3.1 Arithmetic
- Fractions
In a fraction !!:... , n is the numerator and dis the denominator. The denominator of a fraction can never be
d
0, because division by 0 is not defined.
Two fractions are said to be equivalent if they represent the same number. For example, JL and^14 are
36 63
equivalent since they both represent the number l. In each case, the fraction is reduced to lowest terms
9
by dividing both numerator and denominator by their greatest common divisor (gcd). The gcd of 8 and 36
is 4 and the gcd of 14 and 63 is 7.
Addition and subtraction of fractions.
Two fractions with the same denominator can be added or subtracted by performing the required
operat10n. wit. h h t e numerators, 1. eavmg t e h d enommators. t e h same. F or examp^1 e, 3 4 3+4^7
5
+
5
= -
5
- =
5
and 2 l =^5 -^2 = l. If two fractions do not have the same denominator, express them as equivalent
7 7 7 7 3 4
fractions with the same denominator. For example, to add
5
and
7
, multiply the numerator and
denominator of the first fraction by 7 and the numerator and denominator of the second fraction by 5,
o b tammg..^21 d^20. l.^21 +^20 _^41
35
an
35
, respective y,
35 35
For the new denominator, choosing the least common multiple (1cm) of the denominators usually lessens
the work. For l + 1, the 1cm of 3 and 6 is 6 (not3 x 6 = 18), so l + 1 = l x l + 1 = i + 1 = 2-.
36 36326666
Multiplication and division of fractions.
To multiply two fractions, simply multiply the two numerators and multiply the two denominators.
For example -x-=--=-. 2 4 2x4^8
' 3 7 3x7 21
To divide by a fraction, invert the divisor (that is, find its reciprocal) and multiply. For example,
l+i= lxZ= 14 = Z.
3734126
In the problem above, the reciprocal of i is Z. In general, the reciprocal of a fraction !!:... is !!...., where n
and dare not zero.^7 4 d n
Mixed numbers.
A number that consists of a whole number and a fraction, for example, 7 l, is a mixed number: 7 l
2
3 3
means 7 +-.
3
To change a mixed number into a fraction, multiply the whole number by the denominator of the
fraction and add this number to the numerator of the fraction; then put the result over the denominator
of the fraction. For example, 7 -^2 = ---'--------'----(3x7)+2 = -.^23
3 3 3
