Problem 14 - Elementary and Intermediate Algebra

Question

Restaurant Management. Chick-fil-A" recently sold small lemonades for \(\$ 1.29,\) medium lemonades for \(\$ 1.49,\) and large lemonades for \(\$ 1.85 .\) During a lunch-time rush, Chris sold 40 lemonades for a total of \(\$ 59.40 .\) The number of small drinks and large drinks, combined, was 10 fewer than the number of medium drinks. How many drinks of each size were sold?

Step-by-Step Solution

Answer

10 small, 25 medium, and 5 large lemonades were sold.

Step 1: Define Variables

Let - S be the number of small lemonades sold,- M be the number of medium lemonades sold, and- L be the number of large lemonades sold.

Step 2: Write Equations Based on Total Sales and Total Number of Drinks

From the problem, we know:1. The total number of lemonades sold is 40, so: \[ S + M + L = 40 \]2. The total sales amount to \$59.40, hence the equation for sales is: \[ 1.29S + 1.49M + 1.85L = 59.40 \]

Step 3: Write the Equation Based on the Relationship Between the Number of Drinks

The problem also states: 'The number of small drinks and large drinks, combined, was 10 fewer than the number of medium drinks,' which can be written as: \[ S + L = M - 10 \]

Step 4: Solve the System of Equations

We have the following system of equations:1. \[ S + M + L = 40 \]2. \[ 1.29S + 1.49M + 1.85L = 59.40 \]3. \[ S + L = M - 10 \]Using equation (3), substitute \[ S + L \] in terms of \[ M \]: \[ S + L = M - 10 \] or \[ S + L + 10 = M \]. Call this equation 4.

Step 5: Substitute Equation into Step 4 into Other Equations

Substitute \[ M \] in equation (1):\[ S + (S + L + 10) + L = 40 \]This simplifies to: \[ 2S + 2L + 10 = 40 \]\[ 2S + 2L = 30 \]\[ S + L = 15 \]. Call this equation 5.Then substitute \[ M \] in equation (2):\[ 1.29S + 1.49(S + L + 10) + 1.85L = 59.40 \]This simplifies to:\[ 1.29S + 1.49S + 1.49L + 14.9 + 1.85L = 59.40 \]Combine like terms:\[ 2.78S + 3.34L + 14.9 = 59.40 \]\[ 2.78S + 3.34L = 44.50 \]. Call this equation 6.

Step 6: Solve the New System of Equations

Substitute equation (5) into equation (6):\[ 2.78S + 3.34(15-S) = 44.50 \]Distribute and simplify:\[ 2.78S + 50.1 - 3.34S = 44.50 \]Combine like terms: \[ -0.56S + 50.1 = 44.50 \]Solving for \[ S \]:\[ -0.56S = -5.6 \]\[ S = 10 \]From equation (5), \[ 10 + L = 15 \]\[ L = 5 \]From equation (4), \[ M = S + L + 10 \]\[ M = 10 + 5 + 10 = 25 \]

Step 7: Conclude the Number of Each Drink Sold

The number of small lemonades sold is 10, the number of medium lemonades sold is 25, and the number of large lemonades sold is 5.

Key Concepts

linear equations

Linear equations are mathematical expressions that represent straight-line relationships. In our exercise, we have three main linear equations. Each equation corresponds to a different aspect of the problem surrounding small, medium, and large lemonades sold at Chick-fil-A. We can outline these equations as follows:
1. The total number of drinks sold: \[ S + M + L = 40 \] where \( S \) is small, \( M \) is medium, and \( L \) is large drinks.
2. The total sales amount: \[ 1.29S + 1.49M + 1.85L = 59.40 \]
3. The relationship between drink sizes: \[ S + L = M - 10 \]
These equations are crucial because they help set up a system that can be solved through algebraic methods. Understanding how to form and manipulate these linear equations is key to solving real-world problems.

variable substitution

Variable substitution is a method used to solve systems of equations where one equation is simplified and substituted into another. For our problem, we started by isolating \( M \) (medium drinks) from equation 3:
\[ S + L = M - 10 \] Then, we rearrange it to get:
\[ M = S + L + 10 \]
This new expression for \( M \) is then substituted into the other equations. For example, substituting into the total number of drinks equation:
\[ S + (S + L + 10) + L = 40 \]
This simplifies our equation to \[ 2S + 2L + 10 = 40 \] and further to \[ S + L = 15 \]. By using variables substitution, complex systems are reduced to simpler, solvable forms.

solving algebraic equations

Once we have substituted variables, next comes solving the simplified algebraic equations. We ended up with these two main equations:
1. \[ S + L = 15 \]2. \[ 2.78S + 3.34L = 44.50 \]
Starting with the first equation, we simplify further to solve for \( S \). Using \( L = 15 - S \), substitute into the second equation:
\[ 2.78S + 3.34(15 - S) = 44.50 \]
This leads to:
\[ 2.78S + 50.1 - 3.34S = 44.50 \]
By combining and simplifying, we get:
\[ -0.56S + 50.10 = 44.50 \]
\[ -0.56S = -5.60 \]
\[ S = 10 \]
With \( S = 10 \), we find \( L \):
\[ 10 + L = 15 \]\[ L = 5 \]
Finally, solving for \( M \):
\[ M = S + L + 10 \]\[ M = 10 + 5 + 10 \], gives us \( M = 25 \). Understanding and solving these equations unlocks the solution to the problem.

real-world applications of algebra

Algebra is not just theoretical; it has many real-world applications. In this problem, we use algebra to manage sales data for a restaurant, a common practical scenario. By setting up and solving equations, we determined the number of each type of lemonade sold, crucial for inventory and sales strategies.
For example, restaurant managers can:

• Determine how much stock is needed.
• Track sales patterns over time.
• Adjust pricing models based on sales and profit data.

Thus, algebra serves as an essential tool in navigating and solving everyday business challenges. This not only applies to restaurants but also to finance, logistics, engineering, and many other fields.