Chapter 8 - Decimals

Introduction to Decimal Numbers

The word 'decimal' is derived from the Latin word 'decimus' which means 'tenths'.

Decimal numbers are written with a dot between two digits of a number. The dot is known as the decimal point. For example, 23.45, 3.14, 0.02, etc.

A Combination of Whole and Fraction

Decimals are similar to fractions in that they too represent parts of a whole. For example, 3½ is a mixed fraction that can be written as a decimal number 3.5.

The number to the left of the decimal point represents a whole number, and
The number to the right of the decimal point represents a fractional part.

Decimals are used to represent numbers that lie between two consecutive whole numbers.

In a decimal number, the number to the left of the decimal point are read like whole numbers, while the digits to the right of the decimal point are read individually. For example, 47.28 will be read as "forty-seven point two eight".

Fractions to Decimals

Fractions and decimals are closely related. A fraction can be converted to a decimal number by following through the division in the fraction. For example,

${7}/{4}$ = 1.75

A decimal number represents a number that is greater than a whole number, but smaller than its consecutive next whole number, similar to a mixed fraction.

Place Value of Decimals

Place Value of Whole Numbers

We know how the different digits of a whole number have different place values depending on their places in the place value table.

For example, in the whole number 27, the digit 2 is in tens' place and the digit 7 is in ones' place.

As we move to the left, the place value increases from its previous value by a factor of 10.
As we move to the right, the place value decreases from its previous value by a factor of ${1}/{10}$

Place Value of Decimals

While writing a decimal number, the decimal point is used to separate its fractional parts from the whole parts.

Just like whole numbers, decimals too have their own well-defined place value system.

The Tenths' place is formed when we divide a whole into ten equal parts.
The Hundredths' place is formed when we divide a tenth further into ten equal parts.

Whole and Fractional Parts

All place values to the left of the decimal point are called the whole parts.
All place values to the right of the decimal point are called the fractional parts.

Example: Draw the place value table for the decimal number 218.95.

Solution:

$$218.95 = 200 + 10 + 8 + {9}/{10} + {5}/{100}$$

Place value table:

Comparing Decimals

Let's use examples to learn how to compare two decimal numbers.

Example: Compare the decimal numbers, 12.9 and 15.4.

Solution:

First, compare the whole parts of the decimal numbers.
Here, they are 12 and 15.
Clearly, 12 < 15.

Hence, 12.9 < 15.4

If the whole parts of two decimal numbers are equal, then compare their fractional parts.

Example: Compare the decimal numbers, 50.54 and 50.45.

Solution:

The whole parts are same, i.e., 50.
So, let's compare the fractional parts.
Clearly, 54 > 45.

Hence, 50.54 > 50.45

Using Place Value Charts

Place value charts can be used to compare decimal numbers.

It's a two-step process:

Write the numbers in their place value charts.
Then, compare the digits from the leftmost place to the rightmost place.

Example: Compare the decimal numbers, 4.5 and 4.56.

Solution:

Writing the numbers in their place value charts,

Fill the empty place values with 0 as placeholder.

Comparing the digits at each place,

4.5 < 4.56

4.5 is smaller than 4.56.

Representation of Decimal Numbers on the Number Line

To represent a decimal on the number line, it is first converted into its corresponding fraction, which is then represented on the number line.

To mark any decimal number on the number line,

First, locate the two consecutive whole numbers between which the number lies.
Then, divide the gap between the two whole numbers into ten equal parts (or tenths).

Decimal numbers with more than 2 digits after the decimal point are difficult to represent on the number line.

For example, to represent 0.75 on the number line, the gap between 0 and 1 has to be divided into 100 equal parts and it is not an easy task using a pen and paper.

Example: Represent the decimal 5.7 on the number line.

Solution:

Splitting 5.7 into whole and decimal,

5.7 = 5 + 0.7

So, 5.7 lies between 5 and 6.

Next, divide the region between 5 and 6 into 10 equal parts.

On the number line, when the gap between two whole numbers is divided into 10 equal parts, each part represents a tenth of a whole.

Hence, the 7^th line between 5 and 6 represents 5.7.

Example: Represent the decimal 0.5 on the number line.

Solution:

Converting decimal to fraction,

0.5 = ${5}/{10}$

Numerator > Denominator, so it is a proper fraction that lies between 0 and 1.

Divide the region between 0 and 1 into 10 equal parts.

The 5^th line from 0 towards right is ${5}/{10}$ or 0.5 on the number line.

Converting Fractions to Decimals

Converting Decimals to Fractions

To convert a decimal number into a fraction,

First, the number is written without the decimal point.
Treat this number as the numerator of the fraction.
The denominator of the fraction will be 1 followed by as many zeros as the number of digits after the decimal point in the number.

