## Chapter 1 Simplification Decimal Fraction

**Decimal Fraction:**

- You have already studied integers and vulgar fractions in detail.
- In the present article, we shall discuss decimal fractions in detail.

** What is Decimal Fraction :**

- When we express a proper fraction or an improper fraction or a mixed fraction by a decimal point (.), then the fraction is called a
**Decimal Fraction.** - For example, \(\frac{1}{2}\) is a proper fraction; when we express it by a decimal fraction then we write it as 0.5 i.e., \(\frac{1}{2}\) = 0.5.
- Again, \(\frac{13}{4}\) is an improper fraction; it is expressed in decimal fraction as 3.25 i.e.,
- \(\frac{13}{4}\) = 3.25
- In a similar way a mixed fraction 4 \(\frac{1}{2}\) can be expressed in a decimal fraction as 4.5.

**Read And Learn More: WBBSE Notes For Class 6 Maths Chapter 1 Simplification**

**Wbbse Class 6 Maths Solutions**

** Examples of Decimal Fractions :**

0.32, 1.57, 11.004, 102.59, etc. are examples of Decimal Fractions.

## Role of Decimal Point:

- The role of a decimal point in any number is to make out a clear concept about the integral part and the fractional part of the number.
- The left-hand part of the decimal point is called the integral part and the right-hand part of the decimal point including the decimal point is called the fractional part.
- For example, 2.04 is a decimal number and it contains 2 on the left side of the decimal point.
- So the integral part of the number is 2 and the number contains 04 after the decimal point in the right side of the decimal point including the decimal point and so the fractional part is .04.
- If a decimal-number be such that there is no significant digit in the number in the left side of the decimal point then the integral part of the number is taken as 0.

For example, the integral part of the number 0.0025 is 0 and the integral part of the number 0.9015 is zero.

**Wbbse Class 6 Maths Solutions**

** How to write a decimal fraction?**

**Decimal fraction:**

- The last digit i.c., the digit in the extreme right side place of an integral part of a number is called the unit’s place digit.

After this unit’s place digit (the just right side of the unit’s place digit) there is a point (.) written which is called the decimal point. - After writing this decimal point, the digits of the fractional part of the number, then only the complete decimal fraction are written.
- For example, if for a number, the integral part is 245 and the fractional part is 356 then the decimal fraction is 245.356.

** Why the name of the point (.) is the decimal point?**

- The point (.) is used for multiplication or division by 10 only, that’s why the point () is called the decimal point.
- Suppose a decimal point exists in a number.
- If the decimal point is shifted one place towards the right, then the value of the number increases 10 times, which means that the new number becomes 10 times the previous number, or in other words, the previous number is multiplied by 10.
- Again if the decimal point is shifted one place towards the left, then the value of the new number is obtained by dividing the previous number by 10.
- Suppose the given number is 2456.1251.
- If we write 24561.251, this means that 2456.1251 is multiplied by 10.
- If we write 245612.51, this means that 2456.1251 is multiplied by 100.
- If we write 2456125.1, this means that 2456.1251 is multiplied by 1000.

**In a reverse way:**

- If we write 245.61251, this means that 2456.1251 is divided by 10.
- If we write 24.561251, this means that 2456.1251 is divided by 100.
- If we write 2.4561251, this means that 2456.1251 is divided by 1000.

## The Usefulness Of The Decimal System

**The utility of the Decimal system is:**

- Large multiplication or division by 10 or by its multiplier is very easy in the decimal system, unlike multiplication or division by other numbers.

In the decimal system only shifting of decimal point towards right or left can be done. - The units of length, mass and time, etc. can be expressed easily.

**Wbbse Class 6 Maths Solutions**

**The Face-value and Place-value of the digits in Decimal Fractions:**

The face value of a digit in any number is its own value while is the same everywhere.

In other words, the face value of a digit in any number is its absolute value.

For example, in the number 246, the face value of 2 is 2; the face value of 4 is 4 and the face value of 6 is 6.

Again in the number 24.567, the face value of 7 is 7; the face value of 2 is 2, etc.

But the place value of any digit in a number is the product of the digit and the place value of that place where the digit is placed.

**For example**, the place value of 5 in the number 258 is 5 x 10 = 50, because, 5 is placed in the ten’s place.

In order to determine the place value of any digit in any number, we follow the following rule which you have also learned already

For decimal fractions, the place value of any digit after the decimal point can be determined according to the following rule:

**Wbbse Class 6 Maths Solutions**

In the decimal fraction 84.7325.

The place value of 7=7 x \(\frac{1}{10}\)

= \(\frac{7}{10}\)

= 0.7

The place value of 3 = 3 x \(\frac{1}{100}\)

\(\frac{3}{100}\)

= 0.002

The place value of 2 = 2 × \(\frac{1}{1000}\)

= \(\frac{2}{1000}\)

The place value of 5 = 5 x \(\frac{1}{10000}\)

= \(\frac{5}{10000}\)

= 0.0005

** How to read Decimal Fraction:**

- Let 6.25 be a given decimal fraction. How will we read it? We see that the integer before the decimal point is 6 and the number after the decimal point is 25.
- The given decimal fraction can be read as “six decimal two five” or “six decimal two one-tenths five one-hundredths.
- Now let the decimal fraction 0-002 be given. This decimal fraction can be read as “zero decimal zero two” “zero decimal 2 one-thousandth” or simply “two one-thousandths”.
- Similarly, 1247-253 = 1 thousand 2 hundred 4 tens 7 units decimal 2 one-tenths 5 one-hundredths 3 one-thousandths.
- 42.538 4 tens 2 units decimal 5 one-tenths 3 one-hundredths 8 one-thousandths. 0.237 2 one-tenths 3 one-hundredths 7 one-thousandths.

