Arithmetic : Fractions

Proper Fractions, Improper Fractions and Mixed Numbers

A proper fraction is a fraction with a numerator smaller than the denominator, and therefore a value less than 1.
For example, 2/5 and 17/19 are proper fractions.

An improper fraction is a fraction with a numerator greater than or equal to the denominator, and therefore a value greater than or equal to 1.
For example, 5/2 and 19/17 are improper fractions.

A mixed number comprises a natural number and a properfraction, and therefore always has a value greater than 1.
For example, 5¾ (read as 'five and three-fourths') is a mixed number whose fractional part is ¾. Note that 5¾ = 5 + ¾.

MUST-KNOW : When doing calculations with mixed numbers, change the mixed numbers to improper fractions.

To convert a mixed number to an improper fraction, first multiply the natural number by the denominator of the fractional part, then add it to the numerator of the fractional part, and finally use this sum as the numerator of the required fraction. The denominator of the required fraction is the same as the denominator of the fractional part of the mixed number.

After fractions are added, subtracted, multiplied or divided, the result is often expressed as a mixed number if it is an improper fraction.
To convert an improper fraction into a mixed number, first divide the numerator by the denominator to obtain the natural number of the mixed number, and then place the remainder (as numerator) over the denominator to get the fractional part of the mixed number.

Example
If each pizza comprises 4 pieces, then how many pizzas have been consumed at a party where 27 pieces have been eaten?
Solution. 4 pieces make one pizza. So, 27 pieces make 27/4 pizzas.
To convert 27/4 to a mixed number, first divide 27 by 4. The quotient is 6 and the remainder is 3. So, 27/4 = 6¾.
Thus, 6¾ pizzas have been consumed.

Note that any whole number n can be represented as a fraction with a numerator equal to n times the denominator.
For example, 20/5 = 4 and 21/7 = 3.