Problem 51 - Elementary Algebra

Question

A candy store offers mixed candy at \(\$ 3.75\) for every half-pound. How much will 2.6 pounds of candy cost?

Step-by-Step Solution

Answer

The cost is \$19.50.

Step 1: Determine Cost Per Pound

The cost of mixed candy is given as \(\\( 3.75\) for every half-pound. To find the cost per pound, we need to double this amount since there are two half-pounds in one pound. \(3.75 \times 2 = 7.50\). Thus, the cost per pound is \(\\)7.50\).

Step 2: Calculate Total Cost for 2.6 Pounds

Now that we know the cost per pound is \(\$7.50\), we can find the cost for 2.6 pounds of candy by multiplying the cost per pound by the number of pounds. So, \(7.50 \times 2.6 = 19.50\).

Step 3: Final Cost

The total cost for 2.6 pounds of candy is \(\$19.50\).

Key Concepts

Unit Price Calculation

Understanding unit price calculation is essential for solving problems like the one in the exercise. Unit price refers to the cost per single unit of measurement, such as per pound, liter, or other measurements. In this candy store scenario, we need to determine the cost per pound.

• Initially, the price is given for a half-pound: \( \\(3.75 \).
• Since there are two halves in a whole, the price for a full pound is calculated by multiplying the half-pound price by 2: \( 3.75 \times 2 = 7.50 \).

Now, we know the cost per pound is \( \\)7.50 \), which allows us to calculate the cost for any number of pounds the customer wants. For any future calculations, remember that consistency in units is crucial; always convert to the same unit before calculating.

Multiplication of Decimals

Calculating cost often involves multiplying decimals, just like in determining the total price of candy. Let's break it down:

• Identify the numbers to multiply; in our case, it's the unit price, \( 7.50 \), and the total weight, \( 2.6 \) pounds.
• Line up the numbers on a vertical axis as you would with regular multiplication, ignoring the decimal points temporarily.
• Perform the multiplication as with whole numbers, then count the total number of decimal places in both original numbers combined.

In our example, the calculation \( 7.50 \times 2.6 \) involves two decimal places from \( 7.50 \) and one from \( 2.6 \), making a total of three decimal places in the product. Thus, the answer is \( 19.500 \). As a convention, we can drop the trailing zero, leaving us with \( 19.50 \).

Always remember to adjust the decimal place correctly for accurate results.

Real-World Word Problems

Word problems in mathematics, like our candy store exercise, bridge our understanding from theoretical math to its practical application. Here's how to approach them:

• Start by identifying all relevant information and what you are asked to find.
• Convert all given information into a mathematical form; this often involves setting up equations or expressions.
• Solve the problem by applying the appropriate mathematical operations, such as multiplication for unit pricing.

In the candy shop example, we first identified the unit price calculation. Then, we used multiplication to find the total cost. Solving real-world problems requires not only calculations but also an understanding of context, ensuring every answer relates back to the question at hand.

Practical applications like this hones your analytical skills and prepares you for everyday challenges that involve mathematics.