## Conversion Of Decimal Fraction Into Vulgar Fraction

- In order to convert a decimal fraction into a vulgar fraction, omit the decimal point.
- Take the number thus obtained as the numerator of the required vulgar fraction.
- The denominator will be the number obtained by putting as many zeroes as there is a number of digits after the decimal point of the given decimal fraction towards the right of 1.
- Then reduce this fraction to the lowest term.
- If the fraction thus obtained is an improper fraction, then convert it into a mixed fraction.

**Example 1. Convert 2.175 into Vulgar fraction.**

Class 6 Wb Board Math Solution :

**Given :**

** 2.175 **

** **2.175 = \(\frac{2175}{1000}\)

= \(\frac{87}{40}\)

= \(2 \frac{7}{40}\)

Omitting the decimal point from the given decimal fraction 2:175, we get the number 2175. This is taken as the numerator.

There are 3 digits after the decimal point in the given decimal fraction. So the denominator will be 1000 which is obtained by putting 3 zeroes to the right side of 1.

∴ The vulgar fraction = \(\frac{2175}{1000}\)

Reducing it to the lowest term, we get, \(\frac{87}{40}\)

∴ But this is an improper fraction.

Converting it into a mixed fraction, we get 2 \(\frac{7}{40}\)

The required vulgar fraction = 2 \(\frac{7}{40}\)

**Example 2. Convert 0.06235 into Vulgar fraction.**

**Solution:**

**Given: 0.06235**

** **0.06235 = \(\frac{6235}{100000}\)

= \(\frac{1247}{20000}\)

**Example 3. Convert the following decimal fractions into Vulgar fractions: 2.39; 0.0255; 1.3608; 0.045045.**

Class 6 Wb Board Math Solution :

**Given: 2.39; 0.0255; 1.3608; 0.045045**

2.39 = \(\frac{239}{100}\) = 2 \(\frac{39}{100}\)

0.0255 = \(\frac{255}{10000}\)

= \(\frac{51}{2000}\)

1.3068 = \(\frac{13608}{10000}\)

= \(\frac{1701}{1250}\)

= 1 \(\frac{451}{1250}\)

0.045045 = \(\frac{45045}{1000000}\)

= \(\frac{9009}{20000}\)

## Conversion of Vulgar Fraction Into Decimal Fraction

In order to convert a Vulgar fraction into a decimal fraction, divide the numerator of the Vulgar fraction by its denominator.

**Example 1. Express as a decimal fraction.**

**Solution :**

So the required decimal fraction is 0.4016.

**Example 2. Express 3**** \(\frac{7}{50}\) ****as a decimal fraction.**

Class 6 Wb Board Math Solution:

**3**** \(\frac{7}{50}\)**

3 \(\frac{7}{50}\)

= \(\frac{157}{50}\)

= 3.14

**Example 3. Express \(\frac{9}{10}, \frac{73}{100}, \frac{31}{1000}, 2 \frac{3}{1000}\) as decimal fractions.**

**Solution:**

** \(\frac{9}{10}, \frac{73}{100}, \frac{31}{1000}, 2 \frac{3}{1000}\) **

\(\frac{9}{10}\) = 9 ÷ 10

= 0.9

\(\frac{73}{100}\) = 73 ÷ 100

= 0.73

\(\frac{31}{1000}\) = 31 ÷ 1000

= 0.031

2 \(\frac{3}{1000}\) = \(\frac{2003}{1000}\)

= 2003 ÷ 1000

= 2.003

**Example 4. Convert 13 \(\frac{17}{75}\) into decimal fractions up to 3 places of decimal.**

**Solution:**

**13 \(\frac{17}{75}\) **

The given mixed fraction can be converted into decimal fractions up to 3 places of decimal as worked out in example 2.

The given mixed vulgar fraction can also be converted into decimal fractions up to 3 places of decimal alternatively as follows:

The integer contained in the given mixed fraction is 13, this integer will also remain in the decimal fraction.

Therefore, at first, the fraction is to be converted \(\frac{17}{75}\) into a decimal fraction and then put integer 13 to the left of the decimal point.

∴ \(\frac{17}{75}\) = 226 (upto 3 places of decimal).

∴ The required decimal fraction = 13.226 (up to 3 places of the decimal).

## Addition And Subtraction Of Decimal Fractions

The decimal fractions which are to be added or subtracted are to be written one below the other in such a way that their decimal points must be one below the other.

The digits in the units, tens, thousands, tenths, hundredths, thousandths’ place, etc. of one decimal fraction should be written below the digits in the respective places of the other decimal fractions.

Then using the usual procedure of addition and subtraction of integers, the addition and subtraction of the given decimal fractions are done, and put the decimal point in the result is just below the decimal column.

If there are one, two, or three digits after the decimal point in a decimal fraction, that is, if there are tenths, hundredths, or thousandths’ place digits in a decimal fraction after the decimal point, one can place the zeroes according to the requirements after the last digit and then the addition and subtraction can be done.

**Example 1. Add: 289.7, 25.379, 93.25, 7.5278**

Class 6 Wb Board Math Solution :

**Given: 289.7, 25.379, 93.25, 7.5278**

∴ The required sum = 415.8568.

**Example 2. Subtract 87.5923 from 205.31.**

**Solution:**

**Given: 87.5923 And 205.31**

∴ The required result = 117.7177