Problem 24 - Algebra 2 and Trigonometry

Question

Last week, Emily paid \(\$ 8.25\) for \(x\) pounds of apples. This week she paid \(\$ 9.50\) for \((x+1)\) pounds of apples. The price per pound was the same each week. How many pounds of apples did Emily buy each week and what was the price per pound? (Use \(\frac{\text { total cost }}{\text { number of pounds }}=\) cost per pound for each week.)

Step-by-Step Solution

Answer

Emily bought 6.6 pounds last week and 7.6 pounds this week; the price per pound was approximately $1.25.

Step 1: Equation for Last Week

Last week, Emily paid \$8.25 for \(x\) pounds of apples. Therefore, the price per pound last week is given by the formula: \[ \text{Price per pound (last week)} = \frac{8.25}{x} \]

Step 2: Equation for This Week

This week, Emily paid \$9.50 for \((x+1)\) pounds of apples. Therefore, the price per pound this week is given by the formula: \[ \text{Price per pound (this week)} = \frac{9.50}{x+1} \]

Step 3: Set the Prices Equal

Since the price per pound was the same both weeks, we can set the expressions from Step 1 and Step 2 equal to each other: \[ \frac{8.25}{x} = \frac{9.50}{x+1} \]

Step 4: Cross-Multiply to Solve for x

Cross-multiply the equation to eliminate the fractions: \[ 8.25(x + 1) = 9.50x \] Distribute \(8.25\) on the left side: \[ 8.25x + 8.25 = 9.50x \]

Step 5: Solve for x

Re-arrange the equation so all terms involving \(x\) are on one side: \[ 8.25 = 9.50x - 8.25x \] Simplify the expression: \[ 8.25 = 1.25x \] Divide both sides by \(1.25\) to solve for \(x\): \[ x = \frac{8.25}{1.25} = 6.6 \]

Step 6: Calculate Total Pounds Each Week

Last week, Emily bought \(x = 6.6\) pounds. This week, Emily bought \(x + 1 = 6.6 + 1 = 7.6\) pounds.

Step 7: Calculate Price Per Pound

Using last week's data to calculate the price per pound: \[ \text{Price per pound} = \frac{8.25}{6.6} \approx 1.25 \text{ dollars per pound} \]

Key Concepts

Understanding Cost Per Pound

When we talk about the cost per pound, we are referring to how much it costs to purchase one unit of weight, which in this situation is one pound of apples. This is a simple yet important concept in algebraic equations, particularly when comparing costs across different situations. To calculate the cost per pound, use the formula:

• Cost per pound = Total cost / Number of pounds

For example, last week Emily paid $8.25 for x pounds of apples, which gave us: Price per pound (last week) = \( \frac{8.25}{x} \)The formula shows how dividing the total cost by the number of pounds gives us the price per single pound. In this way, even if the total amount or number of pounds changes, we can always find the value of one pound.

Exploring Systems of Equations

A system of equations occurs when we have more than one equation working together to describe a scenario. In this exercise, we have two equations, each representing the price per pound for different weeks.

• The first equation represents last week: \( \frac{8.25}{x} \)
• The second equation represents this week: \( \frac{9.50}{x+1} \)

Since Emily paid the same price per pound each week, we set these two equations equal:\( \frac{8.25}{x} = \frac{9.50}{x+1} \)By doing this, we establish an equation that can help us find a specific solution for x. Cross-multiplying helps to clear up the fractions, leading us towards solving for the unknown variable. Solving a system like this requires re-arranging and simplifying the equations to find a solution that satisfies both conditions.

Steps in Price Calculation

Price calculation in this context relies on using algebra to find out unknown values, based on known situations. The goal is to determine how many pounds of apples were bought each week, as well as the price per pound.First, establish the equation based on known costs and weights. For Emily, the two knowns are:

• Last week: \( 8.25 = x \times \text{price per pound} \)
• This week: \( 9.50 = (x + 1) \times \text{price per pound} \)

These equations lead us into solving for x, which represents the pounds bought last week. By cross-multiplying and rearranging:\( 8.25(x + 1) = 9.50x \)After distributing and simplifying, we determine:

• \( x = 6.6 \), meaning 6.6 pounds were bought last week
• This week she bought one extra pound, totaling 7.6 pounds

Finally, the calculation of the price per pound from last week's data shows us that dividing \(8.25 by 6.6 pounds results in roughly \)1.25 per pound. This is a common pricing strategy to ensure costs are understandable and consistent. It highlights the power of algebra in everyday situations that require careful calculation and planning.