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:

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

Understanding the Context

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:

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

\frac { 9 x + 3 } { 9 } = 9 | Filo

Image Gallery

\frac { 9 x + 3 } { 9 } = 9 | Filo
Solved (e) By expressing \\( \\frac{9 x-17}{(4 x-7)(x-3)} | Chegg.com
Solved By expressing \\( \\frac{9 x-17}{(4 x-7)(x-3)} \\) in | Chegg.com
Find \( \lim \left(\frac{1}{3 x^{4 / | Chegg.com
`int_(0)^(pi//2)(sqrt(tanx)+sqrt(cotx))\ dx` - YouTube

Key Insights

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:

Continue Reading

20k \equiv 40 \pmod{100} \implies k \equiv 2 \pmod{5}
So \( k = 5m + 2 \), \( n = 10(5m + 2) + 6 = 50m + 26 \). Smallest \( n = 26 \). Check: \( 26^2 = 676 \), ends in 76 — valid.
Between 24 and 26, the smaller is 24. Therefore, the smallest positive integer whose square ends in 76 is \( \boxed{24} \).

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Final Thoughts

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

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: