Problem 30 - Prealgebra

Question

Multiply. $$0.03(115,900)$$

Step-by-Step Solution

Answer

The product is 3,477.

Step 1: Understand the Problem

The problem asks us to multiply the decimal 0.03 by the number 115,900. This involves calculating the product of these two numbers.

Step 2: Setup the Multiplication

To make the multiplication easier, consider how decimals work in multiplication. Recognize that multiplying by 0.03 is the same as multiplying by 3 and then dividing the result by 100, because 0.03 = \( \frac{3}{100} \).

Step 3: Multiply by 3

First, multiply 115,900 by 3. This simplifies the multiplication with the decimal by handling only whole numbers at this stage. Calculate: \( 115,900 \times 3 = 347,700 \).

Step 4: Adjust for the Decimal

Since the multiplication in Step 3 does not account for the decimal, divide the result by 100 to adjust. This step aligns with the initial setup of multiplying by 0.03. Calculate: \( \frac{347,700}{100} = 3,477 \).

Step 5: State the Final Answer

Having divided by 100 after multiplying by 3, we identify 3,477 as the final result. This is the product of 0.03 and 115,900.

Key Concepts

Decimal Arithmetic

Decimal arithmetic isn't as intimidating as it may first seem. Imagine dealing with decimals as a slightly different way of looking at whole numbers. Essentially, decimals help us express numbers that represent parts of a whole.
To understand decimal arithmetic, it's crucial to see how decimals align with fractions. For example, the decimal number 0.03 is the same thing as the fraction \( \frac{3}{100} \). So, when we multiply any number by 0.03, we're essentially finding out what three hundredths of that number would be.
It’s also essential to keep track of decimal placement. When you multiply decimals, the number of digits to the right of the decimal point in the product is the sum of the number of digits to the right of the decimal points in the numbers being multiplied. If this feels overwhelming, remember you can always convert the decimal into a fraction like we did with 0.03, and perform your calculations as with whole numbers and fractions.

Step-by-Step Multiplication

Tackling multiplication step-by-step with decimals is a game-changer. Step-by-step approaches simplify complex math into manageable pieces. Let’s break down how to multiply 0.03 and 115,900 in an orderly fashion.
First, we need to understand that multiplying by a decimal such as 0.03 follows the same concept as with fractions. Picture multiplying by 3, and then netting off two decimal places once the multiplication is complete.

• **Setup the Problem**: Recognize what the multiplication asks us to do. Here, when multiplying 115,900 by 0.03, we are simultaneously multiplying by 3 and dividing by 100.
• **Multiplication of Whole Numbers**: Focus on ignoring the decimal initially for ease. Multiply 115,900 by 3: \( 115,900 \times 3 = 347,700 \).
• **Adjust for the Decimal**: Post multiplication, introduce the decimal by dividing by 100, transforming 3 into 0.03. \( \frac{347,700}{100} = 3,477 \).

This step-by-step method makes dealing with decimals more like second nature, helping you tackle any similar problem with confidence.

Basic Multiplication Techniques

Understanding basic multiplication techniques can empower any math learner with skills that extend beyond simple calculations. These techniques form the backbone of complex arithmetic and problem-solving.
Multiplication, in essence, is repeated addition. When multiplying whole numbers, like in our example with 115,900 and 3, you're essentially adding 115,900 three times.
Using pencil and paper might feel old-school, but it’s an excellent tool for visualizing each part of the multiplication process. These techniques often include:

• **Breaking Down Numbers**: Splitting larger numbers can simplify calculations. Consider using the distributive property: \( a(b+c) = ab + ac \).
• **Using Commutative Property**: This property tells us that changing the order of factors doesn’t affect the product, i.e., \( a \times b = b \times a \).
• **Aligning Numbers When Multiplying**: When working with pencil and paper, align numbers by place value, start multiplying from the rightmost digit, and move leftwards.

While these methods are foundational, they are powerful once you apply them to both whole numbers and decimals. They not only help determine results reliably but also build a robust mathematical understanding.