Basic Operations with Decimals

Decimals are an essential part of mathematical operations, and mastering the basic operations with them can significantly enhance your numerical skills. In this article, we’ll explore how to perform addition, subtraction, multiplication, and division with decimal numbers. We’ll provide clear examples and useful tips to ensure you understand each operation with ease.

Addition of Decimals

Adding decimals is similar to adding whole numbers, but with an emphasis on lining up the decimal points. Here’s a step-by-step guide:

  1. Align the Decimal Points: Write the numbers one under the other, ensuring that the decimal points are in the same column.
  2. Add from Right to Left: Start adding from the rightmost digits, carrying over if necessary.
  3. Place the Decimal Point in the Result: Draw a decimal point in the result directly below the decimal points of the numbers being added.

Example 1

Let’s add 12.35 and 7.8:

  12.35
+  7.80
-------

Now, align the decimal points and add:

  • Start from the right: 5 + 0 = 5
  • Next column: 3 + 8 = 11 (write down 1 and carry over 1)
  • Next column: 2 + 7 + 1 (carried over) = 10 (write down 0 and carry over 1)
  • Last column: 1 + 0 + 1 (carried over) = 2

So, 12.35 + 7.80 = 20.15.

Example 2

Now, let’s try a longer decimal: 0.456 and 0.789.

  0.456
+ 0.789
-------

Align the decimals:

  • 6 + 9 = 15 (write down 5, carry over 1)
  • 5 + 8 + 1 = 14 (write down 4, carry over 1)
  • 0 + 7 + 1 = 8

Thus, 0.456 + 0.789 = 1.245.

Subtraction of Decimals

Subtracting decimals requires a similar approach to addition. Here’s how to do it:

  1. Align the Decimal Points: Write the larger number above the smaller number, ensuring the decimal points are aligned.
  2. Subtract from Right to Left: Perform the subtraction from the rightmost digit to the left.
  3. Reinforce the Decimal Point: Place the decimal point in the result directly under the decimal points of the numbers.

Example 3

Let’s subtract 15.9 from 25.6:

  25.6
- 15.9
-------

Align the decimal points:

  • 0 (from 6) - 9 (borrow 1) = 7
  • 5 - 5 = 0
  • 2 - 1 = 1

Therefore, 25.6 - 15.9 = 9.7.

Example 4

Now, consider 3.05 - 1.97:

  3.05
- 1.97
-------

Align the decimals:

  • 5 - 7 (borrow 1) = 8
  • 0 - 9 (borrow 1) = 2
  • 2 - 1 = 1

Thus, 3.05 - 1.97 = 1.08.

Multiplication of Decimals

Multiplying decimals can be simplified by treating the decimals as whole numbers first, and then placing the decimal point in the result afterwards. Here’s how:

  1. Ignore the Decimal Points: Treat the numbers as whole numbers and multiply them normally.
  2. Count the Decimal Places: Count how many digits are to the right of the decimal points in both numbers you are multiplying.
  3. Place the Decimal Point: In your final product, place the decimal point so that the total number of decimal places equals the sum you counted in step 2.

Example 5

Multiply 2.5 by 3.4:

Ignoring the decimals, we have:

  25
× 34
------

Calculating:

  • 25 × 4 = 100
  • 25 × 30 = 750 (or 25 × 3 × 10)

Now, we add these two results:

  100
+750
------
  850

Now, there is one decimal place in 2.5 and one in 3.4, which means we have a total of two decimal places. Therefore, we place the decimal point two places from the right in 850:

So, 2.5 × 3.4 = 8.50 or simply 8.5.

Example 6

Next, let’s multiply 0.6 by 0.4:

Ignoring the decimals, we have:

  6
× 4
------

Calculating:

  6 × 4 = 24

In this case, there is one decimal place in each number (total two). Thus, we place the decimal point two places from the right.

0.6 × 0.4 = 0.24.

Division of Decimals

Dividing decimals involves making adjustments to ensure that the divisor is a whole number. Here’s how you can perform the division:

  1. Move the Decimal Point: If the divisor (the number you are dividing by) is not a whole number, move the decimal point to the right until it is. Move the decimal point of the dividend (the number being divided) the same number of places.
  2. Divide as Usual: Then, proceed to divide as you would with whole numbers.
  3. Place the Decimal Point: The decimal point in the result is placed directly above where it is in the dividend.

Example 7

Let’s divide 4.2 by 0.6:

First, we move the decimal point in 0.6 one place to the right (making it 6). We do the same with 4.2 (making it 42).

This gives us:

  42 ÷ 6

Now divide:

  • 6 goes into 42 seven times, since 6 × 7 = 42.

Thus, 4.2 ÷ 0.6 = 7.

Example 8

Now let’s try dividing 5.25 by 0.15.

We move the decimal points in 0.15 two places to the right (making it 15) and do the same in 5.25 (making it 525).

So we need to solve:

  525 ÷ 15

Dividing:

  • 15 goes into 52 three times (15 × 3 = 45).
  • Subtracting 45 from 52 gives us 7.
  • Carry down the 5, making it 75.
  • 15 goes into 75 five times (15 × 5 = 75).

Thus, 5.25 ÷ 0.15 = 35.

Conclusion

Mastering the basic operations of addition, subtraction, multiplication, and division with decimals is crucial for effective mathematical problem-solving. By practicing the steps outlined in this article, you can become more comfortable and proficient in working with decimal numbers, paving the way for more advanced mathematical concepts.

Don’t hesitate to use these examples as a reference, or practice with different decimal values to further solidify your understanding! Happy calculating!