DECIMAL FRACTIONS

These decimal fractions when expressed in the decimal form are known as decimal numbers or simply decimals.

Examples:  $(8/10), (64/100), (329/1000)$ are all decimal fractions which in decimal form can be written as $0.8, 0.64,$ and $3.29$ respectively.

A Decimal has two parts namely whole number part and decimal part. These parts are separated by a Dot (·) called the decimal point. The digit lying to the left of the decimal point form the whole number part. The decimal point together with the digits lying to its right form the decimal part.

Example: In the decimal $57.612$, the whole number part – is $57$ and the decimal part is $.612$.

DECIMAL PLACES

The number of digits contained in the decimal part of a decimal gives the number of its decimal places.

Examples:

The decimal $5.48$ has $2$ decimal places.
The decimal $7.067$ has $3$ decimal places.

Like Decimals: Decimals having the same number of decimal places are called like decimals.

Examples: $0.2, 33.4, 867.6, 4211.5$ are like decimals, each having one decimal place.

Unlike Decimals: Decimals having different number of decimal places are called unlike decimals.

Examples: $0.421, 3.31, 36.1, 391$ are all unlike decimals.

To Convert Unlike Decimals to Like Decimals

Out of the given unlike decimals find the decimal which has the largest number of decimal places, say n. Convert each of the remaining decimals to the one having n decimal places by annexing the required number of zeros to the extreme right of the decimal part.

Remarks: Annexing any number of zeros to the extreme right of the decimal part of a decimal does not change its value, i.e., $4.4 = 4.40 = 4.400,$ etc.

Example: Convert the decimals $13.42, 0.123, 9.6, 1.97$ into like decimals.

Solution: The decimal $0.123$ has the largest number of decimal places, i.e., $3$. So, we convert each of the given decimals into the one having $3$ decimal places. Thus, we write: $13.42 = 13.420, 0.123 = 0.123, 9.6 = 9.600$ and $1.97 = 1.970$

To Write a Decimal in an Expanded Form

Example:  Arrange the digits of the decimal 395.174 in the place value chart. Hence, write 395.174 in the expanded form.

Solution: We may arrange the digits of 395.174 in place-value chart, as shown below:

 Hundred Tens One Decimal Point Tenths Hundredths Thousandths 3 9 5 . 1 7 4

Therefore,

$\displaystyle 395.174 = 300 + 90 + 5 + 0.1 + 0.07 + 0.004$

$\displaystyle = 300 + 90 + 5 + \frac{1}{10} + \frac{7}{100}+ \frac{4}{1000}$

$\displaystyle = 3 \times 10^2 + 9 \times 10^1 + 5 \times 1 + 1 \times \frac{1}{10} + 7 \times \frac{1}{10^2} + 4 \times \frac{1}{10^3}$

Comparison of Two Decimals

1. Convert the given decimals into like decimals.
2. First compare their whole number parts. The decimal with the greater whole number part is greater.
3. If the whole number parts are equal, compare the tenths digits. The decimal with bigger digit- in tenths place is greater.
4. If the tenths digits are also equal, compare the hundredths digits and so on.

Example: Compare $17.63$ and $13.9$

Solution: We shall first convert the given decimals into like decimals. We thus get the decimals as $17.63$ and $13.90$. Now, we compare their whole number parts. Clearly, $17 > 13$; Therefore $17.63 > 13.90$

Example: Write the following decimals in descending order: $9.03, 4.85, 0.974, 7.5, 4.92 \text{ and } 0.7.$

Solution: Converting the given decimals into like decimals, we can write them as: $9.030, 4.850, 0.974, 7.500, 4.920$ and $0.700$. Clearly, $9.030 > 7.500 > 4.920 > 4.850 > 0.974 > 0.70$. Hence, the given decimals in descending order are: $9.03, 7.5, 4.92, 4.85, 0.974, \text{ and } 0.7$

OPERATIONS ON DECIMALS

1. Convert the given decimals into like decimals.
2. Write the addends under each other with decimal points in the same vertical column.
3. Add the numbers as whole numbers and in the result, place the decimal point just under all decimal points.

Example: Add: $24.6, 8.57, 0.9, 136.2358$ and $3.07$.

