}{27} \) into \( \frac{x}{9} \), extraneous elements are removed. Each division by a common factor brings us closer to a more manageable form.
• Fractions like \( \frac{x}{9} \) and \( \frac{2}{9} \) with the same denominator allow us to directly set the numerators equal.
• Through simplification, the chance of mistakes reduces as there's less complexity in numbers. You deal with direct integers (x and 2) rather than complicated expressions, making equations approachable.

Simplifying fractions is a key step in reasoning mathematically about relationships between numbers. When fractions are simplified, as here, math becomes less about computation skills and more about understanding properties and relationships, which leads us easier and faster to the solution.