Class 6 — Mathematics (NCERT)

Decimals: Place Value, Addition and Subtraction

Class 6

✓Understand the concept of decimal numbers and their relation to fractions.
✓Identify the place value of digits in decimal numbers up to hundredths.
✓Represent decimal numbers on a place value chart.
✓Perform addition of decimal numbers.
✓Perform subtraction of decimal numbers.

Key concepts

Decimal Numbers

Decimal numbers are another way of representing fractions where the denominator is 10, 100, 1000, and so on. They are used to represent parts of a whole. A decimal number has two parts: a whole number part and a decimal part, separated by a decimal point. For example, 3.5 means 3 whole and 5 tenths.

Place Value of Decimals

Just like whole numbers, each digit in a decimal number has a place value. The digits to the left of the decimal point represent the whole number part (Ones, Tens, Hundreds, etc.). The digits to the right of the decimal point represent the fractional part.\n\n- The first digit to the right of the decimal point is in the 'Tenths' place (value 1/10).\n- The second digit to the right is in the 'Hundredths' place (value 1/100).\n- The third digit to the right is in the 'Thousandths' place (value 1/1000), and so on.\n\nFor example, in 25.37:\n- 2 is in the Tens place (value 20)\n- 5 is in the Ones place (value 5)\n- 3 is in the Tenths place (value 3/10)\n- 7 is in the Hundredths place (value 7/100)

Addition of Decimals

To add decimal numbers, we need to align the decimal points vertically. This ensures that digits with the same place value are added together. If necessary, we can add trailing zeros to make the number of decimal places equal. Then, add the numbers as we would add whole numbers, placing the decimal point in the sum directly below the decimal points of the numbers being added.

Subtraction of Decimals

To subtract decimal numbers, similar to addition, we must align the decimal points vertically. We can add trailing zeros to the number with fewer decimal places to make the number of decimal places equal. Then, subtract the numbers as we would subtract whole numbers, placing the decimal point in the difference directly below the decimal points of the numbers.

Key facts to remember

1Decimal numbers represent parts of a whole and are written with a decimal point.
2Digits to the left of the decimal point are whole numbers (Ones, Tens, Hundreds).
3Digits to the right of the decimal point are fractional parts (Tenths, Hundredths, Thousandths).
4The place value of the first digit after the decimal point is Tenths (1/10).
5The place value of the second digit after the decimal point is Hundredths (1/100).
6To add or subtract decimals, always align the decimal points vertically.
7Adding trailing zeros to the right of the last decimal digit does not change the value of the decimal number (e.g., 5.2 = 5.20 = 5.200).

Worked examples

Example 1

Write the number represented by the following place value chart:\n| Hundreds | Tens | Ones | . | Tenths | Hundredths |\n| :------- | :--- | :--- | :- | :----- | :--------- |\n| 3 | 0 | 7 | . | 4 | 5 |

IIdentify the digits in the whole number part: 3 Hundreds, 0 Tens, 7 Ones. This forms the number 307.

IIIdentify the digits in the decimal part: 4 Tenths, 5 Hundredths. This forms the decimal part .45.

IIICombine the whole number part and the decimal part with a decimal point.

Answer

307.45

Example 2

Add 15.34 and 8.7.

IWrite the numbers vertically, aligning the decimal points:

II 15.34

III+ 8.7

IV-------

VAdd a trailing zero to 8.7 to make the number of decimal places equal:

VI 15.34

VII+ 8.70

VIII-------

9Add the digits column by column, starting from the rightmost digit:

10 - Hundredths place: 4 + 0 = 4

11 - Tenths place: 3 + 7 = 10 (Write 0, carry over 1 to the Ones place)

12 - Ones place: 5 + 8 + 1 (carry over) = 14 (Write 4, carry over 1 to the Tens place)

13 - Tens place: 1 + 1 (carry over) = 2

14Place the decimal point in the sum directly below the decimal points of the numbers being added.

Answer

24.04

Always align the decimal points carefully before adding.

Example 3

Subtract 12.5 from 30.25.

IWrite the numbers vertically, aligning the decimal points. The larger number (30.25) goes on top:

II 30.25

III- 12.5

IV-------

VAdd a trailing zero to 12.5 to make the number of decimal places equal:

VI 30.25

VII- 12.50

VIII-------

9Subtract the digits column by column, starting from the rightmost digit:

10 - Hundredths place: 5 - 0 = 5

11 - Tenths place: 2 - 5. We cannot subtract 5 from 2, so we borrow 1 from the Ones place (0 becomes 9, 3 becomes 2). The 2 becomes 12. So, 12 - 5 = 7.

12 - Ones place: 9 - 2 = 7 (after borrowing from Tens place)

13 - Tens place: 2 - 1 = 1 (after 3 became 2 due to borrowing)

14Place the decimal point in the difference directly below the decimal points of the numbers.

Answer

17.75

Remember to borrow from the next higher place value when a digit is smaller than the one being subtracted.

Common mistakes

✗Not aligning the decimal points correctly during addition or subtraction.
✗Forgetting to add trailing zeros to make the number of decimal places equal before performing addition or subtraction, leading to incorrect alignment.
✗Incorrectly borrowing or carrying over digits, especially when zeros are involved.
✗Placing the decimal point in the wrong position in the final answer.
✗Confusing place values (e.g., thinking the first digit after the decimal is 'ones' or 'units').

Exam tips

★Always write down the numbers neatly, ensuring the decimal points are perfectly aligned vertically.
★Use a place value chart for understanding and writing decimal numbers if you find it helpful.
★When adding or subtracting, add trailing zeros to ensure all numbers have the same number of decimal places. This helps prevent alignment errors.
★Double-check your calculations, especially borrowing and carrying over, by working backwards or using estimation.

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