Decimals, Expanded Form, Operations on Decimals

# Decimals, Expanded Form, Operations on Decimals

## Decimals

A fraction with denominator 10, 100, 1000, etc. are known as a decimal fraction.

Decimal fraction with denominator 10 is known as one-tenth (1/10), written as 0.1 in decimals and is read as zero point one.
The decimal fraction with denominator 100 is known as one-hundredth (1/100), written as 0.01 and is read as zero point zero one.
The decimal fraction with denominator 1000 is known as one-thousandth (1/1000), written as 0.001 and is read as zero point zero zero one.
A decimal number has a whole number part and a fractional part (decimal part) which are separated by a decimal point.
For example, in 46.835, 46 is the whole number part and 835 is the fractional part or decimal part.

### Expanded Form of Decimals

Consider the decimal 726.385
The expanded form of 726.385 is 726.385 = 7 × 100 + 2 × 10 + 6 × 1 + 3 × 1/10 + 8 × 1/100 + 5 × 1/1000 and can be read as seven hundred twenty-six point three eight five.

Example: Expand the following.
a. 495.672                                b. 754.23

Solution:
a. 495.672 = 4 × 100 + 9 × 10 + 5 × 1 + 6 × 1/10 + 7 × 1/100 + 2 × 1/1000
b. 754.23 = 7 × 100 + 5 × 10 + 4 × 1 + 2 × 1/10 + 3 × 1/100

## Converting Decimals into Fractions

To convert decimals into fractions, write the number without the decimal point in
the numerator and write 10 or powers of 10 in the denominator according to the decimal places in the decimal and write this fraction in the simplest form.

Example: 0.6 = 6/10 = 3/5; 0.02 = 2/100 = 1/50; 3.125 = 3125/1000 = 25/8

## Converting Fractions into Decimals

1. When the denominator of the given fraction is 10 or a multiple of 10, count from extreme right to the left, mark the decimal point after as many digits of the numerator
as there are zeroes in the denominator.
For example, 46/10 = 4.6; 7054/100 = 70.54; 563/1000 = 0.563

2. When the denominator can be expressed as multiples of 10, multiply numerator and
denominator of a fraction by a suitable number such that the denominator becomes 10
or power of 10 and then follow the above steps.
For example, 1/2 = (1 × 5)/(2 × 5) = 5/10 = 0.5

## Like and Unlike Decimals

All the decimals having the same number of decimal places are called like decimals and all the decimals having different number of decimal places are called unlike decimals.
For example,
a. 4.57, 18.78, 127.09 are like decimals.          b. 7.5, 4.85, 2.605 are unlike decimals.

The unlike decimals can be converted into like decimals by adding the required number of zeroes in the extreme right of the decimal.

Example: Convert 2.43, 4.6, 3.475 and 8.4237 into like decimals.

Solution: 2.43 can be written as 2.4300, 4.6 can be written as 4.6000 and 3.475 can be
written as 3.4750.
Thus, 2.4300, 4.6000, 3.4750 and 8.4237 are like decimals.

## Comparison of Decimals

While comparing two decimals, first we compare the whole number parts of both
the decimals. The decimal with greater whole number part is greater.
When whole number parts of both the decimal numbers are the same, then we compare the tenths part. If tenths parts of both the decimal numbers are the same, then we compare the hundredths part and so on.

Example: Compare and arrange the following numbers in ascending order:
76.63, 62.335, 63.36, 62.33

Solution: 62.330 < 62.335; 62.335 < 63.36; 63.36 < 76.63
Ascending order: 62.33, 62.335, 63.36, 76.63

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