Understanding Engineering Mathematics

this way is thepercentage. Thus we usually express 32/100 as ‘32 percent’ rather than as
its equivalent, ‘8 out of 25’!
Fractions are multiplied ‘top by top and bottom by bottom’ as you might expect:

m

n

×

p

q

=

mp

nq

(n,q= 0 ) e.g.

3

2

×

5

11

=

15

22

withpandqalso any integers. There may, of course, be common factors to cancel down,
as for example in

3

2

×

6

7

=

9

7

The inverse orreciprocalof a fraction is obtained by turning it upside down:

1

/(m

n

)

=

n

m

e.g. 1

/( 3

2

)

=

2

3

where bothmandnmust be non-zero. So dividing by a vulgar fraction is done by inverting
it and multiplying:

(

p

q

)

÷

(m

n

)

=

(

p

q

)/(

m

n

)

=

p

q

×

n

m

=

np

mq

e.g.

7

2

÷

14

6

=

7

2

×

6

14

=

3

2

Multiplication and division of fractions are therefore quite simple. Addition and subtrac-
tion are less so.
Two fractions with the same denominator are easily added or subtracted:

m

n

±

p

n

=

m±p

n

e.g.

3

2

−

1

2

=

3 − 1

2

=

2

2

= 1

So to add and subtract fractions in general we rewrite them all with the samecommon
denominator, which is thelowest common multipleof all the denominators. For example

3

4

−

4

3

=

3 × 3

12

−

4 × 4

12

=

9 − 16

12

=−

7

12

In the example, 12 is the LCM of 3 and 4.

An electrical example – resistances in parallel

Three resistancesR 1 ,R 2 ,R 3 connected in parallel are equivalent to a single resistanceR
given by

1

R

=

1

R 1

+

1

R 2

+

1

R 3

So, for example ifR 1 = 2 ,R 2 =^12 ,R 3 =^32 then

1

R

=

1

2

+ 2 +

2

3

(units of inverse ohms)