18
Formulae
CHAPTER
A formula is a way of describing a fact or a rule. A formula can be written using algebraic expressions.
A formula must have an sign.
In Section 9.6 the area (A) of a trapezium was found using
a
A 12(a b)h
h
This is called an algebraic formula.
b
A appears just once and only on the left-hand side of the formula,
so A is called the subject of the formula.
The letters a and b represent the lengths of the parallel sides of the trapezium and h represents the
height of the trapezium.
18.1 Using an algebraic formula
The formula A 12(a b)h can be used to work out the area of a trapezium.
The value of A is calculated by substituting values of a, b and h into this formula.
Example 1
a
The area, A cm2, of this hexagon is given by the formula
A h(a b)
h
b
Find the area of the hexagon when a 5 cm, b 9 cm and h 4 cm.
Solution 1
A h(a b)
4(5 9)
4 14 56
h
Substitute a 5, b 9 and h 4
a
Area of hexagon 56 cm2.
Example 2
Use the formula F 3n2 5n to work out the value of F when n 2
Solution 2
F 3n2 5n
3 (2)2 5 (2)
Substitute n 2
3 4 10
(2)2 4 and 5 (2) 10,
negative negative positive.
F 22
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Formulae
CHAPTER 18
Example 3
This formula can be used to change temperatures in degrees Fahrenheit (F ) to temperatures in
degrees Celsius (C ).
C 59(F 32)
Change 4 degrees Fahrenheit to degrees Celsius.
Solution 3
C 59(F 32)
59(4 32)
Substitute F 4
59 36
Work out the brackets first.
20
4°F 20°C
Exercise 18A
1 Sunita’s pay is worked out using the formula
P rh
P pay, r rate of pay for each hour worked and h number of hours worked.
Use this formula to work out Sunita’s pay when she works for 5 hours at a rate of £8 an hour.
2 T c 3d
Use this formula to work out the value of T, when
b c 10, d 6
a c 12, d 9
yc
3 The formula m can be used to work out the gradient of a straight line.
x
Use this formula to work out the value of m when
b x 3, y 1 and c 10
a x 2, y 11 and c 4
4 The formula v u at is used to work out velocity, v.
Use this formula to work out the value of v when
b u 4, a 5 and t 2
a u 5, a 10 and t 3
c u 20, a 2 and t 8
5 This formula can be used to change temperatures in degrees Celsius (C ) to temperatures in
degrees Fahrenheit (F ).
F 95 C 32
Use this formula to find the value of F when
b C0
a C 30
6 Use the formula
P 3x
to work out the value of P when
b x 12
a x3
284
c C 40
18.2 Writing an algebraic formula
CHAPTER 18
7 W 5t 2 t
Find the value of W when
a t3
b t 4
8 P kr2 2rs
Work out the value of P when
a k 10, r 4 and s 5
b k 3, r 5 and s 10
9 The cooking time, T (minutes), to cook a joint of meat is given by the formula
T 20w 30
where w (kg) is the weight of the joint of meat.
Use this formula to work out the cooking time when
b w 2.5
a w4
10 This formula is used to work out the distance (d ) travelled, when the time taken is t and the
average speed is s.
d st
Use this formula to work out the value of d
a when s 30 and t 2
b when s 5.5 and t 6
c when s 20 and t 3.4
l
11 T 2 g
Work out the value of T when 3.14, l 12 and g 9.8
12 s ut 12 at 2
Find the value of s when u 24, t 6 and a 2.4
18.2 Writing an algebraic formula
The diagram shows a rectangle.
a the length of the rectangle.
b the width of the rectangle.
a
b
The perimeter of a shape is the total distance around the edges of the shape.
The perimeter of this rectangle a b a b.
Collecting the like terms together gives the perimeter 2a 2b.
Let P represent the perimeter of the rectangle.
b
a
This can then be written as the algebraic formula
P 2(a b)
When writing an algebraic formula it is important to define what each letter stands for.
