Lotion Mixture Problem Solving Ounces Of Oil And Glycerin

Let's embark on a mathematical journey to unravel the composition of a unique lotion. This lotion is meticulously crafted from a blend of oil and glycerin, each contributing its unique properties. Our goal is to determine the precise amounts of oil and glycerin required to achieve the desired formulation. The lotion is a blend of oil and glycerin, and we aim to determine the quantities of each component in the final mixture.

Setting the Stage The Cost and Quantity Puzzle

We know that the oil blend comes at a cost of $1.50 per ounce, while glycerin is priced at $1.00 per ounce. The final product, four ounces of this luxurious lotion, carries a total price tag of $5.50. This information sets the stage for our mathematical exploration. We have a system of equations waiting to be deciphered, a puzzle that will reveal the secrets of the lotion's recipe.

Defining Our Variables The Building Blocks of the Equation

To begin our mathematical quest, let's define our variables. Let 'x' represent the number of ounces of the oil blend and 'y' represent the number of ounces of glycerin. These variables will serve as the foundation upon which we construct our equations.

Constructing the Equations The Language of Mathematics

Now, let's translate the given information into the language of mathematics. We know that the total volume of the lotion is four ounces, which gives us our first equation:

• x + y = 4

This equation elegantly captures the relationship between the quantities of oil and glycerin, ensuring that their combined volume equals the desired four ounces.

Next, we consider the cost aspect. The oil blend costs $1.50 per ounce, so 'x' ounces of oil will cost 1.50x dollars. Similarly, 'y' ounces of glycerin at $1.00 per ounce will cost 1.00y dollars. The total cost of the four-ounce lotion is $5.50, leading us to our second equation:

• 1. 50x + 1.00y = 5.50

This equation beautifully represents the cost relationship, ensuring that the combined cost of oil and glycerin matches the lotion's selling price.

Solving the System The Quest for the Solution

With our equations in place, it's time to embark on the quest for the solution. We have a system of two equations with two unknowns, a classic mathematical puzzle. We can employ various methods to solve this system, such as substitution or elimination. Let's choose the substitution method for this exploration.

The Substitution Method A Step-by-Step Approach

From the first equation, x + y = 4, we can easily isolate 'y' by subtracting 'x' from both sides:

• y = 4 - x

Now, we have an expression for 'y' in terms of 'x'. We can substitute this expression into our second equation, replacing 'y' with '(4 - x)':

• 1. 50x + 1.00(4 - x) = 5.50

This substitution transforms our second equation into an equation with only one unknown, 'x'. We can now simplify and solve for 'x'.

Unraveling the Value of 'x' A Journey Through Algebra

Let's simplify the equation and embark on an algebraic journey to unravel the value of 'x':

• 1. 50x + 4 - x = 5.50

Combine the 'x' terms:

• 2. 50x + 4 = 5.50

Subtract 4 from both sides:

• 3. 50x = 1.50

Finally, divide both sides by 0.50:

• x = 3

Eureka! We have discovered the value of 'x'. The lotion contains 3 ounces of the oil blend. This is a crucial piece of the puzzle, bringing us closer to understanding the lotion's composition.

Unveiling the Value of 'y' Completing the Picture

Now that we know the value of 'x', we can easily find the value of 'y'. Recall our expression for 'y':

• y = 4 - x

Substitute the value of 'x' (3) into this equation:

• y = 4 - 3

• y = 1

Thus, we have unveiled the value of 'y'. The lotion contains 1 ounce of glycerin. We have now completed the picture, knowing the exact quantities of both oil and glycerin in the lotion.

The Final Composition A Triumph of Mathematics

Our mathematical exploration has led us to the final composition of the lotion. The four-ounce lotion is a blend of:

• 3 ounces of the oil blend
• 1 ounce of glycerin

This discovery is a triumph of mathematics, showcasing the power of equations to solve real-world problems. We have successfully decoded the lotion's recipe, revealing the precise quantities of its key ingredients. This blend ensures the lotion's desired consistency, moisturizing properties, and overall quality. This exploration highlights the practical applications of mathematical concepts in everyday scenarios, from crafting lotions to solving complex scientific problems.

Table Representation: Ounces and Cost per Ounce

This table provides a clear and concise summary of the lotion's composition, showcasing the quantities and costs associated with each ingredient. It serves as a visual representation of our mathematical findings, further solidifying our understanding of the lotion's formulation. The meticulous balance of oil and glycerin ensures the lotion's effectiveness, making it a testament to the art and science of cosmetic formulation. This mathematical exploration not only solves a specific problem but also underscores the broader applicability of mathematics in various fields.

In conclusion, by translating the word problem into a system of equations and solving it, we have successfully determined the exact quantities of oil blend and glycerin required to create the lotion. This exercise demonstrates the power of mathematical problem-solving and its relevance in everyday applications.