Adding and Subtracting Decimals is a unit of instruction that shows students how to read, write, compare, order, add and subtract decimals. Other topics include estimating decimal sums and differences, and solving word problems.
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How to Add Decimals
Adding decimals is a fundamental arithmetic operation. Here’s a step-by-step explanation of how to add decimals:
Step 1: Line up the decimals.
Begin by aligning the decimal points of the numbers you want to add. This means that the decimal points should be vertically aligned, making it easier to add the corresponding place values.
Step 2: Add the digits to the right of the decimal.
Start from the rightmost digit (ones place) and add the numbers in each place value column one by one. Begin with the ones place and move to the tenths, hundredths, thousandths, and so on.
Step 3: Carry over if necessary.
As you add, if the sum of digits in a column exceeds 9, you may need to carry over to the next column. For example, when adding 7 + 8 in the tenths place, you get 15. In this case, write down 5 in the tenths place and carry over the 1 to the next column (hundredths).
Step 4: Continue adding.
Keep adding digits in each place value column, including any carried-over values from the previous step. Proceed from right to left until you’ve added all the digits.
Step 5: Write the result with the decimal point.
After adding all the digits, including any carried-over values, place the decimal point in the result directly below the aligned decimal points in the numbers you were adding.
Step 6: Simplify (if necessary).
Finally, check if the result can be simplified. Remove any trailing zeros to express the answer in its simplest form if applicable.
How to Subtract Decimals
Subtracting decimals is another fundamental arithmetic operation. Here’s a step-by-step explanation of how to subtract decimals:
Step 1: Line up the decimals.
Begin by aligning the decimal points of the numbers you want to subtract. Just like when adding decimals, this ensures that the decimal points are vertically aligned for each number.
Step 2: Subtract the digits to the right of the decimal.
Start from the rightmost digit (ones place) and subtract the numbers in each place value column one by one. Begin with the ones place and move to the tenths, hundredths, thousandths, and so on.
Step 3: Borrow if necessary.
As you subtract, if a digit in the top number is smaller than the digit below it, you may need to borrow from the next higher place value column. For example, when subtracting 5 from 3 in the tenths place, you can’t do it directly. In this case, borrow 1 from the ones place, making it 13 – 5.
Step 4: Continue subtracting.
Keep subtracting digits in each place value column, including any borrowed values from the previous step. Proceed from right to left until you’ve subtracted all the digits.
Step 5: Write the result with the decimal point.
After subtracting all the digits, including any borrowed values, place the decimal point in the result directly below the aligned decimal points in the numbers you were subtracting.
Step 6: Simplify (if necessary).
Finally, check if the result can be simplified. Remove any trailing zeros to express the answer in its simplest form if applicable.
How to Order and Compare Decimals
Ordering and comparing decimals is an essential skill in mathematics and everyday life. It involves arranging decimals in ascending (from smallest to largest) or descending (from largest to smallest) order, as well as determining which decimal is greater or smaller in value. Here’s a detailed explanation of how to order and compare decimals:
Ordering Decimals
Step 1: Understand the Decimal Place Value.
Each digit to the right of the decimal point has a specific place value. The place values decrease from left to right: tenths, hundredths, thousandths, and so on. Understanding these place values is crucial for comparing and ordering decimals.
Step 2: Arrange the Decimals Based on Place Values.
Start by comparing the digits in the leftmost place value (e.g., tenths place). Arrange the decimals from smallest to largest or vice versa based on these digits.
If the digits in the leftmost place value are equal, move on to the next place value (e.g., hundredths) and compare those digits. Continue this process until you’ve compared all the digits.
Step 3: Fill in Leading Zeros (if necessary).
When comparing decimals, you may encounter situations where some decimals have fewer digits to the right of the decimal point than others. In such cases, consider the missing digits as zeros to make comparisons easier.
Comparing Decimals
Step 1: Compare the Leftmost Digits.
Start by comparing the digits in the leftmost place value (e.g., tenths place). Determine which decimal has the larger or smaller digit in that place value.
Step 2: If the Leftmost Digits Are Equal, Move to the Next Place Value.
If the digits in the leftmost place value are equal, compare the digits in the next place value (e.g., hundredths). Continue this process until you find a difference or have compared all the digits.
Step 3: Consider Leading Zeros (if necessary).
When comparing decimals with different numbers of digits to the right of the decimal point, remember that zeros to the right of significant digits do not affect the comparison. Focus on the first non-zero digit in each decimal.
Step 4: Determine Greater or Smaller.
Based on your comparisons, determine which decimal is greater or smaller. If you find a difference in the leftmost place value, that will determine the order. If you’ve compared all the digits and they are equal, the decimals are equal in value.
Examples For You
Let’s compare and order the decimals 0.56, 0.007, 0.59, and 0.6.
Ordering Decimals:
Step 1: Understand the Decimal Place Value.
The place values from left to right are tenths, hundredths, and thousandths.
Step 2: Arrange the Decimals Based on Place Values.
Start with the tenths place: 0.56, 0.007, 0.59, 0.6.
Now, compare the hundredths place: 0.007, 0.56, 0.59, 0.6.
Finally, compare the thousandths place: 0.007, 0.56, 0.59, 0.6.
So, the ascending order is 0.007, 0.56, 0.59, 0.6.
Comparing Decimals:
Step 1: Compare the Leftmost Digits.
In the tenths place, we have 0.5, 0.0, 0.5, and 0.6. The largest is 0.6.
Step 2: If the Leftmost Digits Are Equal, Move to the Next Place Value.
In the hundredths place, we have 0.05, 0.00, 0.09, and 0.00. The largest is 0.09.
Step 3: Consider Leading Zeros.
When comparing 0.007 and 0.009, we can ignore the leading zeros and compare 7 and 9.
Step 4: Determine Greater or Smaller.
After comparing all the digits, 0.009 is greater than 0.007.
So, the order from smallest to largest is 0.007, 0.56, 0.59, 0.6.