Decimals

Decimals are fractions whose denominators are 10, 100, 1000 etc. These types of fractions are called decimal fractions or simply decimals.

• Tenths (Decimal fractions with 10 as denominator)
• Hundredths (Decimal fractions with 100 as denominator)
• Thousandths (Decimal fractions with 1000 as denominator)
• Mixed Decimals
• Conversion of Decimals into Common Fractions
• Conversion of Fractions into Decimals
• Expanded Form of Decimals
• Like and Unlike Decimals
• Conversion of Unlike Decimals into Like Decimals
• Order Relations in Decimals
• Addition of Decimals
• Subtraction of Decimals
• Multiplication of Decimals
• Dividing Decimals
• Multiplying and Dividing Decimals by 10 and 100
• Using Decimals in the Units of Length, Mass and Capacity

Tenths (Decimal fractions with 10 as denominator)

Decimal fractions with 10 as denominator are called Tenths.
To represent a fraction with 10 as denominator, as a decimal fraction, just put a dot[.] to the left of the digit in the ones place of the numerator.
Observe the following table.

Absence of digit to the left or right of the decimal point[.] can be represented by 0.
.5 can be written as 0.5

Hundredths (Decimal fractions with 100 as denominator)

Decimal fractions with 100 as denominator are called hundredths.
To represent a fraction with 100 as denominator, as a decimal fraction, just put a dot[.] to the left of the digit in the tens place of the numerator.
Observe the following table.

Thousandths (Decimal fractions with 1,000 as denominator)

Decimal fractions with 1,000 as denominator are called thousandths.
To represent a fraction with 1,000 as denominator, as a decimal fraction, just put a dot[.] to the left of the digit in the hundreds place of the numerator.
Observe the following table.

Mixed Decimals

When there is a whole number (1, 2, 3, ...) is placed to the left of the decimal point[.], then that decimal is called a mixed decimal.
A decimal has a whole number part and a decimal part. The absence of a whole number part is represented by 0.
For example,
1.5 is read as one point five.
2.45 is read as two point four five.
10.005 is read as ten point zero-zero five.

The number of digits after the decimal point is known as the number of decimal places.

Conversion of Decimals into Common Fractions

To convert a decimal into a fraction, first count the number of effective decimal points (don't count leading zeroes to the right of the decimal). The number formed by 1 and as many zeroes as the number of decimal points forms the denominator of the fraction. The decimal number without decimal points and leading zeroes at the right forms the numerator of the fraction.

Conversion of Fractions into Decimals

To convert a fraction into a decimal, first we need to count the number of zeroes coming in the denominator after 1. We need to place the decimal point at that place between the digits of the numerator, by counting from the right and stops when it reaches the count obtained in the previous step.
We can add zeroes to the left, if the numerator is short of digits.

Expanded Form of Decimals

We write the expanded form of decimals with the help of the place value chart.
Here is the place value chart for decimal numbers up to 3-places of decimal.

Example:

Write the place value of each digit in 745.238

Ans:
 7 hundreds
 4 tens
 5 ones
 2 tenths
 3 hundredths
 8 thousandths

Like and Unlike Decimals

Decimals having the same number of decimal places are called Like Decimals.
For example, 1.25, 3.52, 2.01 are like decimals.

Decimals having different number of decimal places are called Unlike Decimals.
For example, 1.232, 3.55, 2.1 are unlike decimals.

Conversion of Unlike Decimals into Like Decimals

We have already learnt that leading zeroes to the right of a decimal number doesn't change its value. Using this principle, we can convert any unlike decimal into like decimal by adding the required number of zeroes to the end of a decimal.
1.25, 5.5, 2.586 are unlike decimals.
1.250, 5.500, 2.586 are the same above decimals and they are like decimals too.

Order Relations in Decimals

How to compare decimals?

Follow these steps.

