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Chapter 1: Problem 104

Perform each operation. SEE EXAMPLES 10,11 , and \(12 .\) (OBJECTIVE 5 ) $$4.21 \cdot 2.73$$

Short Answer

Expert verified
The result of multiplying 4.21 and 2.73 is 11.4813

Step by step solution

01

Multiplication like Integer Numbers

First, treat both numbers as if they were whole numbers. The decimal points don't exist for now. So, 421 multiplied by 273 equals 114,813.
02

Count the Decimal Places

Next, count the total number of digits after decimal in the original numbers. Here, there are two digits after the decimal in 4.21 and two in 2.73. Hence, the total number is four.
03

Placement of Decimal in Answer

The last step is to place the decimal in the answer from step 1. Since there were four places in the original numbers, you count four places from the right in your answer. So, the final result is 11.4813.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Step-by-Step Math Solutions
Understanding math problems can be easier when you have a clear, step-by-step solution. This approach will help you make sense of decimal multiplication by breaking it down into manageable steps. Let's take the example of multiplying two decimal numbers: 4.21 and 2.73.

First, it's important to strip away anything that might complicate things, such as the decimal points. Treat the numbers like whole numbers temporarily. For 4.21 and 2.73, think of them as 421 and 273.
  • Multiply these whole numbers: 421 multiplied by 273.
  • This gives a product of 114,813.
After you've multiplied, don't forget about the decimals. The next step, which we'll discuss later, is vital for placing the decimal back correctly in the final answer, turning it into the precise solution you need.
Multiplication Techniques
Multiplication can seem tricky with decimals, but a simple technique simplifies the process. Any decimal multiplication starts by ignoring the decimals.
This method is effective because it leverages the default multiplication techniques learned with whole numbers.

Consider multiplying 421 by 273, similar to multiplying larger numbers using columns:
  • First, multiply the numbers as you would with basics.
  • Add zeros to shift numbers appropriately when multiplying by tens, hundreds, etc.
Once you have those products, add them up. This solid integer product will guide you to the correct decimal result later. It鈥檚 crucial to focus on precision and order when adding the intermediate products. This practice ensures greatness when you come back to re-instate the decimals in the final step.
Decimal Places Counting
Counting decimal places correctly is crucial when multiplying. This ensures that the final product reflects the number of decimals across both numbers multiplied.

Let's break it down using our example of 4.21 and 2.73:
  • Identify how many digits follow the decimal point in each number: 4.21 has two digits, 2.73 also has two.
  • Add these to determine the total number of decimal places in your result. Here, that鈥檚 4 total decimal places.
Once you have your total number of places, you apply them to your final product (from the integer multiplication). Count from right to left, four places, ensuring that the decimal point lands accurately within your answer number "114,813" to make it "11.4813." This placement is important as it defines the scale and accuracy of your final solution.

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