Decimal fraction is related to addition, subtraction, multiplication, division, comparison, recurring of decimal fractions.

**Examples**:

**1.** \(\frac{1}{10} = 1\) tenth = .1

**2.** \(\frac{1}{100} = 1\) hundredth = .01

**3.** \(\frac{99}{100} = 99\) hundredths = .99

**4.** \(\frac{7}{1000} = 7\) thousandths = .007

**Addition & subtraction**:

**Addition** and **subtraction** is done by placing the given numbers under each other that the decimal points lie in one column.

**Example 1**:

Evaluate: 12.1212 + 101.32 – 306.76 = ? (**B.S.R.B, 2003**)

**Solution**:

- Given Expression = (12.1212 + 17.0005) – 9.1102 = (29.1217 – 9.1102) = 20.0115.

**Example 2**:

Evaluate: 48.95 – 32.006 (**I.B.P.S. 2002**)

**Solution**:

- 48.95 – 32.006 = 16.944

**Example 3**:

Evaluate: 792.02 + 101.32 – 306.76 (**NABARD, 2002**)

**Solution**:

- 792.02 + 101.32 = 893.34

893.34 – 306.76 = 586.58

**Multiplication**

**Multiplication** is done by two methods.

- One is by a power of 10 i.e. we need to shift the decimal to the right as many places as in the power of 10.
- Other is by multiplying the given numbers without decimal points and after obtaining the answer put the decimal as sum of the number of decimal places in the given numbers.

**Example 1**:Â

Evaluate: 0.002 Ã— 0.5 = ? (**Bank P.O 2003**)

**Solution**:

- 2 x 5 = 10. Sum of decimal places = 4.

∴ 0.002 x 0.5 = 0.0010 = 0.001.

**Example 2**:Â

Evaluate: 16.02 x 0.001 = ? (**Bank P.O 2002**)

**Solution**:

- 1602 x 1 = 1602. Sum of decimals places = 5.

∴ 16.02 x 0.001 = 0.000196.

**Example 3**:

Evaluate: 3.14 x \(10^{6}\)

**Solution**:

- 3.14 x \(10^{6}\) = 3.140000 x 1000000 = 3140000.

**Division**

**Division** is done by two methods.

- One is by a decimal fraction i.e. we need to multiply both dividend and divisor by power of 10 to make divisor a whole number.
- Other is by dividing the given numbers without decimal points and after obtaining the answer put the decimal as many places of decimal as there are in dividend.

**Example 1**:

Evaluate: \(2.5 \div 0.0005\)

**Solution**:

- \(\frac{2.5}{0.0005}\) = \(\frac{25 \times 10000}{0.0005 \times 10000}\) = \(\frac{25000}{5}\) = 5000.

**Example 2**:

Evaluate: \(0.006 \div ?\) = 0.6

**Solution**:

- Let \(\frac{0.006}{x}\) = 0.6. Then, \(x\) = \(\frac{0.006}{0.6}\) = \(\frac{0.006 \times 10}{0.6 \times 10}\) = \(\frac{0.06}{0.6}\) = 0.01.

**Example 3**:

Evaluate: \(? \div .025\) = 80

**Solution**:

- Let \(\frac{x}{.025}\) = 80. Then, \(x\) = 80 x .025 = 2.

**Recurring**:

**Recurring** means repeating a figure or a set of figures continuously in a decimal fraction.

**Examples**:

**1**. \(\frac{5}{6}\) = 0.83333….. = 0.8\(\overline{3}\).

**2**. Express as vulgar fractions: 0.\(\overline{37}\) = \(\frac{37}{99}\).

**3**. Express as vulgar fractions: 0.\(\overline{053}\) = \(\frac{53}{999}\).

Convert 0.75 to a fraction.

\(\frac{0.75}{1}\)

\(\frac{0.75 \times 100}{1 \times 100}\) = \(\frac{75}{100}\)

\(\frac{75}{100}\) = \(\frac{15}{20}\) = \(\frac{3}{4}\) (Divide by 5)

Here \(\frac{75}{100}\) is called a decimal fraction and \(\frac{3}{4}\) is called a common fraction.

