Enter the correct numerator to match the new denominator.
$\frac{-8}{13}=\frac{\square}{-26}$
(Simplify your answer.)
We are asked to find the missing numerator \(x\) in the equation \(\frac{-8}{13} = \frac{x}{-26}\). This problem involves equivalent fractions, where two fractions represent the same value even if they have different numerators and denominators. The property of equivalent fractions states that if \(\frac{a}{b} = \frac{c}{d}\), then there exists a constant \(k eq 0\) such that \(c = a \times k\) and \(d = b \times k\). In our case, we have \(\frac{-8}{13} = \frac{x}{-26}\). To find the value of \(x\), we need to determine the relationship between the denominators.
We have the relationship between the denominators as \(13 \times k = -26\). Solving for \(k\):
\[k = \frac{-26}{13} = -2.\]
Since the denominator of the first fraction is multiplied by \(-2\) to get the denominator of the second fraction, we must multiply the numerator of the first fraction by the same constant to keep the fractions equivalent:
\[x = -8 \times (-2).\]
Performing the multiplication:
\[x = 16.\]
Therefore, the equation is \(\frac{-8}{13} = \frac{16}{-26}\). We can verify this by simplifying the right-hand fraction: \(\frac{16}{-26} = \frac{16 \div (-2)}{-26 \div (-2)} = \frac{-8}{13}\), which confirms our result is correct.
The missing numerator \(x\) that satisfies the equation \(\frac{-8}{13} = \frac{x}{-26}\) is \(16\). \(\boxed{16}\).
Equivalent fractions are used in various real-life scenarios, such as cooking or construction. For instance, if a recipe calls for \(\frac{3}{4}\) of a cup of flour but you only have a \(\frac{1}{8}\) cup measuring spoon, you need to find an equivalent fraction to determine how many scoops are needed: \(\frac{3}{4} = \frac{x}{8}\). Since \(4 \times 2 = 8\), you multiply the numerator by 2 to get \(x = 3 \times 2 = 6\). Thus, you need 6 scoops. Understanding how to find missing parts of equivalent fractions helps maintain the correct proportions in any ratio-based task.