Solution: Converting the given decimals into like decimals, we get them as: $24.6000, 8.5700, 0.9000, 136.2358$ and $3.0700$.  Adding them column wise, we get:

Subtraction of Decimals

1. Convert both the decimals into like decimals.
2. Write the subtrahend (the number to be subtracted) under the minuend (the number from which subtraction is to be done) such that their decimal points are in the same vertical
3. Subtract as in whole numbers and in the result, place the decimal point just under the decimal points in the above numbers.

Example: Subtract $19.56$ from $25.2$.

Solution: Converting the given decimals into like decimals, we may write them as $19.56$ and $25.20$. Subtracting column wise, we get:

Multiplication of Decimals

Multiplication of a Decimal by $10, 100, 1000$, etc.: On multiplying a decimal by $10, 100, 1000$, etc. the decimal point is shifted to the right by as many places as the number of zeros in the multiplier.

Example Multiply:

• $93 \text{ by } 10$
• $584 \text{ by } 1000$
• $932 \text{ by } 100$
• $4 \text{ by } 100$

Solution:

• $93 \times 10 = 79.3$ [Decimal point is shifted 1 place to the right]
• $932 \times 100 = 1893.2$ [Decimal point is shifted 2 places to the right]
• $584 \times 1000 = 46584$ [Decimal point is shifted 3 places to the right]
• $4 \times 100 = 976.40 \times 100 = 97640$ [Decimal point is shifted 2 places to the right]

Multiplication of a Decimal by a Whole Number

1. Without taking the decimal point into consideration, multiply the given decimal by the given whole number (just like the multiplication of two whole numbers).
2. In the product, put the decimal point in such a way that the resulting decimal contains as many decimal places as there are in the given decimal.

Example:  Multiply: $74.53 \text{ by } 16$

Solution:

Multiplication of two or more Decimals

1. Without taking the decimal points (of the given decimals) into consideration, multiply the given decimals (just like the multiplication of whole numbers).
2. In the product, put the decimal point in such a way that the resulting decimal contains as many decimal places as the sum of the decimal places in all the given decimals.

Example: Find the products: $9.76 \text{ and } 1.2$

Solution: We have

$976 \times 12 = 11712$

Therefore $9.76 \times 1.2 = 11.712$ [Taking (2 + 1) decimal places in the product]

Division of Decimals

Dividing a Decimal by $10, 100, 1000,$ etc.: On dividing a decimal by $10, 100, 1000$, etc., the decimal point is shifted to the left by as many places as the number of zeros in the divisor.

Examples: Divide:

• $91.5 \text{ by } 10$
• $662.19 \text{ by } 100$

Solution:

• $91.5 \div10 = 9.15$ (Decimal point is shifted one place to left)
• $662.19 \div 100 = 6.6219$ (decimal point is shifted two places to left)

Dividing a decimal by a whole number: We make ordinary division and mark the decimal point in the quotient as soon as we cross over the decimal point in the dividend

Dividing a decimal by a Decimal

1. Convert the divisor into a whole number by multiplying both the dividend and the divisor by a suitable power of 10
2. Divide the new dividend by the whole number as the divisor

Simplification of Expressions Involving Decimals

Use of BODMAS Rule (remember the word BODMAS)

We simplify the expressions by applying the operations strictly in the order

1. Brackets
2. Of
3. Division
4. Multiplication
6. Subtraction

Removal of Brackets: Follow this order

1. Bar or Vinculum ()
2. Parenthesis ( )
3. Curly Brackets { }
4. Square Brackets [ ]

CONVERSION OF A FRACTION INTO A DECIMAL

To Convert a Fraction into a Decimal by Division Method

1. Divide the numerator by the denominator.
2. Complete the division. Let a non-zero remainder be left.
3. Insert a decimal point in the dividend and the quotient.
4. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Divide again just like whole numbers.
5. Repeat Step 4 till either the remainder is zero or the requisite number of decimal places has been obtained.

Rounding off of Decimals

The process of obtaining the value of a decimal correct to the required number of decimal places is called rounding off and the value obtained is called the rounded off or corrected value of the decimal.

To Round off a Decimal to the Required Number of Decimal Place

1. Retain as many digits after the decimal point as are required and omit the remaining digits.
2. Out of the omitted digits, if the first digit is 5 or more, then increase the last retained digit by 1, otherwise do not make any change.

Example:  Write $2.6483$ to $2$ decimal places:

Solution: In the given decimal $2.6483$, we retain $2$ digits after decimal point and omit the other digits. So, we get $2.64$. The first omitted digit is $8$, which is greater than $5$. So, we increase the last retained digit by $1$

Therefore the rounded off value of the given decimal is $2.65$.