**Example 2**

Convert 0.625 to a fraction

**Solution**:

\(\frac{0.625}{1}\)

\(\frac{0.625 \times 1000}{1 \times 1000}\) = \(\frac{625}{1000}\)

\(\frac{625}{1000}\) (Divide by 25) = \(\frac{25}{40}\) (Divide by 5) = \(\frac{5}{8}\)

**Example 3**

Convert 2.35 to a fraction

**Solution**:

- When there is a whole number part, put the whole number aside and bring it back at the end.

So here, Put the 2 aside and just work on 0.35

\(\frac{0.35}{1}\)

\(\frac{35 \times 100}{1 \times 100}\) = \(\frac{35}{100}\)

\(\frac{35}{100}\) = \(\frac{7}{20}\) (Divide by 5)

Bring back 2 to make a mixed fraction : 2 \(\frac{7}{20}\)

**Comparison** is the arrangement of given fractions in ascending order and descending order. so first we need to convert the given fractions into decimal numbers and then it is easy to arrange them accordingly.

**Example 1**

Arrange the fractions \(\frac{1}{2}; \frac{3}{4}; \frac{5}{6}\) in ascending order?

**Solution**:

- by converting, 0.5; 0.75; 0.833

Ascending order is 0.833 > 0.75 > 0.5

Therefore, \(\frac{5}{6} >\frac{3}{4} >\frac{1}{2}\).

**Example 2**

Arrange the fractions \(\frac{3}{5}; \frac{4}{7}; \frac{8}{9} and \frac{9}{11}\) in their descending order. (**R,.B.I 2003**)

**Solution**:

- Clearly, \(\frac{3}{5}\) = 0.5, \(\frac{4}{7}\) = 0.571, \(\frac{8}{9}\) = 0.88, \(\frac{8}{11}\) = 0.818.

Now, 0.88 > 0.818 > 0.6 > 0.571.

∴ \(\frac{8}{9}\), \(\frac{9}{11}\), \(\frac{3}{5}\), \(\frac{4}{7}\)

.

**Example 3**

Arrange the fractions \(\frac{5}{8}; \frac{7}{12}; \frac{13}{16}, \frac{16}{29} and \frac{3}{4}\) in Ascending order of magnitude.

**Solution**:

- Converting each of the given fractions into decimal form, we get:

\(\frac{5}{8}\) = 0.625, \(\frac{7}{12}\) = 0.5833, \(\frac{13}{16}\) = 0.8125, \(\frac{16}{29}\) = 0.5517 and \(\frac{3}{4}\) = 0.75.

Now, 0.5517 < 0.5833 < 0.625 < 0.75 < 0.8125.

∴ \(\frac{16}{29}\) < \(\frac{7}{12}\) < \(\frac{5}{8}\) < \(\frac{3}{4}\) < \(\frac{13}{16}\).

- Line with a point = a+b

- Square Area = \(a^{2}\)

- Rectangle Area = ab

- Draw a line with a point which divides a, b

- Total distance of this line = a+b
- Now find out the square of a+b ie. \((a+b)^{2}\)

- The above diagram represents â€“ a + b is a line and the square are is

\(a^{2}\) + \(b^{2}\) + 2ab

**Prove \((a+b)^2 = a^2 + b^2 +2ab\) in geomentry**:

Hence proved \((a+b)^{2}\) = \(a^{2}\) + \(b^{2}\) + 2ab

**Prove \((a+b)^2 = a^2 + b^2 +2ab\) in Algebra**:

\((a+b)^{2}\) = (a+b)*(a+b)

= (a*a + a*b) + (b*a + b*b)

= (\(a^{2}\) + ab) + (ba + \(b^{2}\))

= \(a^{2}\) + 2ab + \(b^{2}\)

Hence proved \((a+b)^{2}\) = \(a^{2}\) + \(b^{2}\) + 2ab

**Example 1**:

Expand the term \((3x + 4y)^{2}\) using the identity \((a+b)^{2}\) = \(a^{2}\) + \(b^{2}\) + 2ab

