Problem 67 - Algebra 1: Concepts and Skills

Question

The saline solution that you use to clean your contact lenses is on sale for $4.99 a bottle. You decide to stock up and buy 4 bottles. Use the distributive property to mentally calculate the total cost of the bottles of saline.

Step-by-Step Solution

Answer

The total cost of 4 bottles of saline solution is $19.96.

Step 1: Break Down the Price

To simplify the multiplication, separate the price of $4.99 into two parts, $4.00 and $0.99.

Step 2: Multiply the Easier Part

Four bottles at $4.00 each costs \(4 * 4.00 = 16.00\).

Step 3: Multiply the Other Part

Four bottles at $0.99 each costs \(4 * 0.99 = 3.96\).

Step 4: Add Up

The total cost is the sum of the two products calculated in Step 2 and Step 3: \(16.00 + 3.96 = 19.96\).

Key Concepts

Mental Math Multiplication

Mental math multiplication is a skill that can be incredibly useful in everyday life, allowing you to quickly calculate numerical problems without the need for a calculator. To master mental multiplication, it's often helpful to memorize multiplication tables, understand number patterns, and learn shortcuts that can simplify calculations.

For instance, in our exercise, instead of multiplying \(4.99 directly by 4, we 'break down' the number into two more manageable parts: \)4.00, which is easy to multiply, and \(0.99, which is close enough to \)1 to make the subsequent multiplication almost as straightforward. By reducing the complicated number into simpler components, you can mentally compute each part separately and then combine them for the final total, simplifying the mental multiplication process.

Breaking Down Numbers

Breaking down numbers is a critical technique in mental math that involves separating complex numbers into more manageable parts. The goal is to simplify calculations by working with round numbers or numbers that are easy to multiply or add.

For example, the number \(4.99 is decomposed into \)4.00 and \(0.99 in our saline solution exercise. This smart division leverages the ease with which we can multiply whole numbers, as well as capitalize on near-whole numbers like \)0.99. Once the original number is broken down, each part can be multiplied or added separately, allowing us to proceed with simpler arithmetic operations that are less prone to error and faster to compute.

Multiplying Decimals

Multiplying decimals can often appear daunting, yet with the distributive property and a good grasp of place value, it becomes a much simpler task. The strategy involves breaking the decimals into whole numbers and fractions of whole numbers, multiplying each separately, and then combining the results.

In our example, multiplying \(4.00 by 4 is straightforward since there are no decimal places to consider. However, multiplying \)0.99 by 4 might seem more complicated. To simplify, remember that \(0.99 is just \)1.00 minus \(0.01, which means multiplying by 4 yields \)3.96 — just \(4.00 minus \)0.04. This technique turns the process into a simple subtraction away from a much easier multiplication, allowing for faster and more accurate calculations.