We are asked to find the unknown value represented by the box \(\Box\) in the equation \(\frac{2}{5} = \frac{\Box}{15}\). This is an equation of equivalent fractions, where two ratios are equal to each other. To solve for \(\Box\), we need to determine what number should be placed in the numerator so that both fractions represent the same portion of a whole.
To maintain the equality \(\frac{2}{5} = \frac{\Box}{15}\), the two fractions must be proportional. We can observe the relationship between the denominators of the two fractions: the denominator of the first fraction is \(5\), and the denominator of the second fraction is \(15\). Since \(15 = 5 \times 3\), we see that the denominator has been multiplied by \(3\). To keep the fractions equivalent, we must multiply the numerator of the first fraction by the same number, \(3\). Thus, the missing numerator is \(2 \times 3\).
We calculate the value of the numerator: \(\Box = 2 \times 3 = 6\). Therefore, the equation becomes \(\frac{2}{5} = \frac{6}{15}\). Since \(2 \times 15 = 30\) and \(5 \times 6 = 30\), the cross-multiplication confirms that the fractions are indeed equal. The missing value is \(\boxed{6}\).
Imagine you are baking cookies and have a recipe that requires 2 cups of sugar for every 5 cups of flour. If you decide to use 15 cups of flour to make a larger batch, you need to adjust the amount of sugar proportionally. Using equivalent fractions, you can determine that \(\frac{2}{5} = \frac{6}{15}\), so you will need 6 cups of sugar to keep the proportions correct for your delicious cookies!