Problem 7 - Elementary Algebra Within Reach

Question

In Exercises \(7-10\), use a system of linear equations to solve the problem. The selling price of a watch is \(\$ 108.75\). The markup rate is \(45 \%\) of the wholesale cost. Find the wholesale cost.

Step-by-Step Solution

Answer

The wholesale cost of the watch is \$75.

Step 1: Setup the important equations

Let \(x\) be the wholesale cost of the watch. Since the selling price is the sum of the wholesale cost and the markup, we can write this as: \(x + 0.45x = 108.75\)

Step 2: Solve for \(x\)

Merge the like terms on the left side of the equation: \(1.45x = 108.75\). Now, in order to isolate \(x\), you need to divide both sides by \(1.45\). This will give us the value of \(x\), which represents the wholesale cost.

Step 3: Calculate the cost

The calculation \(108.75 \div 1.45 = 75\). Therefore, the wholesale cost of the watch is \$75.

Key Concepts

Understanding Markup Rate

When talking about retail, the markup rate is a common term. It's the percentage added to the wholesale cost of a product to determine the selling price. This is done because businesses need to make a profit.

For example, if the markup rate is 45%, it means that 45% of the wholesale cost is added to the original cost to get the final selling price.

• Formula: Markup = Markup rate × Wholesale cost
• A 45% markup on a $50 wholesale cost adds an additional $22.50, leading to a selling price of $72.50.

Thus, understanding markup helps in pricing products correctly and ensuring profitability.

Deciphering Wholesale Cost

Wholesale cost is what the retailer pays for a product before it is marked up for selling. It's the base cost required for purchasing from the manufacturer or supplier.

• This cost is crucial for determining the final price for consumers.
• Being aware of the wholesale cost helps businesses strategize to balance affordability and profit margins.

In our exercise, if the wholesale cost is given as "x," and we know the markup rate, we can easily calculate the selling price. Alternatively, knowing the selling price and markup rate allows us to find the unknown wholesale cost by setting up an equation.

Solving Equations to Find Costs

To solve problems like the one in our exercise, we use a system of linear equations. This involves setting up an equation based on known variables.
Let's break it down:

• You know the selling price of the watch.
• You know the markup rate, which helps ascertain the part of the cost that is markup.

By setting up the equation, "wholesale cost + markup = selling price," we can substitute the markup with its equivalent in terms of the wholesale cost (here, 0.45x).

Solving the equation involves combining like terms, as seen where 1.45x equals the selling price. This step allows you to solve for the unknown variable, providing the sought-after wholesale cost.

Such exercises strengthen skills in algebra and practical mathematics, vital for anyone involved in commerce or budgeting.