Terminating Decimals: While expressing a fraction a decimal by the division method, if the division comes to an end after a finite number of steps, then such a decimal is a terminating decimal.

$\displaystyle \text{ Example } \frac{1}{2} = 0.5 \text{ or } \frac{1}{4} = 0.25$

Non-Terminating Decimals: While expressing a fraction into a decimal by the division method, if the division process continues indefinitely and a zero remainder is never obtained, then such a decimal is known as a non-terminating decimal

$\displaystyle \text{ Example } \frac{1}{3} = 0.33333...$

Repeating or Recurring Decimals: If in a decimal, a digit or a set of digits in the decimal part is repeated continuously, then such a number is called a recurring or repeating decimal.

$\displaystyle \text{ Example } \frac{1}{3} = 0.333... = 0.\dot{3} \text{ or } \frac{5}{6} = 0.8333... = 0.8\dot{3}$

Pure Recurring Decimals: A decimal in which all the digits in the decimal part are repeated, is called a pure recurring

$\displaystyle \text{ Example } \frac{1}{3} = 0.\overline{3} \text{ or } \frac{3}{7} = 0.428571428571... = 0.\overline{428571}$

Mixed Recurring Decimals: A decimal in which some of the digits in the decimal part are repeated and the rest are not repeated, is called a mixed recurring decimal.

$\displaystyle \text{ Example } \frac{5}{6} = 0.8333... = 0.8\dot{3} \text{ or } 0.5833... = 0.58\dot{3}$

CONVERSION OF A DECIMAL INTO A FRACTION (RATIONAL NUMBER)

To Convert a Terminating Decimal into a Fraction, follow the following steps:

1. Write the given decimal without the decimal point as
2. Take 1 annexed with as many zeros as is the number of decimal places in the given decimal as de
3. Reduce the above fraction in the simplest

$\displaystyle \text{ Example } 13.484 = \frac{13484}{1000} = \frac{3371}{250} = 13\frac{21}{250}$

To Convert a Recurring Decimal into a Fraction

The method of converting a recurring decimal into a fraction will be clear from the following example.

Example: Express $\displaystyle 0.666...$ as a fraction

Solution:

Let, $\displaystyle x = 0.666 ...$

Therefore $\displaystyle 10x = 6.666...$

Subtracting the two we get

$\displaystyle 9x = 6$

$\displaystyle x = \frac{2}{3}$

To Convert a Pure Recurring Decimal into Vulgar Fraction (Short Cut Method)

Write the repeated digits only once in the numerator and take as many nines in the denominator as the number of repeating digits.

Example:

$\displaystyle \dot{4} = \frac{4}{9}$

$\displaystyle \overline{54} = \frac{54}{99}$

$\displaystyle 1.\overline{074} = 1 + \frac{74}{999} = 1\frac{74}{999}$

Converting a Mixed Recurring Decimal into Vulgar Fraction (Short Cut Method)

In the numerator take the difference between the number formed by all the digits in the decimal part (taking repeated digits only once) and the number formed by the digits which are not repeated.

In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.

Example:

$\displaystyle 0.5\dot{3} = \frac{53-5}{90} = \frac{48}{90} = \frac{8}{15}$

Remarks: Order relation in fractions can be established by converting them into decimals.

H.C.F. AND L.C.M. OF DECIMALS

To Find the H.C.F. and L.C.M of Given Decimals follow the following steps:

1. Convert the given decimals into like decimals.
2. Find the H.C.F. or L.C.M (as the case may be) of the numbers without the decimal points.
3. In the result, mark the decimal point to have as many decimal places as there are in each decimal, obtained in Step 1.

Example:  Find the H.C.F. and L.C.M. of $0.54, 1.8$ and $7.2$

Solution:  Converting the given decimals into like decimals, we get them as: $0.54, 1.80$and $7.20$. (Each of these decimals has 2 decimal places)

We shall first find the H.C.F. and L.C.M of $54, 180$ and $720$.

H.C.F of $54, 180$ and $720 = 18$

Therefore so H.C.F. of $0.54, 1.8$ and $7.2 = 0.18$ [Taking 2 decimal places]

Also, L.C.M of $54, 180$ and $720 = 2160$

Therefore L.C.M. of $0.54 , 1.8$ and $7.2 = 21.60$ [Taking 2 decimal places]