TRANSPOSITION OF FORMULAE
General rules are that whatever is done to one side of the equal sign must be done to
the other side. Note that it does not matter which side of the equals sign, the subject
letter is.
ADDITION/SUBTRACTION
Ex 1: Consider the equation:
a+b = c+d
To make “a” the subject, subtract “b” from the both sides.
a +b−b = c+ d −b
∴a = c + d − b
Ex 2: Now consider the equation:
a −b = c + d
To make “a” the subject, add “b” to both sides.
a −b+b = c + d +b
∴a = c + d + b
Simply it can be seen that the end result is such that “b” just moves over to the other
side but changes its polarity sign.
MULTIPLICATION/DIVISION
Ex 3: Consider the equation:
a×b = c×d
To make “a” the subject, divide both sides by “b”.
a×b c×d
=
b
b
∴a =
c×d
b
Ex 4: Now consider the equation:
a c
=
b d
To make “a” the subject, multiply both sides by “b”.
a×b c×b
=
b
d
1
This gives:
a=
c×b
d
Simply it can be seen that the end result is such that “b” just moves diagonally across
the equal sign but keeps the same polarity sign.
CROSS-MULTIPLICATION
Using the examples given, the whole process can be simplified as follows:
a
b
=
c
d
The arrows show the direction of movement that can take place. They do not indicate
that the letters must be exchanged.
Ex 5:
a −b e
=
c+d f
The same rules applies for cross-multiplication ensuring that (a - b) and (c + d) are
treated as single terms.
To make “a” the subject, multiply both sides by “(c + d)” or just simply move (c + d)
diagonally across the equal sign.
a −b e
=
c+d f
a −b =
e(c + d )
f
Now add “b” to both sides or just simply move “b” to the other side of the equal sign
and change its polarity sign.
a=
e(c + d )
+b
f
2
POWERS & ROOTS
To remove square root signs it is necessary to square it and to remove a square term it
is necessary to square root it.
Ex 6:
Z = R2 + X 2
To make “R” the subject, it is necessary to square both sides first thus removing the
square root sign.
Z2 =
(
R2 + X 2
2
) =R +X
2
2
Now subtract X 2 from both sides or simply move it to the other side and change its
polarity sign:
Z 2 − X 2 = R2
Now take the square root of both sides:
± Z 2 − X 2 = R2 = R
Z2 − X 2 = R
Notice that positive and negative values of
Further exercises are on the worksheets.
SUBJECT LETTER APPEARS MORE THAN ONCE
Sometimes the subject letter may appear more than once, hence several operations are
needed to transpose the formula. Consider the following examples:
Ex 7: Tranpose the formula: ab = ac + de and make "a" the subject
Gathering all the "a" terms to one side:
Factorising:
ab − ac = de
a ( b − c ) = de
Transposing for "a"
a=
de
b−c
3
Ex 8:
ab
cd
=
and make "x" the subject
x x− y
Tranpose the formula:
Cross-multiplying:
( x − y )( ab ) = cdx
Expanding brackets: abx − aby = cdx
Gathering all the "x" terms to one side:
Factorising:
abx − cdx = aby
x ( ab − cd ) = aby
Transposing for "x":
x=
aby
( ab − cd )
Ex 9: Tranpose the formula:
squaring both sides:
D
=
d
2
⎛ D ⎞ ⎛
⎜ ⎟ = ⎜⎜
⎝ d ⎠ ⎝
D2 f
∴ 2 =
d
f
f+p
and make "p" the subject
f −p
2
f + p ⎞
f+p
⎟⎟ =
f − p ⎠
f −p
+p
−p
cross multiplying gives:
D2 ( f − p ) = d 2 ( f + p )
expanding brackets:
D2 f − D2 p = d 2 f + d 2 p
Gathering the "p" terms to one side:
Factorising :
D2 f − d 2 f = d 2 p + D2 p
f ( D2 − d 2 ) = p (d 2 + D2 )
cross multiplying gives:
f ( D2 − d 2 )
(d
2
+ D2 )
=p
-------------------------------------------------------------------------------------------------------
4