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References

**Types of fractions**

**Proper fraction**

A fraction whose numerator is less than the denominator is called the proper fraction. A proper fraction is a fraction that represents a part of a whole.

**Improper fraction**

A fraction whose numerator is greater than the denominator is called an improper fraction. An improper fraction is a combination of a whole and a proper fraction.

**Mixed fraction**

An improper fraction can be expressed as a mixed fraction. The numerator of the improper fraction is divided by the denominator to obtain the quotient and the remainder. The mixed fraction is written as $\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}$.

**Multiplication of fractions**

To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction.

Ex: 2 Ã— $\frac{\text{2}}{\text{3}}$ = $\frac{\text{4}}{\text{3}}$.

While multiplying a whole number by a mixed fraction, change the mixed fraction into an improper fraction and multiply the whole number by the numerator of the fraction.

Ex: 3 Ã— 2 $\frac{\text{3}}{\text{4}}$ = 3 Ã— $\frac{\text{11}}{\text{4}}$ = $\frac{\text{33}}{\text{4}}$.

To multiply two fractions, multiply their numerators and denominators.

Ex: $\frac{\text{3}}{\text{4}}$ Ã— $\frac{\text{2}}{\text{5}}$ = $\frac{\text{3 \xc3\u2014 2}}{\text{4 \xc3\u2014 5}}$ = $\frac{\text{6}}{\text{20}}$.

When two proper fractions are multiplied, the product is less than each of the individual fractions.

When two improper fractions are multiplied, the product is greater than each of the individual fractions.

**Fraction as an operator â€˜of â€™**

Fraction acts as an operator of. 'of' represents multiplication.

Ex: $\frac{\text{1}}{\text{3}}$ of 90 = $\frac{\text{1}}{\text{3}}$ Ã— 90 = 30.

**Reciprocal of a fraction**

To obtain the reciprocal of a fraction, interchange the numerator with the denominator.

Ex: The reciprocal of $\frac{\text{3}}{\text{7}}$ is $\frac{\text{7}}{\text{3}}$.

**Division of fractions**

To divide a whole number by a fraction, find the reciprocal of the fraction and then multiply it by the whole number.

Ex: 2 Ã· $\frac{\text{3}}{\text{4}}$ = 2 Ã— $\frac{\text{4}}{\text{3}}$ = $\frac{\text{8}}{\text{3}}$.

To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number.

Ex: $\frac{\text{2}}{\text{7}}$ Ã· 6 = $\frac{\text{2}}{\text{7}}$ Ã— $\frac{\text{1}}{\text{6}}$ = $\frac{\text{1}}{\text{21}}$.

To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.

Ex: $\frac{\text{1}}{\text{2}}$ Ã· $\frac{\text{3}}{\text{4}}$ = $\frac{\text{1}}{\text{2}}$ Ã— $\frac{\text{4}}{\text{3}}$ = $\frac{\text{4}}{\text{6}}$.

To divide a whole number by a mixed fraction, convert the mixed fraction into an improper fraction and multiply the whole number by the reciprocal of the improper fraction.

Ex: 5 Ã· 6$\frac{\text{1}}{\text{4}}$ = 5 Ã· $\frac{\text{25}}{\text{4}}$ = 5 Ã— $\frac{\text{4}}{\text{25}}$ = $\frac{\text{4}}{\text{5}}$.

**Types of fractions**

**Proper fraction**

A fraction whose numerator is less than the denominator is called the proper fraction. A proper fraction is a fraction that represents a part of a whole.

**Improper fraction**

A fraction whose numerator is greater than the denominator is called an improper fraction. An improper fraction is a combination of a whole and a proper fraction.

**Mixed fraction**

An improper fraction can be expressed as a mixed fraction. The numerator of the improper fraction is divided by the denominator to obtain the quotient and the remainder. The mixed fraction is written as $\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}$.

**Multiplication of fractions**

To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction.

Ex: 2 Ã— $\frac{\text{2}}{\text{3}}$ = $\frac{\text{4}}{\text{3}}$.

While multiplying a whole number by a mixed fraction, change the mixed fraction into an improper fraction and multiply the whole number by the numerator of the fraction.

Ex: 3 Ã— 2 $\frac{\text{3}}{\text{4}}$ = 3 Ã— $\frac{\text{11}}{\text{4}}$ = $\frac{\text{33}}{\text{4}}$.

To multiply two fractions, multiply their numerators and denominators.

Ex: $\frac{\text{3}}{\text{4}}$ Ã— $\frac{\text{2}}{\text{5}}$ = $\frac{\text{3 \xc3\u2014 2}}{\text{4 \xc3\u2014 5}}$ = $\frac{\text{6}}{\text{20}}$.

When two proper fractions are multiplied, the product is less than each of the individual fractions.

When two improper fractions are multiplied, the product is greater than each of the individual fractions.

**Fraction as an operator â€˜of â€™**

Fraction acts as an operator of. 'of' represents multiplication.

Ex: $\frac{\text{1}}{\text{3}}$ of 90 = $\frac{\text{1}}{\text{3}}$ Ã— 90 = 30.

**Reciprocal of a fraction**

To obtain the reciprocal of a fraction, interchange the numerator with the denominator.

Ex: The reciprocal of $\frac{\text{3}}{\text{7}}$ is $\frac{\text{7}}{\text{3}}$.

**Division of fractions**

To divide a whole number by a fraction, find the reciprocal of the fraction and then multiply it by the whole number.

Ex: 2 Ã· $\frac{\text{3}}{\text{4}}$ = 2 Ã— $\frac{\text{4}}{\text{3}}$ = $\frac{\text{8}}{\text{3}}$.

To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number.

Ex: $\frac{\text{2}}{\text{7}}$ Ã· 6 = $\frac{\text{2}}{\text{7}}$ Ã— $\frac{\text{1}}{\text{6}}$ = $\frac{\text{1}}{\text{21}}$.

To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction.

Ex: $\frac{\text{1}}{\text{2}}$ Ã· $\frac{\text{3}}{\text{4}}$ = $\frac{\text{1}}{\text{2}}$ Ã— $\frac{\text{4}}{\text{3}}$ = $\frac{\text{4}}{\text{6}}$.

To divide a whole number by a mixed fraction, convert the mixed fraction into an improper fraction and multiply the whole number by the reciprocal of the improper fraction.

Ex: 5 Ã· 6$\frac{\text{1}}{\text{4}}$ = 5 Ã· $\frac{\text{25}}{\text{4}}$ = 5 Ã— $\frac{\text{4}}{\text{25}}$ = $\frac{\text{4}}{\text{5}}$.