Find the missing fraction.
$-\frac{6}{11} \cdot\left(\frac{x}{y}\right)=-\frac{1}{11}$
We are given the equation \(-\frac{6}{11} \cdot \left(\frac{x}{y}\right) = -\frac{1}{11}\). Our objective is to determine the value of the fraction \(\frac{x}{y}\) that satisfies this equality. This is a linear algebraic equation in terms of the unknown quantity \(\frac{x}{y}\).
Let the unknown fraction be represented by \(F = \frac{x}{y}\). The equation then becomes \(-\frac{6}{11} \cdot F = -\frac{1}{11}\). To isolate \(F\), we need to eliminate the coefficient \(-\frac{6}{11}\) from the left side of the equation. We can achieve this by multiplying both sides of the equation by the multiplicative inverse (reciprocal) of \(-\frac{6}{11}\), which is \(-\frac{11}{6}\).
Multiplying both sides by \(-\frac{11}{6}\), we get:
\[F = \left(-\frac{1}{11}\right) \cdot \left(-\frac{11}{6}\right)\]
Performing the multiplication, the negative signs cancel out, and we multiply the fractions:
\[F = \frac{1 \cdot 11}{11 \cdot 6}\]
Simplifying the expression by canceling the common factor of \(11\) in the numerator and the denominator, we find:
\[F = \frac{1}{6}\]
Thus, the value of the fraction \(\frac{x}{y}\) is \(\frac{1}{6}\).
This type of equation is commonly found in problems where you need to solve for a missing ratio or a proportion. For instance, if you have a recipe that requires 6 parts of flour to 11 parts of sugar, and you need to scale the recipe to use only 1/11 of the original sugar amount, you would use this type of proportional reasoning to find the necessary flour amount. Understanding how to solve such equations allows for quick adjustments in daily tasks involving ratios, such as cooking, mixing proportions, or scaling project materials.