Problem 19 - Thinking Mathematically

Question

Calculators were purchased at \(\$ 65\) per dozen and sold at \(\$ 20\) for three calculators. Find the profit on six dozen calculators.

Step-by-Step Solution

Answer

The profit on sales of six dozen calculators is computed through a series of calculation steps - finding cost per calculator, total cost for 72 calculators, selling price per calculator, total selling price for 72 calculators and finally the profit.

Step 1: Calculate cost per calculator

We first determine the cost of each calculator by dividing the given cost price for a dozen calculators by 12 (since there are 12 items in a dozen). So the cost price for each calculator is \(\frac{$65}{12}\)

Step 2: Calculate the total cost of 6 dozen calculators

Multiplying the cost price per calculator from step 1 by the total number of calculators which is 6 dozens or 72 calculators. Therefore, total cost = Cost per calculator * 72

Step 3: Calculate selling price per calculator

We know that three calculators are sold for $20. So, the selling price for each calculator can be calculated as \(\frac{$20}{3}\)

Step 4: Calculate the total selling price of 6 dozen calculators

Multiply the selling price per calculator from step 3 by the total number of calculators which is 72. Therefore, total selling price = Selling price per calculator * 72

Step 5: Calculate profit

Profit is calculated by subtracting the total cost price from the total selling price. Therefore, Profit = Total selling price – Total cost price

Key Concepts

Cost Price Calculation

When we're talking about cost price or the amount spent to acquire something, it's crucial to break it down to the level of a single unit. In this exercise, calculators are bought at $65 per dozen. But, we need to know the cost for each one calculator to proceed to further calculations. Remember, a dozen means 12 units. So to find the cost price per calculator, you simply divide the total cost for a dozen, which is $65, by 12. This gives us a much-needed individual cost. When working with larger quantities, such as six dozen calculators, you'll need to scale the cost price accordingly by multiplying the individual cost by the total number of calculators. In this case, you have six dozen calculators, translating into 72 calculators. Thus, multiply the single unit cost by 72 to find the total cost for six dozen calculators.

Selling Price Calculation

Once you know the cost of each calculator, you can shift to determining how much you're selling them for, also known as the selling price. Here, we're given that three calculators are sold for $20. To find the selling price for one calculator, divide $20 by 3. This way, you'll know how much you're making per calculator, which is useful for projecting profits. After calculating how much one calculator sells for, calculate the total selling price for a larger batch—like six dozen calculators. Remember that six dozen equals 72 calculators. So, you multiply the selling price of one calculator by the total number, which is 72 in our scenario, to get the total selling price for six dozen calculators.

Basic Arithmetic Operations

Understanding basic arithmetic operations is key to solving profit calculation problems efficiently. You'll use division to find individual unit prices from bulk prices, like dividing the cost of a dozen calculators by 12 or the price for three calculators by 3. Multiplication comes in when you need to scale things up. To find total costs or profits for multiple units, you multiply the unit cost or unit selling price by the number of units you have, such as multiplying the price per calculator by 72. Finally, subtraction is used to calculate the profit, a critical step in this exercise. Here, you subtract the total cost from the total selling price to find how much profit or loss you have made. This simple operation tells you whether your sales strategy is effective or needs adjustments.