Decimals: Operations and Comparisons Quiz

Decimals: A Guide to Fractional Numbers

Decimals are a fundamental part of our numeric system, allowing us to represent and work with fractional amounts. In this article, we'll explore the essential aspects of decimals, including decimal places, comparison of decimals, and the four main operations (addition, subtraction, multiplication, and division).

Decimal Places

In decimals, each digit to the right of the decimal point represents a tenth of the previous digit's value. For instance, in the decimal 12.345, the first digit, 1, represents whole units, the second digit, 2, represents tenths, the third digit, 3, represents hundredths, and so on.

Comparing Decimals

To compare decimals, follow these steps:

1. Compare the digits to the left of the decimal point. If they are equal, move to the next step.
2. Compare the digits in the first decimal place. If they are equal, move to the next step.
3. Continue comparing digits of the decimal places, moving from left to right, until you find a difference. The decimal with a greater value in the compared place is the larger number.

Decimal Addition

To add decimals, follow these steps:

1. Line up the decimal points of both numbers.
2. Add the digits from right to left, carrying over to the next column as needed.

For example, to add 12.34 and 7.89:

1. Line up the decimal points: 12.34 | 7.89
2. Add the digits: 1 + 7 = 8, carry 1 | 3 + 8 = 11, carry 1 8.1 | 1.1
3. Add the carried digits: 1 + 1 = 2 ____ | 2.1

The result is 12.34 + 7.89 = 20.43.

Decimal Subtraction

To subtract decimals, follow these steps:

1. Line up the decimal points of both numbers.
2. Subtract the digits from right to left, borrowing from the previous column as needed.

For example, to subtract 12.34 from 19.78:

1. Line up the decimal points: 19.78 | 12.34

2. Subtract the digits: 1 - 1 = 0, no borrow | 9 - 2 = 7 9.78 | 7.34

3. Subtract the tens digit: 1 - 0 = 1, borrow 1 from the next column

    9.7   |   7 _
     8 - 1 + 0.01 = 7.01
      ____   |   7.34 - 0.01 = 7.33
   

The result is 19.78 - 12.34 = 7.44, rounded to two decimal places.

Decimal Multiplication

To multiply decimals, follow these steps:

1. Line up the decimal points of both numbers.
2. Multiply the digits from right to left, adding zeros as needed to align the number of digits.
3. Add any decimal places that may result.

For example, to multiply 12.34 by 7.89:

1. Line up the decimal points: 12.34 | 7.89

2. Multiply the digits: 3 * 8 = 24, add zeros to align: 2400 4 * 7 = 28, add zero: 280 1 * 9 = 9, no zeros needed

    2400   |   280
      _____|_______
   

3. Multiply the numbers: 2400 * 28 = 67200 280 * 7 = 1960

    67200   |   1960
      _____|_______
   

4. Add any decimal places: 67200 + 196 = 67396

The result is 12.34 * 7.89 = 97.732, rounded to two decimal places.

The decimal number system allows us to represent and manipulate fractional parts of whole numbers, providing accuracy and precision in various fields, such as science, engineering, and finance. Understanding decimals and their properties is essential for working with and communicating numerical information.