Problem 59 - Algebra A Combined Function

Question

A new self-tanning lotion for everyday use is to be sold. First, an experimental lotion mixture is made by mixing 800 ounces of everyday moisturizing lotion worth \(\$ 0.30\) an ounce with self-tanning lotion worth \(\$ 3\) per ounce. If the experimental lotion is to cost \(\$ 1.20\) per ounce, how many ounces of the selftanning lotion should be in the mixture?

Step-by-Step Solution

Answer

400 ounces of self-tanning lotion should be in the mixture.

Step 1: Define the Variables

Let \( x \) be the amount of self-tanning lotion in ounces we need to find. The total mixture will then be \( 800 + x \) ounces.

Step 2: Set Up the Cost Equation

The total cost must equal the cost per ounce times the total amount of lotion. This is expressed by the equation: \\[ 0.30 \times 800 + 3.00 \times x = 1.20 \times (800 + x) \] \where \( 0.30 \times 800 \) is the cost of moisturizing lotion and \( 3.00 \times x \) is the cost of the self-tanning lotion.

Step 3: Simplify and Solve the Equation

First, calculate the left side: \\[ 0.30 \times 800 = 240 \] \This results in the equation: \\[ 240 + 3x = 1.20 \times (800 + x) \] \Next, expand the right side of the equation: \\[ 240 + 3x = 960 + 1.20x \] \Subtract \( 1.20x \) from both sides to keep \( x \) on one side: \\[ 240 + 3x - 1.20x = 960 \] \Simplify: \\[ 240 + 1.8x = 960 \] \Subtract 240 from both sides: \\[ 1.8x = 720 \] \Finally, divide by 1.8 to find \( x \): \\[ x = \frac{720}{1.8} = 400 \]

Key Concepts

Equation Setup

Setting up equations is a powerful tool when tackling mixture problems. In this exercise, we're tasked with finding the amount of self-tanning lotion needed for a desired cost mixture. We need an equation that represents the relationship between the amounts and costs of the two types of lotions involved.

• The total cost of the mixture must equal the sum of the individual costs of its components, represented through an equation.
• We know the target cost, the cost of the everyday lotion, and the cost of the self-tanning lotion. Combining them into one equation helps resolve the problem.

By creating an equation that accounts for all these factors, we set a foundation to solve for the unknown quantity.

Variable Definition

Defining variables concisely is crucial in solving equations, especially in mixture problems. Here, we define the unknown quantity, which is essential for forming the equation.

• Let \( x \) be the number of ounces of self-tanning lotion. This definition provides clarity and simplifies further calculations.
• The total amount of the mixture then becomes \( 800 + x \) because we start with 800 ounces of moisturizing lotion and add the unknown \( x \) ounces.

Assigning a variable like this not only makes the equation setup more straightforward but also makes solving the equation achievable.

Cost Calculations

Calculating costs accurately is central to determining the mixture's price. With known prices for each component, you can compute individual and total costs.

• Calculate the cost of the everyday lotion: \( 0.30 \times 800 \), which results in \( 240 \).
• Define the cost of the self-tanning lotion: \( 3.00 \times x \).
• The total cost set against the desired cost per ounce leads to the equation: \( 0.30 \times 800 + 3x = 1.20 \times (800 + x) \).

Understanding these calculations allows you to see how each product's cost contributes to the desired final price.

Algebraic Manipulation

After setting up the equation, algebraic manipulation helps isolate the variable and find its value. This involves simplifying the equation step by step.

• First, simplify quantities on either side of the equation. Calculate any known values, such as \( 0.30 \times 800 = 240 \).
• Expand and simplify the equation: \( 240 + 3x = 1.20 \times (800 + x) \).
• Bring all terms involving \( x \) to one side to isolate the variable.
• Subtract constants from both sides, simplify further, and solve for \( x \).
• This results in \( x = \frac{720}{1.8} = 400 \), deriving the solution for \( x \), the ounces of self-tanning lotion needed.

Taking these steps in algebraic manipulation ensures the solution is precise and error-free.