Example 4
In some football matches, 3 points are awarded for a win, 1 point is awarded for a draw and no points
are awarded for a loss.
Write down an algebraic formula that can be used to work out the total points awarded to a football
team. You must define the letters used.
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Formulae
CHAPTER 18
Solution 4
Let P the total number of points awarded to each team.
Let w the number of matches won.
Let d the number of matches drawn.
The number of points awarded for wins 3 the number of matches won 3 w 3w.
The number of points awarded for draws =1 the number of matches drawn =1 d d.
The formula for the total number of points awarded to each team is P 3w d.
Exercise 18B
1 A florist sells t tulips at 40p each and d daffodils at 30p each. If C pence is the total cost of the
flowers sold, write down, in terms of t and d, a formula for C.
2 A shop sells eating apples and baking apples.
Write down an algebraic formula that can be used to work out the total number of apples that
the shop sells. You must define the letters used.
3 Dan hires a car for his holidays. He pays a fixed charge of £75 plus £30 for each day that he has
the car.
a Write down a formula that Dan can use to work out the total cost of hiring the car.
You must define the letters used.
b Use your formula to work out the cost of hiring this car for 7 days.
4 a Write down a formula that can be used to work out the perimeter of an equilateral triangle.
You must define the letters used.
b Use your formula to work out the perimeter of an equilateral triangle of side 5 m.
5 Andrew earns p pounds per hour. He works for t hours. He earns a bonus of b pounds.
a Write down a formula for the total amount he earns, T pounds.
b p 5, t 38 and b 20
Use your formula to work out the value of T.
6 A box of nails costs £1.50 A box of screws costs £2
Bob buys n boxes of nails and s boxes of screws. The total cost is C pounds.
a Write a formula for C in terms of n and s.
b Work out the value of C when n 4 and s 3
7 Jean sits a mathematics exam, an English exam and a science exam. Her mean mark is x%.
Write down a formula that can be used to work out the value of x. You must define the letters
used.
8 A bus can seat p passengers and 8 passengers are allowed to stand.
a Write a formula that can be used to work out the total number, T, of passengers that can
travel on b buses.
b
286
i Work out the value of T when b 6 and p 60
ii What does your value of T tell you?
18.3 Changing the subject of a formula
CHAPTER 18
18.3 Changing the subject of a formula
The subject of the formula appears just once and only on the left-hand side of the formula.
p is the subject of the formula
p 2q 3
The formula can be rearranged, using the balance method, as follows
p 3 2q 3 3
p 3 2q
p 3 2q
2
2
p3
q
2
Subtract 3 from both sides.
Divide both sides by 2
p3
This is written as q 2
The subject of the formula p 2q 3 has now been changed to make q the subject of the formula.
Example 5
Make R the subject of the formula V IR.
Solution 5
V IR
V IR
I
I
V
R
I
V
R I
Divide both sides by I and then cancel.
Write the subject (R ) on the left-hand side.
Example 6
P 2(a b) can be used to work out the perimeter, P, of a rectangle of length a and width b.
Make b the subject of the formula.
Solution 6
P 2(a b)
P 2a 2b
Expand the brackets.
P 2a 2b
P 2a
b 2
Subtract 2a from both sides to get only the b term on one side.
Divide both sides by 2 and write b on the left-hand side.
287
Formulae
CHAPTER 18
Example 7
3 4x
Make x the subject of the formula y .
5
Solution 7
3 4x
y 5
3 4x
Multiply both sides by 5, then cancel.
y 5 5
5
5y 3 4x
Now rearrange this to get a positive x term on one side.
5y 4x 3 4x 4x
Add 4x to both sides.
5y 4x 3
5y 4x 5y 3 5y
Subtract 5y from both sides to get 4x on its own.
4x 3 5y
4x 3 5y
4
4
3 5y
x 4
Divide both sides by 4, then cancel.
