$\frac{\text{ }}{3} = \frac{6}{9}$
The problem presents the equation \(\frac{x}{3} = \frac{6}{9}\) and asks us to find the value of \(x\). We are dealing with a simple rational equation where both sides represent the same ratio. To find \(x\), we need to isolate it by performing algebraic operations. Our approach will be to simplify the right-hand side of the equation and then use properties of fractions to solve for \(x\).
First, we focus on the right-hand side of the equation: \(\frac{6}{9}\). We observe that both the numerator (6) and the denominator (9) share a common factor of 3. By dividing both the numerator and the denominator by 3, we can simplify the fraction:
\[ \frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \]
Now, the original equation \(\frac{x}{3} = \frac{6}{9}\) can be rewritten as:
\[ \frac{x}{3} = \frac{2}{3} \]
Now that we have the equation in the form \(\frac{x}{3} = \frac{2}{3}\), we can isolate \(x\). Since the denominators on both sides are the same, the numerators must also be equal for the equation to hold true. Alternatively, multiplying both sides of the equation by 3 gives:
\[ 3 \times \frac{x}{3} = 3 \times \frac{2}{3} \]
This results in:
\[ x = 2 \]
Thus, the value of \(x\) that satisfies the original equation is 2.
Ratios are used everywhere in cooking to keep recipes consistent. If a recipe says to use 6 cups of water for every 9 cups of rice, but you only have a smaller container that allows for 3 cups of rice, you can use the equation \(\frac{x}{3} = \frac{6}{9}\) to determine exactly how many cups of water (\(x = 2\)) you need for your batch. Understanding how to solve these simple equations ensures your meals turn out just right every time!