**Solution**:

- \((3x + 4y)^{2}\) = \((3x)^{2}\) + \((4y)^{2}\) + 2(3x)(4y) = \(9x^{2}\) + \(16y^{2}\) + 24xy

∴ \((3x + 4y)^{2}\) = \(9x^{2}\) + \(16y^{2}\) + 24xy

**Example 2**:

Expand the term \((\sqrt{2}x + 4y)^{2}\) using the identity \((a+b)^{2}\) = \(a^{2}\) + \(b^{2}\) + 2ab

**Solution**:

- \((\sqrt{2}x + 4y)^{2}\) = \((\sqrt{2}x)^{2}\) + \((4y)^{2}\) + 2\((\sqrt{2}x)\)(4y) = \(2x^{2}\) + \(16y^{2}\) + 8\(\sqrt{2}\)xy

∴ \((\sqrt{2}x + 4y)^{2}\) = \(2x^{2}\) + \(16y^{2}\) + 8\(\sqrt{2}\)xy

**Example 3**:

Expand the term \((x + \frac{1}{x})^{2}\) using the identity \((a+b)^{2}\) = \(a^{2}\) + \(b^{2}\) + 2ab

**Solution**:

- \((x + \frac{1}{x})^{2}\) = \(x^{2}\) + \(\frac{1}{x^{2}}\) + 2\((x)(\frac{1}{x})\) = \(x^{2}\) + \(\frac{1}{x^{2}}\) + 2

∴ \((x + \frac{1}{x})^{2}\) = \(x^{2}\) + \(\frac{1}{x^{2}}\) + 2

- Given that \(\frac{52416}{312} = 168\)

or it can also be written as \(\frac{52416}{312}\)=312

given thatÂ Â \(\frac{52.416}{0.0168}\)

â‡’\(\frac{524160}{168}\)

â‡’ \(\frac{52416}{168}\) Ã— 10

â‡’ 312 Ã— 10 = 3120

Therefore the quotient = 3120

**2. Find the unknown number from 654.485 + 16.42 + ? = 936.5489**

**Solution**:

- Take the unknown number as \(“x”\)

by placing \(“x”\) in the given equation,

â‡’654.485 + 16.42 + \( x \) = 936.5489

â‡’670.905 + \( x \) = 936.5489

â‡’\( x \) = 936.5489 – 670.905

Therefore \( x \) = 265.6439

**3. Calculate \(\frac{125.36}{25.6}\)?**

**Solution**:

- Given that \(\frac{125.36}{25.6}\)

multiply both dividend and divisor by power of 10 i.e.

â‡’\(\frac{12536 Ã— 10}{256 Ã— 10}\)

â‡’\(\frac{125360}{2560} \) = 48.96

Therefore, \(\frac{125.36}{25.6}\) = 48.96

**4. Find the descending order of \(\frac{15}{3}; \frac{6}{10}; \frac{2}{7}; \frac{4}{11}\)?**

**Solution**:

- Given that \(\frac{15}{3}; \frac{6}{10}; \frac{2}{7}; \frac{4}{11}\)

Now,Â fractions areÂ converted into decimals i.e.

\(\frac{15}{3}\) = 5

\(\frac{6}{10}\) = 0.6

\(\frac{2}{7}\) = 0.28

\(\frac{4}{11}\) = 0.36

0.28 < 0.36 < 0.6 < 5

Therefore, descending order is \(\frac{2}{7}; \frac{4}{11}; \frac{6}{10}; \frac{15}{3}\)

**5. Convert theÂ decimals 0.625, 0.8125, 0.5833, 0.75 into simple form of fractions?**

**Solution**:

- Given decimals are 0.625, 0.8125, 0.5833, 0.75

0.625 = \(\frac{625}{1000} = \frac{5}{8}\)

0.8125 = \(\frac{8125}{10000} = \frac{13}{16}\)

0.5833 = \(\frac{5833}{10000} = \frac{7}{12}\)

0.75 = \(\frac{75}{100} = \frac{3}{4}\)