Question

If 2 cookies cost \(y\) dollars, how many dollars do \(x\) cookies cost? (A) \(2 x y\) (B) \(\frac{x y}{2}\) (C) \(\frac{2 x}{y}\) (D) \(\frac{2 y}{x}\)

Short answer

(B) \(\frac{x y}{2}\)

Step-by-step solution

Find the cost of one cookie

To find the cost of one cookie, divide the cost of 2 cookies (\(y\) dollars) by 2. \[ \text{cost of one cookie} = \frac{y}{2} \]

Find the cost of \(x\) cookies

Now, we multiply the cost of one cookie by \(x\) to find the cost of \(x\) cookies. \[ \text{cost of x cookies} = x \cdot \frac{y}{2} = \frac{x y}{2} \]

Check if the result matches with any of the given options

We found the cost of \(x\) cookies to be \(\frac{x y}{2}\). This matches with option (B): \[ \boxed{(B) \ \frac{x y}{2}} \]

Key concepts

Ratio and Proportion

Understanding how ratios and proportions work is crucial when dealing with quantities and their comparisons. A ratio is essentially a way to compare two quantities, showing how many times one value contains another. Proportions, on the other hand, state that two ratios are equal. In our cookies problem, the ratio comes from comparing the cost of a certain number of cookies to the number of cookies itself.

In the given exercise, the ratio is set up as '2 cookies per y dollars'. To establish the cost for one cookie, you proportionally reduce the term by dividing both the number of cookies and the cost by 2, maintaining the equality of ratios. This uses the basic principle of proportions where if a/b = c/d, then ad = bc. Hence, scaling up or down to find the cost for a different number of cookies keeps the ratios equivalent, ensuring that the proportion stays consistent.

Basic Algebra

Basic algebra involves using letters to represent numbers in equations and formulas, which allows us to solve a wide range of problems. In the context of the cookie problem, we have two variables, 'x' and 'y'. The variable 'y' represents the known cost of 2 cookies, and 'x' represents the unknown quantity of cookies we're interested in.

To solve for 'x' cookies, we perform algebraic operations. We start by finding the unit price of a single cookie, which is \(y/2\) dollars - a simple division. Then, algebra comes into play by multiplying this unit price by 'x' to scale up to the cost of 'x' cookies. This is a fundamental concept in algebra known as solving for an unknown by establishing a relationship between known quantities through equations.

Math Problem Solving

Math problem solving is a critical skill that combines understanding concepts, applying techniques, and thinking logically to come to a solution. It usually involves several steps, from comprehending the problem to performing operations and checking the answer. The steps provided in the solution to our cookie problem model such a process.

Firstly, we understand the problem and identify the variables involved. We then find a relationship between the variables—the cost per cookie. After that, we perform calculations and check the results against the given options. Effective problem solving not only requires executing the steps correctly but also having the foresight to verify the results, ensuring accuracy and understanding. By approaching math problems systematically, you can break them down into manageable parts and solve complex questions with confidence.