Decimal Fractions

GETTING STARTED

Let us take a number 234.89

☛ The hundreds digit is 2.
Place value = 2 x100 =200. Thus the place value is 200

☛ The tens digit is 3.
Place value = 3 x 10 =30. Thus the place value is 30

☛ The units digit is 4.
Place value = 4 x1 =4. Thus the place value is 4

☛ The multiplier keeps decreasing in multiples of 10 from right to left.
As we move beyond the units place, the multiplier becomes 1/10, 1/100 and so on.

☛ A fraction whose multipliers are negative powers of 10 (1/10 , 1/100 …) is known as decimal fraction.

 ☛ The place value of 8 in the number is 8 x 1/10  = 0.8

The decimal point is placed after as many zeros in denominator from right. Here denominator is 10, so decimal point is placed after 1 digit from right

☛ Similarly, place value of 9 is 9 x 1/100 = 0.09

A decimal fraction can be converted into vulgar fraction and vice versa

To convert a decimal fraction into vulgar fraction,

We write the entire number (without any decimal point) and divide it by the number obtained by writing 1

followed by as many zeroes as the numbers after decimal point. It is then reduced to its lowest form

Eg. Convert 2.35 into vulgar fraction:

2.35 = 235/100 [Dividing 235 by 100 … (Since two digits are after decimal point)]
= 47/20 

 

    is a mixed fraction since it has both whole number and a fraction part.

It is to be noted that adding zeroes to the extreme right of decimal fraction does not change its value.

Eg. 0.4 is same as 0.400

1.3 is same as 1.30

Mathematical Operations in Decimal fractions:

Addition/ Subtraction of decimal fractions: ☛ Write the two numbers one after the other in such a way that the decimal point lies in one column and add/subtract them the usual way.  

Multiplying a decimal fraction by 10 and its powers:
☛ When we multiply a decimal fraction by a power of 10, we shift the decimal as many places to right as the number of zeroes in the multiplier.

   Eg. Multiply 2.084 with 100

   There are 2 zeroes in multiplier. So, shifting decimal 2 places to right, we get, 208.4

Multiplying two/three decimal fractions:

☛ When two or more decimal fractions are multiplied, the numbers are multiplied normally without considering the decimal.

☛ Then the decimal point is placed after as many digits from the right as the sum of decimal places in given numbers.

Eg. 2.34 x 3.2 x1.1

Multiplying 234, 32 and 11 we get, 82368

Number of Decimal place in 2.34 = 2
Number of Decimal place in 3.2 =1
Number of Decimal place in 1.1 =1

Total sum of decimal places = 4.

Placing the decimal point after 4 places from right, we get, 8.2368

☛ In lengthier calculations, it is possible to get confused. Let us say we have to check if the answer is 8.2368 or 82.368

 In order to check, multiply only the integer values.
 2 x 3 x1 =6.

Thus the answer should have a number nearer to 6.

Hence the first one-   8.2368 is appropriate and second one can be ruled out.

Dividing Decimal Fractions:

☛ When dividing a decimal fraction by an integer, we divide it normally without considering the decimal point.

☛ Then, we place the decimal point after the number of decimal places in dividend

Eg. 36.235 divided by 5

On dividing 36235 by 5 we get, 7247 and placing the decimal point after 3 digits (decimal places in 36.235 = 3) from right, we get, 7.247.

☛ When dividing a decimal fraction by another decimal fraction, we convert the decimal fraction in denominator into an integer by multiplying by powers of 10.

Correspondingly, numerator is also multiplied by the same power of 10. Then it is divided as the above case.

Eg. 23.835 divided by 1.5

Multiplying 1.5 by 10 to make it an integer and the same with the numerator – 23.835

We get, 238.35 divided by 15
Solving, we get, 15.89

Decimal Fractions classified into types as following:

Decimal fractions can be either classified as non-recurring (terminating) or recurring.

☛ Terminating decimal fractions:
The fractions terminate within a few numbers after the decimal point

eg. 0.375, 0.3

 ☛ Recurring decimal fractions:

If a figure or a set of figures repeat themselves continuously, then it is a recurring decimal fraction.

They are of two types – Pure recurring decimal and Mixed recurring decimal.

Pure recurring decimal have the entire part after the decimal point repeating.

 Eg.

   (0.33333….)

Only a few digits repeat in the mixed decimal.

Eg.  

  (0.375757575…)

Here only 75 is repeated

Converting Recurring Decimal fraction to Vulgar Fractions:

In order to convert a pure decimal fraction into vulgar fraction, we divide the part that is being repeated by a number formed by as many 9’s as the repeating no.

Eg1.

3 is the recurring no. in the given example.

Thus, we divide 3 by 9 and simplify it to the lowest form.

3/9 = 1/3 is the required vulgar fraction

Eg 2:

The recurring numbers are 375 and it will be the numerator.

Denominator will be the number formed with as many 9s as the numbers in numerator ie. 999

Required vulgar fraction = 375/999 = 125/333

In order to convert a mixed decimal fraction into a vulgar fraction, ☛ Numerator = Number formed by attaching recurring number to the right of non-recurring number – nonrecurring number ☛ Denominator = Number formed by as many 9s as recurring unit attached to as many no. of zeros as the non-recurring number.

Eg.

Numerator = 375 – 3= 372 (Non-recurring unit is 3 and the recurring unit is ‘75’)

Denominator = 990 (Two digits recurring – 7 and 5, so two 9 and One digit non-recurring -3, so one 0)

Reqd Fraction = 372/990 = 124/330 =62/165