$\frac{2}{3}=\frac{}{21}$
The problem asks us to solve for the variable \(x\) in the equation \(\frac{2}{3} = \frac{x}{21}\). This is a proportion, which represents the equality of two ratios. To solve for an unknown in a proportion \(\frac{a}{b} = \frac{c}{d}\), we can utilize the cross-multiplication property, which states that \(a \cdot d = b \cdot c\). Applying this to our equation, we set the product of the numerator of one ratio and the denominator of the other equal to the product of the denominator of the first and the numerator of the second.
Applying the cross-multiplication property to \(\frac{2}{3} = \frac{x}{21}\), we get:
\[2 \cdot 21 = 3 \cdot x\]
Performing the multiplication on the left side:
\[42 = 3x\]
To isolate \(x\), we divide both sides of the equation by \(3\):
\[x = \frac{42}{3}\]
Performing the division:
\[x = 14\]
We have determined that \(x = 14\). We can verify this by substituting the value back into the original equation: \(\frac{2}{3} = \frac{14}{21}\). Simplifying the fraction \(\frac{14}{21}\) by dividing both the numerator and denominator by their greatest common divisor, which is 7, yields \(\frac{14 \div 7}{21 \div 7} = \frac{2}{3}\). The equation holds true. The final answer is \(\boxed{14}\).
Proportions are essential in everyday life for scaling measurements. For instance, if a recipe calls for 2 cups of flour for every 3 cups of sugar, and you decide to use 21 cups of sugar, you can use the same proportional logic to find that you need \(\frac{2}{3} = \frac{x}{21}\), leading to \(x = 14\) cups of flour. This helps ensure that the ratios of ingredients remain consistent regardless of the scale of the dish you are preparing.