Step 1: Define Variables

Let:

• $x$: Number of pounds of soap Sandra makes and sells.
• $C$: Total cost of making $x$ pounds of soap.
• $S$: Total sales from selling $x$ pounds of soap.

Step 2: Set Up Equations

1. Total cost ($C$) equation: $C = 90 + 7x$ The fixed cost of $90 represents the equipment, and $7 per pound represents the variable cost of materials.

2. Total sales ($S$) equation: $S = 8x$

3. Break-even condition: $C = S$

Step 3: Solve Using Substitution

From the equations, set the total cost equal to the total sales: $90 + 7x = 8x$

Solve for $x$: $90 = x$

Step 4: Calculate Total Sales and Total Costs

1. When $x = 90$, substitute $x$ into the equations for $C$ and $S$:
   • Total cost: $C = 90 + 7(90) = 90 + 630 = 720$
   • Total sales: $S = 8(90) = 720$

Final Answer:

• Sandra's total sales will be $720.
• Sandra's total costs will be $720.
• Sandra will sell 90 pounds of soap to break even.

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Do you want more details or have any questions?

Related Questions:

1. What would Sandra's profit be if she sold 100 pounds of soap?
2. How would the equations change if the cost per pound of materials increased to $9?
3. What is Sandra's profit margin per pound of soap sold after breaking even?
4. How long would it take Sandra to break even if she sells 15 pounds of soap per week?
5. How does reducing the fixed cost of equipment to $80 impact the break-even point?

Tip: To easily analyze a business's profitability, focus on fixed vs. variable costs and calculate the break-even point!