Fill in the blank to make an equivalent fraction with the given denominator.

[tex]\[ \frac{4}{3x} = \frac{\square}{9x^2y} \][/tex]



Answer :

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]