# 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 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**