For example,

$$0.43 = {43}/{100} \: and \: 0.8 = {8}/{10}$$

Similarly,

$$9.12 = {912}/{100} = 9{12}/{100}$$

Example: Convert the decimal 34.36 into a fraction.

Solution:

Rewriting 34.36,

= 34 + 0.36

= 34 + ${36}/{100}$

= 34 ${36}/{100}$

= ${3436}/{100}$

There is another method of converting decimals to fractions.

As per place values,

34.36 = 3 × 10 + 4 × 1 + ${3}/{10}$ + ${6}/{100}$

= 30 + 4 + ${3}/{10}$ + ${6}/{100}$

= ${3436}/{100}$ = 34${36}/{100}$

Converting Fractions to Decimals

To convert a fraction into its decimal equivalent, long division is done till 0 is obtained as the remainder. For example,

${3}/{2}$ = 1.5

Similarly,

${3}/{4}$ = 0.75

If the denominator of a fraction is a power of 10 (like 100 or 1000), then it is quite easy to convert the fraction to its decimal equivalent.

Take the numerator and place a decimal point from the right after as many places as the number of 0's in the denominator.

For example,

${34}/{100}$ = 0.34 ; ${6}/{10}$ = 0.6 ; ${764}/{1000}$ = 0.764

If the denominator of a fraction is not a power of 10, then find its equivalent fraction with a denominator that is a power of 10.

For example,

${13}/{25}$ = ${13 × 4 }/{25 × 4}$ = ${52}/{100}$ = 0.52

Real-life Applications of Decimals

It is often that we have to deal with numbers that are not whole numbers. In such cases, decimals are unavoidable. For example, the prices of items we purchase are in decimals like the cost of hair oil may be ₹86.75

Decimals are used in finance, while measuring lengths, while finding the weight of an object, and in many more places.

Decimals in Finance

In finance, decimals are used to write a value in different denominations.

Example:

There are 100 paise in ₹1. So,

1 paisa= ${1}/{100}$ of a rupee

39 paisa = ${39}/{100}$ of a rupee = ₹0.39

Similarly,

293 paise = 200 paise + 93 paise

= ₹2 and 93 paise = ₹2.93

Decimals in Measurements

Decimals play a major role in measuring lengths. The most common units of measuring length are kilometres (km), metres (m), centimetres (cm), and millimetres (mm).

1 km = 1000 m

1 metre = 100 cm

1 cm = 10 mm

Example: Convert 23 cm into metres.

Solution:

100 cm = 1 m

⇒ 1 cm = ${1}/{100}$ m

⇒ 23 cm = ${23}/{100}$ m = 0.23 m

Example: Convert 2.5 m into centimetres.

Solution:

2.5 m = 2 m + 0.5 m

= (2 × 100) cm + (${5}/{10}$ × 100) cm

= 200 cm + 50 cm

= 250 cm

Example: Express 45 gram in kilograms.

Solution:

"Kilo" refers to 1000.

1000 gm = 1 kg

⇒ 1 gm = ${1}/{1000}$ kg

⇒ 45 gm = ${45}/{1000}$ kg = 0.045 kg

Addition of Decimals

Addition is the most basic arithmetic operation that is used to find the sum of two numbers.

Column Method of Addition

While writing the numbers, the digits of the same place value must lie in the same column. For example,

Adding Two Decimal Numbers

While adding two decimal numbers,

Place the numbers one below the other.
Align their decimal points.
Then, add the numbers digit wise from right to left.

Example: Add the decimal numbers 5.47 and 2.16.

Solution:

First, place the digits in their corresponding columns as per their place value, and align the decimal points:

Start the addition from the rightmost place value.

7 + 6 = 13

So, we write 3 and carry the 1 to the tenths place.

Next, add the tenths column,

4 + 1 = 5

Adding the carry from the previous addition, we get

5 + 1 = 6

Finally, adding the ones column,

5 + 2 = 7

Therefore, 5.47 + 2.16 = 7.63

If a decimal number has less number of digits than the other number, then 0's are added as placeholders before the whole number part and after the fractional part. It does not change the value of the decimal number.

For example,

7.65 = 7.650 = 07.650

Subtraction of Decimals

The column method is used to subtract two decimal numbers.

The steps are as follows:

The numbers are written one below the other such that their decimal points are aligned.
Start from the rightmost column and keep subtracting till all columns are subtracted.
If subtraction is not possible, borrow from the next higher place value and subtract.

Example: Subtract 5.47 − 2.16

Solution :

Write the decimal numbers in a place value table with their decimal points aligned.