Problem 45 - Prealgebra

Question

The problems below are a review of some of the concepts we covered previously. Find the following products. (Multiply.) $$3.18 \times 1.2$$

Step-by-Step Solution

Answer

The product of 3.18 and 1.2 is 3.816.

Step 1: Set Up the Problem

Identify the numbers to be multiplied, which are 3.18 and 1.2, and write them in a vertical multiplication format.

Step 2: Multiply Without Decimals

Ignore the decimals initially and multiply 318 by 12. Calculate the product:\[318 \times 12 = 3816\]

Step 3: Account for Decimals

Count the total number of decimal places in the original numbers. There are two decimal places in 3.18 and one decimal place in 1.2. Therefore, the total number of decimal places is three.

Step 4: Place the Decimal in the Product

Starting from the right, move the decimal point in the product 3816 three places to the left to adjust for the total decimal places from the original numbers. This gives us 3.816 as the final product.

Key Concepts

Understanding Place Value in Decimal Multiplication

Place value is a fundamental concept that helps us understand how to work with numbers, especially decimals. Each digit in a number has a place value, which determines its value based on its position. For example, in the number 3.18:

• The digit '3' is in the units place, representing 3 whole units.
• The digit '1' is in the tenths place, representing one-tenth or 0.1.
• The digit '8' is in the hundredths place, representing eight hundredths or 0.08.

When multiplying decimals, understanding place value helps us correctly set up and solve the problem. We often rewrite the decimals as whole numbers (ignoring the decimal points) for multiplication. Once multiplied, we use place value to correctly place the decimal point in the final product. This ensures the multiplication reflects the original importance of each digit's value.

The Steps of Vertical Multiplication

Vertical multiplication is a method where we align numbers vertically and multiply large numbers systematically. It is particularly useful for multiplying numbers with more than one digit. Here's a quick guide:

• Align the numbers by the right and write them on top of each other, just like you would for simple arithmetic.
• Multiply each digit in the bottom number by each digit in the top number, starting from the rightmost digit.
• Move one place to the left for each new digit and add the results accordingly.

For example, in multiplying 318 by 12, you first multiply 8 by 2, then 1 by 2, and so on, to get partial products. Finally, you add these partial products to find the total product before considering the decimals. This method helps ensure accuracy when working with large or decimal numbers.

The Importance of Rounding Decimals

Rounding decimals makes numbers easier to work with, especially when dealing with lengthy calculations or approximating results. It involves adjusting a decimal to a nearby convenient number. Here’s how it works:

• Look at the digit in place value immediately to the right of the place to which you want to round.
• If this digit is 5 or more, round up. If it's less than 5, keep the digit the same.

For instance, rounding 3.816 to the nearest tenth would result in 3.8, as the digit in the hundredths place is 1, which is less than 5. Rounding is crucial not just for simplification but also when precise decimal values exceed the necessary level of detail for a given calculation or measurement. Whether for everyday calculations or complex mathematical operations, rounding helps manage and interpret numbers effectively.