A recipe recommends 1 banana for every 2 smoothies.
Equation: $y=\frac{1}{2} x$
Is the relationship proportional?
The relationship described is between the number of smoothies \(x\) and the number of bananas \(y\) required for a recipe. The problem provides the equation \(y = \frac{1}{2}x\), which states that the number of bananas is half the number of smoothies. In mathematics, a relationship between two variables \(x\) and \(y\) is defined as proportional if it can be expressed in the form \(y = kx\), where \(k\) is a non-zero constant known as the constant of proportionality.
Comparing our equation \(y = \frac{1}{2}x\) to the standard form \(y = kx\), we can identify the constant of proportionality \(k\). Here, \(k = \frac{1}{2}\) (or \(0.5\)). Since \(\frac{1}{2}\) is a constant value that does not change regardless of the number of smoothies prepared, the ratio of bananas to smoothies, \(\frac{y}{x} = \frac{1}{2}\), remains constant for all \(x > 0\). This confirms that the relationship is indeed proportional.
Since the equation fits the form \(y = kx\), we conclude that the relationship between the number of smoothies and the number of bananas is proportional. Therefore, the answer is Yes.
Proportional relationships are common in everyday life, such as when you are doubling a cookie recipe to make more treats. If the original recipe calls for 2 cups of flour for 12 cookies, the ratio of flour to cookies is \(\frac{2}{12} = \frac{1}{6}\). To make 24 cookies, you would use \(\frac{1}{6} \times 24 = 4\) cups of flour. Understanding this proportionality allows you to scale ingredients, adjust budgets, or even calculate travel distances based on speed and time.