To solve the problem of finding the equivalent fraction, let's start by analyzing the given expression:
You have an original fraction:
[tex]\[ \frac{4}{3x} \][/tex]
You must find a new fraction with the same value but with a different denominator, specifically:
[tex]\[ \frac{\square}{9x^2 y} \][/tex]
Let's denote the new numerator by \( N \). Thus, we need:
[tex]\[ \frac{4}{3x} = \frac{N}{9x^2 y} \][/tex]
First, our goal is to determine the relationship between the denominators of these two fractions. The original denominator is \(3x\) and the new denominator is \(9x^2 y\).
To find \(N\), we will equate the two fractions and then solve for \(N\):
The fraction equality implies:
[tex]\[ \frac{4}{3x} = \frac{N}{9x^2 y} \][/tex]
Cross-multiply to solve for \( N \):
[tex]\[ 4 \cdot (9x^2 y) = N \cdot (3x) \][/tex]
Simplify the left side:
[tex]\[ 36x^2 y = 3x N \][/tex]
To isolate \( N \), divide both sides by \( 3x \):
[tex]\[ \frac{36x^2 y}{3x} = N \][/tex]
Simplify the right side:
[tex]\[ N = \frac{36x^2 y}{3x} = 12x y \][/tex]
So, the numerator \( N \) is \( 12xy \).
Therefore, the completed equivalent fraction is:
[tex]\[ \frac{12xy}{9x^2 y} \][/tex]
So, the blank should be filled with \( 12xy \), making the fraction:
[tex]\[ \frac{4}{3x} = \frac{12xy}{9x^2 y} \][/tex]
