Problem 53 - Prealgebra

Question

A bag of potting soil sells for \(\$ 2,\) and a bag of fertilizer sells for \(\$ 13 .\) What is the expression for the total cost of 4 bags of soil and 2 bags of fertilizer? \(\mathbf{A})(4 \times 2)+(2 \times 13)\quad\) \(\mathbf{B}) \)(4 \times 13)+(2 \times 2)\(\quad$$\mathbf{C}) 4(2+13)\quad$$\mathbf{D})(2+13)(4+2)\)

Step-by-Step Solution

Answer

Option A, \((4 \times 2) + (2 \times 13)\), is correct.

Step 1: Understand the Problem

We need to calculate the total cost of purchasing 4 bags of potting soil and 2 bags of fertilizer, given their individual prices.

Step 2: Identify Cost per Bag

The cost of one bag of potting soil is $2, and the cost of one bag of fertilizer is $13.

Step 3: Calculate Cost of Soil Bags

To find the total cost for the soil, multiply the price of one bag by the number of bags: \(4 \times 2 = 8\).

Step 4: Calculate Cost of Fertilizer Bags

To find the total cost for the fertilizer, multiply the price of one bag by the number of bags: \(2 \times 13 = 26\).

Step 5: Sum the Costs

Add the total cost of the soil and the fertilizer: \(8 + 26 = 34\).

Step 6: Identify the Correct Expression

The expression matching the total cost is \((4 \times 2) + (2 \times 13)\), which corresponds to option A.

Key Concepts

Price Calculations

Price calculations are a fundamental aspect of everyday purchases and are essential to understanding how to budget effectively. In our exercise, we are looking to determine the total cost of purchasing specific items: potting soil and fertilizer. Each of these items has a fixed price per bag.
To figure out the total price, you multiply the price per bag by the number of bags you intend to purchase, which gives you the cost for each type.

• Potting soil costs \( \\(2 \) per bag
• Fertilizer costs \( \\)13 \) per bag

Once you know the cost per type, you simply add these amounts together for the final price. This teaches us not only how to calculate total prices but also the importance of knowing each item's price in order to stay within a budget.

Basic Multiplication

Multiplication in our exercise allows us to quickly determine the total cost of multiple units of the same item. Instead of adding the price of each individual bag separately, we can use multiplication for efficiency. Here’s a step-by-step look at how it’s applied:
1. **Determine the quantity needed:** We need 4 bags of soil and 2 bags of fertilizer.2. **Multiply the quantity by the unit price:** - Potting Soil: \( 4 \times 2 = 8 \) (This means the total for the soil is \\(8) - Fertilizer: \( 2 \times 13 = 26 \) (So, the total for the fertilizer is \\)26)3. **Sum the results:** Add the two results to get the grand total.
This process shows just how valuable multiplication is in simplifying and expediting calculations, making it a critical skill in algebra and everyday life.

Addition in Algebra

Addition, especially in algebraic expressions, helps us combine different quantities to find a total sum. After we determine the cost of each item through multiplication, addition helps us find the accumulated cost of all items combined.

• First, calculate the cost of the soil bags: \( 8 \) (from \( 4 \times 2 \)).
• Next, calculate the cost of the fertilizer bags: \( 26 \) (from \( 2 \times 13 \)).
• Finally, add these results: \( 8 + 26 = 34 \).

The arithmetic step of adding these two subtotals together gives us the complete total for all items purchased. Addition not only combines totals in calculation, but it also helps us understand how combined separate amounts matter in overall budgeting and problem-solving. Mastering addition alongside multiplication is foundational in solving algebraic expressions effectively.