Problem 17 - Algebra 2 and Trigonometry

Question

Ian often buys in large quantities. A few months ago he bought several cans of frozen orange juice for \(\$ 24 .\) The next time lan purchased frozen orange juice, the price had increased by \(\$ 0.10\) per can and he bought 1 less can for the same total price. What was the price per can and the numbers of cans purchased each time?

Step-by-Step Solution

Answer

The original price per can was $2.40, and Ian initially bought 10 cans.

Step 1: Define variables

Let the original price per can be \( x \) dollars and the number of cans purchased initially be \( y \). We know from the problem statement that \( xy = 24 \).

Step 2: Express new conditions

When Ian buys the cans the second time, the price per can is \( x + 0.10 \) dollars, and the number of cans bought is \( y - 1 \). The total price remains \( 24 \), so the equation becomes \((x + 0.10)(y - 1) = 24\).

Step 3: Substitute initial equation

From the first equation, \( y = \frac{24}{x} \). Substitute this expression for \( y \) into the second condition: \((x + 0.10)(\frac{24}{x} - 1) = 24\).

Step 4: Simplify and set equation

Simplify the equation: \((x + 0.10)\left(\frac{24 - x}{x}\right) = 24\). Simplifying further, we get \( 24 - x + 2.4 = \frac{24x}{x} \).

Step 5: Solve for x

The equation simplifies to \(24x - x^2 + 2.4x = 24x \). Cancelling \(24x\) from both sides results in \(- x^2 + 2.4x = 0\). Factor out \(x\) to get \(x(x - 2.4) = 0\). This gives solutions \(x = 0\) or \(x = 2.4\). Only \(x = 2.4\) is viable since \(x\) represents the price.

Step 6: Calculate number of cans

Since \(x = 2.4\), substitute \(x\) back into \(xy = 24\) to find \(y\): \(2.4y = 24\). Solving gives \(y = 10\).

Step 7: Verify second condition

Verify the solution by substituting \(x = 2.4\) and \(y = 10\) into \((x + 0.10)(y - 1) = 24\). Now it becomes \((2.5)(9) = 24\), which holds true.

Key Concepts

Price and Quantity Relationship

Understanding the relationship between price and quantity is essential, especially in real-life situations like shopping or budgeting. In many scenarios, the cost of a product varies with the number of items you purchase.
When Ian purchased frozen orange juice, the price per can and the number of cans he bought were directly related to the total amount he spent. This is called a price and quantity relationship.
To break it down:

• If the price per item increases, and you want to maintain the same total cost, you might have to buy fewer items.
• If the price decreases, you could purchase more items with the same amount of money.
• Understanding these dynamics can help you make informed decisions on how to allocate your funds efficiently.

In the exercise, Ian's situation illustrated how even a small change in price (like a $0.10 increase per can) can affect the quantity of items one can afford with a fixed budget.

Linear Equations

Linear equations are equations that make straight lines when graphed. These equations typically involve variables raised to the power of one. They form the basis of many problems in algebra.
In Ian's problem, the linear equations come from the relationship between price and quantity. When Ian initially bought the cans, the equation expressing this relationship was \[xy = 24\].
This is where:

• \(x\) represents the price per can in dollars,
• \(y\) represents the number of cans bought.

This equation tells us that the product of the price of each can and the number of cans equals the total amount he spent. Understanding linear equations like this helps you visualize how two different quantities relate to each other numerically, like price and total cost.

Systems of Equations

Solving systems of equations involves finding the values of variables that satisfy all the equations in a given system simultaneously. They can be used to solve problems involving multiple variables, like those found in many financial scenarios.
Ian’s exercise required setting up a system of equations to find two unknowns: the original price per can and the number of cans purchased.
These equations were:

• The initial purchase equation: \(xy = 24\)
• The second purchase condition: \((x + 0.10)(y - 1) = 24\)

By solving this system, you can understand not only each variable individually but also how they interact within the context.

Problem-Solving Steps

When tackling algebra word problems, having a structured approach is incredibly beneficial. Ian's scenario involved seven manageable steps which can be adapted to various problems.
Here’s a concise method for how to address similar problems:

• Define the variables: Identify what each variable will represent in the problem.
• Formulate equations: Translate the words and conditions from the problem into algebraic expressions.
• Substitute known values or expressions: This helps in narrowing down the unknowns.
• Simplify: Work through the equations to make them easier to solve.
• Solve: Derive solutions for the variables using algebraic manipulation.
• Verify: Ensure your solutions fit all the conditions of the original problem.

This methodology not only brings clarity but also builds confidence in solving complex word problems by breaking them down into simpler tasks.