Solving a proportion of th Solve for \( x \). \[ \frac{15}{x+4}=\frac{9}{x} \]

To solve for \( x \), you can start cross-multiplying. This means you multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the numerator of the second fraction multiplied by the denominator of the first fraction: \[ 15 \cdot x = 9 \cdot (x + 4) \] Expanding both sides gives: \[ 15x = 9x + 36 \] Now, subtract \( 9x \) from both sides: \[ 6x = 36 \] Finally, divide both sides by 6: \[ x = 6 \] So, the solution is \( x = 6 \). Proportions are like magic scales that keep things balanced! This concept has been around since ancient civilizations; the Babylonians were the first to mathematically explore relationships like these. Imagine how they would analyze trade ratios—keeping everyone happy and fair in their bustling markets! And speaking of proportions, a common mistake is forgetting to properly distribute terms when dealing with fractions! It’s easy to get tangled up in signs and numbers. Always double-check your arithmetic—as a little miscalculation can change the whole solution. So grab your calculator, take a deep breath, and check those steps!