Solving the Proportion: \frac{3}{7} = \frac{9}{x}

Understanding how to solve equations involving fractions is a fundamental skill in algebra. One common type of problem students encounter is solving proportions—statements that two ratios are equal. In this article, we’ll walk through how to solve the equation:

\[\frac{3}{7} = \frac{9}{x}\]

This equation presents a proportion where a fraction with numerator 3 and denominator 7 equals another fraction with numerator 9 and unknown denominator \( x \). Let’s break down the steps to find \( x \) and explain the logic behind it.

What Is a Proportion?

A proportion sets two fractions equal to each other, meaning the ratio on the left matches the ratio on the right. The equation:

\[\frac{a}{b} = \frac{c}{d}\]

can be solved by cross-multiplication, so long as \( b \) and \( d \) are not zero.

Step-by-Step Solution

Start with the given equation:

\[\frac{3}{7} = \frac{9}{x}\]

To eliminate denominators, cross-multiply:

\[3 \cdot x = 7 \cdot 9\]

Simplify both sides:

\[3x = 63\]

Now divide both sides by 3 to isolate \( x \):

\[x = \frac{63}{3} = 21\]

Final Answer

\[\boxed{x = 21}\]

This means that when \( x = 21 \), the proportion \(\frac{3}{7} = \frac{9}{21}\) holds true.

Why This Works: A Quick Check

Plug \( x = 21 \) back into the original proportion:

\[\frac{3}{7} = \frac{9}{21}\]

Simplify \( \frac{9}{21} \) by dividing numerator and denominator by 3:

\[\frac{9 \div 3}{21 \div 3} = \frac{3}{7}\]

Both sides are equal—so our solution is verified!

Applications and Why It Matters

Proportions like \(\frac{3}{7} = \frac{9}{x}\) appear in real-life scenarios such as scaling recipes, mixing solutions, or comparing ratios in finance and science. Mastering how to solve them builds a strong foundation for more advanced algebra and problem-solving skills.

Conclusion

Solving \(\frac{3}{7} = \frac{9}{x}\) involves cross-multiplication and isolating the unknown variable. By following straightforward algebraic steps—multiply across, simplify, and solve—it’s easy to find \( x = 21 \). With consistent practice, you’ll confidently tackle similar ratio problems in math and everyday life.

---

Keywords: solve proportion, fraction equation, solve for x, cross multiplying, algebra basics, solving 3/7 = 9/x