Comparing Decimal Numbers:

Comparing decimal numbers involves determining which number is greater, less, or equal to another. The process is similar to comparing whole numbers, but with the additional consideration of digits after the decimal point. Here’s a step-by-step guide on how to compare decimal numbers:

Steps to Compare Decimal Numbers:

1. Align the Decimal Points: When comparing two or more decimal numbers, it’s essential to align the decimal points vertically. This makes it easier to compare digits in the same place value positions (tenths, hundredths, thousandths, etc.).Example:

   3.27
   3.75
   

   Aligning the decimal points looks like this:

   3.27
   3.75
   

2. Compare the Whole Numbers: Start by comparing the digits to the left of the decimal point (the whole numbers). The number with the larger whole number is the greater number.Example:

   4.56
   3.89
   

   • 4 is greater than 3, so 4.56 > 3.89.
3. Compare the Tenths Place: If the whole numbers are the same, move to the first digit to the right of the decimal point, which represents the tenths place (the first position after the decimal point). The number with the larger digit in this place is greater.Example:

   6.42
   6.35
   

   • Both numbers have 6 in the ones place.
   • In the tenths place, 4 is greater than 3, so 6.42 > 6.35.
4. Compare the Hundredths Place: If the numbers are still tied after comparing the tenths place, move to the hundredths place (the second digit to the right of the decimal point). The larger digit in this place will determine which number is greater.Example:

   5.674
   5.632
   

   • Both numbers have 5 in the ones place and 6 in the tenths place.
   • In the hundredths place, 7 is greater than 3, so 5.674 > 5.632.
5. Add Zeroes if Necessary: If one decimal number has fewer digits after the decimal point, you can add zeroes to the end of the shorter number to make the comparison easier. This does not change the value of the number because adding zeroes after the decimal point does not alter its value.Example:

   7.5
   7.50
   

   • These are essentially the same number, so 7.5 = 7.50.
6. Final Decision: Once you compare all the place values (from left to right: ones, tenths, hundredths, thousandths, etc.), the first place where the digits differ determines the relationship between the numbers. The number with the larger digit in that place is greater.

Example Comparisons:

1. Comparing 8.62 and 8.75:
   • Both numbers have 8 in the ones place.
   • In the tenths place, 6 is less than 7.
   • Therefore, 8.62 < 8.75.
2. Comparing 9.35 and 9.3:
   • 9.35 can be written as 9.350 for easier comparison.
   • Both numbers have 9 in the ones place.
   • In the tenths place, both numbers have 3.
   • In the hundredths place, 5 is greater than 0, so 9.35 > 9.3.
3. Comparing 12.045 and 12.05:
   • Both numbers have 12 in the ones place.
   • In the tenths place, 0 is the same.
   • In the hundredths place, 4 is less than 5.
   • Therefore, 12.045 < 12.05.

Important Tips:

• Always align the decimal points before comparing.
• If the numbers have different lengths, consider adding zeroes to the shorter number.
• The place value is crucial: tenths are more significant than hundredths, and hundredths are more significant than thousandths.