## Introduction :

Fractions are written in three different forms. They are proper fractions, improper fractions and mixed fractions.Proper fractions are those in which the numerator is smaller then the denominator.

Improper fractions are those in which the numerator is greater than the denominator.

Mixed fractions are those in which are written in the form Quotient $\frac{Remainder}{Divisor}$.

Then the mixed for is written as Quotient $\frac{Remainder}{Divisor}$.

### Example :$ \frac{13}{3}$

#### Solution:

$\frac{13}{3}$When we divide 13 by 3, we see that 4 x 3 = 12

(i.e) there are 4 threes in 13 and we get 13 - 12=1 as the remainder.

Hence the mixed fraction is $4\frac{1}{3}$

While adding improper fractions we can follow the following steps.

**Step 1**. First we need to take LCM of the denominators .

**Step 2.**Multiply the numerator and the denominator of each fraction by the appropriate numbers so that the product of the numbers in the denominator of each fraction is the LCM we obtained from step 1.

### Example: Add : $\frac{11}{5}+\frac{17}{2}+\frac{23}{10}$

#### Solution:

$\frac{11}{5}+\frac{17}{2}+\frac{23}{10}$ = $\frac{11\times 2}{5\times 2}+\frac{17\times 5}{2\times 5}+\frac{23}{10}$[ LCM of 2, 5 and 10 is 10, we multiply the numerator and the denominator by

the appropriate numbers so that the denominator is 10 ]

= $\frac{22}{10}+\frac{85}{10}+\frac{23}{10}$ [ writing each fraction with denominator as 10 ]

= $\frac{22 + 85 + 23}{10}$ [ combining numerators as per the sign ]

= $\frac{130}{10}$

= 13 [ simplest form by dividing the numerator and the denominator by the common factors ]

While subtracting improper fractions, we follow the same procedure as we do for addition.

**Step 1**. First we need to take LCM of the denominators .

**Step 2**. Multiply the numerator and the denominator of each fraction by the appropriate numbers so that the product of the numbers in the denominator of each fraction is the LCM we obtained from step 1.

### Example : Subtract $\frac{14}{3}-\frac{12}{5}$

**Solution:**

$\frac{14}{3}-\frac{12}{5}$ = $\frac{14\times 5}{3\times 5}-\frac{12\times 3}{5\times 3}$

[ LCM of 3 and 5 is 15, we multiply the numerator and the denominator

by the appropriate numbers so that the denominator is 15 ]

=$ \frac{70}{15}- \frac{36}{15}$ [ writing each fraction with denominator as 15 ]

=$\frac{70-36}{15}$ [ combining the numerators as per the sign ]

=$\frac{34}{15}$ , Final Answer in simplest form.

While multiplying the improper fractions we need to follow the following steps.

**Step 1**. Group the numerators and the denominators with the multiplication signs.

**Step 2**. Split the numerator and denominators into product of prime factors.

**Step 3**. Cancel the common factors in the numerator and denominator.

**Step 4**. Combine the products of the numerator and the denominator.

### Example: $\frac{60}{14}\times \frac{21}{15}$

#### Solution:

$\frac{60}{14}\times \frac{21}{15}$ = $\frac{60\times21}{14\times 15}$ [ grouping the numerators and the denominators ]=$\frac{2\times 2\times 3\times 5\times 7\times3\times 2}{2\times 7\times 3\times 5}$

[ writing each number as the product of prime factors ]

=$\frac{2\times 3}{1}$ [ dividing the numerator and the denominator by the common factors ]

=6 Final answer in simplest form

While dividing improper fractions we follow the following steps:

**Step 1**. Multiply the first fraction by the reciprocal of the second fraction.

**Step 2**. Group the numerators and the denominators with the multiplication signs.

**Step 3**. Split the numerator and denominators into product of prime factors.

**Step 4**. Cancel the common factors in the numerator and denominator.

**Step 5**. Combine the products of the numerator and the denominator.

### Example :$ \frac{39}{15}\div \frac{52}{35}$

#### Solution:

$\frac{39}{15}\div \frac{52}{35}$ = $\frac{39}{15}\times \frac{35}{52}$[ multiplying the first fraction by the reciprocal of the second fraction ]

= $\frac{3\times 13\times 5\times 7}{3\times 5\times 2\times 2\times 13}$

[ writing each number as the product of prime factors ]

= $\frac{7}{2\times 2}$

= $\frac{7}{4}$

= 1 $\frac{3}{4}$

1. Add: $\frac{12}{7}+\frac{11}{21}$

2. Subtract $\frac{23}{10} - \frac{21}{15}$

3. Multiply: $\frac{34}{45}\times \frac{125}{51}$

4. Subtract: $\frac{220}{75}\div \frac{132}{55}$

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