• Convert the given decimals into like decimals.
• Compare the whole number part. Decimal with the greater whole number part is the greater decimal.
• If the whole number parts are equal, compare the decimal parts starting from the tenths.
• If the tenths digits are equal, then compare the hundredths, thousandths digits and so on. Decimal digit with greater value is the greater decimal number.

(i) Compare 3.5 and 2.6
Between 3.5 and 2.6, the whole number part of 3.5 is greater than 2.6. So, 3.5 > 2.6
(ii) Compare 20.45 and 20.54
Between 20.45 and 20.54, the whole number parts are equal. The digit at the tenths of 20.54 is greater than 20.45. So, 20.45 < 20.54

Addition of Decimals

Follow the below steps to add two or more decimals.

• If the given decimals are unlike decimals, convert them into like decimals.
• Write the decimals in columns in such a way that the decimal points[.] come in the same column.
• Add the numbers just like ordinary numbers.
• Place the decimal point[.] at the same place of the sum as the decimal numbers.

Example:

(i) Add 15.852 and 8.56

 15.852
+  8.560
  24.412

(ii) Add 25.5 and 108.86

   25.50
+  108.86
  134.36

Subtraction of Decimals

Follow the below steps to subtract one decimal from another.

• If the given decimals are unlike decimals, convert them into like decimals.
• Write the decimals in columns in such a way that the decimal points[.] come in the same column.
• Subtract the numbers just like ordinary numbers.
• Place the decimal point[.] at the same place of the result as the decimal numbers.

Example:

(i) Subtract 26.32 from 47.16

 47.16
-  26.32
  20.84

(ii) Subtract 25.56 from 51.2

  51.20
-  25.56
 25.64

Multiplication of Decimals

Follow the below steps to multiply two or more decimals.

• Write the decimals in the simplest form. Remove any zeroes that comes at the right of the decimal part.
• Multiply the decimal numbers just like ordinary numbers.
• Count the number of decimals in all the numbers that we just multiplied.
• Put the decimal point[.] at the exact place in the product by counting from the right and stop when it reaches the count got in the previous step.

Example:

(i) Multiply 2.5 and 5.5

Number of decimals = 1 + 1 = 2
25 x 55 = 1375
2.5 x 5.5 = 13.75

(ii) Multiply 12.50 and 30.25

Number of decimals = 1 + 2 = 3
125 x 3025 = 378125
12.5 x 30.25 = 378.125

Dividing Decimals

Let's learn how to divide a decimal by a whole number.

Follow the below steps

1. We divide the decimal number just like as we divide a whole number.
2. Insert the decimal point[.] in the quotient immediately when we encounter the decimal point in the dividend.
3. Continue the division process until we cover all the numbers in the decimal.
4. If the remainder is still a non-zero number, continue the division by adding more zeroes to the right of the decimal part of the dividend.

Example:

Divide 16.62 by 3.

   5.54
3  16.62
  15  
   16 
   15 
   12 
   12 
   0

16.62 ÷ 3 = 5.54

Multiplying and Dividing Decimals by 10 and 100

(i) Multiplying Decimals by 10 and 100

Observe the following

425.0 x 10 = 4250
42.5 x 10 = 425
4.25 x 10 = 42.5

123.00 x 100 = 12300
12.30 x 100 = 1230
1.23 x 100 = 123

We can see that when multiplying a decimal with 10, the decimal point is moving one place to the right. Similarly, when multiplying a decimal with 100, the decimal point is moving two places to the right.

(ii) Dividing Decimals by 10 and 100

Observe the following

425 ÷ 10 = 42.5
42.5 ÷ 10 = 4.25

123 ÷ 100 = 1.23
12.3 ÷ 100 = 0.123

We can see that when dividing a decimal by 10, the decimal point is moving one place to the left. Similarly, when dividing a decimal with 100, the decimal point is moving two places to the left.

Using Decimals in the Units of Length, Mass and Capacity

We have learnt about measurements in the previous classes. We need decimals when we convert smaller units into bigger units.

Observe the below table

Using the above information, we can do any type of conversions. See the table below.