Problem 7 - Prealgebra

Question

Multiply the decimals. (75.3)(0.4)

Step-by-Step Solution

Answer

The product of 75.3 and 0.4 is 30.12.

Step 1: Ignore the Decimals and Multiply as Whole Numbers

First, treat 75.3 and 0.4 as whole numbers by temporarily removing the decimal points. Multiply 753 by 4: \[753 \times 4 = 3012.\]

Step 2: Count the Total Decimal Places

The original numbers, 75.3 and 0.4, have a total of two decimal places. One in 75.3 and another one in 0.4.

Step 3: Place the Decimal Point in the Product

From the product 3012 obtained in Step 1, move the decimal point two places to the left (because there are a total of two decimal places in the original numbers):\[30.12.\]

Step 4: Check the Result

Verify if the placement of the decimal point makes sense in the context of the initial numbers. Both numbers are decimals that should result in another decimal when multiplied.

Key Concepts

Place Value

Understanding place value is essential for grasping how to work with decimals in multiplication. Each digit in a number has a specific position, which determines its value. When we talk about place value, remember:

• To the left of a decimal point, place values are powers of ten (like tens, hundreds, thousands, etc.).
• To the right of a decimal point, place values are fractions of ten (like tenths, hundredths, thousandths, etc.).

In our exercise, for the number 75.3, the '7' is in the tens place, '5' is in the ones place, and '3' is in the tenths place. Conversely, for 0.4, the '4' is in the tenths place. When multiplying decimals, the total number of decimal places in the product is the sum of the decimal places in the factors. This ensures the correct place value of the result. Thus, ensuring that your final product matches the appropriate decimal positions is crucial.

Whole Numbers

Whole numbers are numbers without fractions or decimals, simply put, they are numbers like 1, 2, 300, etc. Although the exercise focuses on decimals, understanding whole numbers is key because we convert decimals into whole numbers to simplify multiplication. In the problem, we first multiply as whole numbers by disregarding the decimal points in 75.3 and 0.4, treating them instead as 753 and 4. This step simplifies the complexity of the calculation and allows us to apply straightforward multiplication skills. This approach gives us a temporary product, 3012, which we'll adjust according to the decimal places in the original numbers. Viewing decimals this way helps demystify the multiplication process and reinforces why whole number multiplication is foundational for understanding decimals.

Arithmetic Operations

Arithmetic operations, including multiplication, are crucial in understanding how decimals interact. Multiplication extends beyond whole numbers to decimals, but the fundamental principles stay the same. Following arithmetic steps, begin by multiplying the equivalent whole numbers you derived from the decimals without focusing on decimal points. This simplifies the operation initially, producing a whole number product. Next, reintroduce decimals by shifting the decimal point in the product based on the total number of decimal places found in the numbers you multiplied. This step is critical because it involves accurately restoring the decimals' influence on the final answer without overlooking the arithmetic rules. In our exercise, multiplying 753 by 4 gives 3012, after which we adjust for the two decimals places from 75.3 and 0.4, yielding the final result, 30.12. Hence, practicing arithmetic operations with decimals sharpens your overall math skills.