Example 8
a Given that t is positive, rearrange the formula W u at2 to make t the subject.
b Find the value of t when W 120, u 80 and a 10
Solution 8
a W u at2
W u u at2 u
W u at2
W u at2
a
a
Wu
t2
a
Wu
t a
120 80
b t 10
t 4
t2
Rearrange to get only the t term on one side.
Subtract u from both sides…
… to get the t term on one side.
Divide both sides by a, then cancel.
Take the square root of both sides to find t.
Substitute W 120, u 80 and a 10 into the formula t t is positive so ignore the 2 solution.
Exercise 18C
1 Make x the subject of the formula
b y 4x
a yx7
d y 3x 5
e y1x
2 Make w the subject of the formula A lw.
288
c y 2x 1
f y 6 2x
Wu
.
a
18.4 Expressions, identities, equations and formulae
CHAPTER 18
3 Make t the subject of the formula s ut u2.
4 Make m the subject of the formula W 3n 2m.
5 A formula to find velocity is given by v u at.
Rearrange this formula to make
b a the subject of the formula
a u the subject of the formula
c t the subject of the formula.
6 Make P the subject of the formula
b D 4(2P 3Q)
a D 3(P Q)
c D 2(5Q P)
ab
7 Make a the subject of the formula c 3
e 3f
8 Make f the subject of the formula g 2
5(b 3n)
9 Make n the subject of the formula a 2
10 The area, A, of a circle is given by the formula A r2, where r is the radius.
Make r the subject of the formula.
11 a Make u the subject of the formula v2 u2 2as.
b Find the positive value of u when v 20, a 10 and s 12.8
18.4 Expressions, identities, equations and formulae
We know from Section 5.1 that 3x 6 is called an expression.
The expression 3x 6 can be factorised to give 3(x 2).
3x 6 3(x 2) is called an identity because the left-hand side, 3x 6, says the same as the
right-hand side, 3(x 2). Another example of an identity is 5x x 4x.
From Section 10.1, we know that 3x 6 0 is called an equation, which can be solved to find the
value of x.
P 3x 6 is called a formula. The value of P can be worked out if the value of x is known.
A formula has at least two letters.
Example 9
Here is a mixture of some terms and some signs.
m
4m
5m
2
n
Using some of the above, write down an example of
a an expression
b an identity
c an equation
d a formula.
Solution 9
a 4m n
An expression does not have an sign.
b 4m 5m m
An identity is true for all values of the letter.
c 5m 2 m
An equation can be solved. In this case the solution is m 12
d n 5m 2
A formula must have at least two letters and an sign. Here n is the
subject of the formula.
289
Formulae
CHAPTER 18
Exercise 18D
1 Write down whether each of the following is an expression or an identity or an equation or a
formula.
a 4x x x x x
b y Ax
c 3m 2m 5m
d y32
e 5s t
bh
f A 2
g x 2 4x 3
h p3 q2
C 3r 2s
x
k 4
3
m c c d d d 2c 3d
j 6( y 4) 6y 24
i
l y mx c
n q2 q 3 0
o 7r2 3s
2 Here is a mixture of some terms and some signs.
3p
5
p
2p q
Using some of the above, write down an example of
a an expression
b an identity
c an equation
d a formula.
18.5 Further changing the subject of a formula
To change the subject of some formulae, further steps including factorisation and squaring of
algebraic expressions are required.
Example 10
Make t the subject of the formula v ut 2nt.
Solution 10
v ut 2nt
v t (u 2n)
There is more than one term involving t so
factorise the right-hand side.
v
t (u 2n)
u 2n
u 2n
Divide both sides by (u 2n), then cancel.
v
t u 2n
290
All the t terms are on the same side.
Write the new subject, t, on the left-hand side.
18.5 Further changing the subject of a formula
CHAPTER 18
Example 11
1 A
Make A the subject of the formula r 2 Solution 11
1 A
A
Rearrange so that only is on one side.
r 2 1 A
Multiply both sides by 2
2 r 2 2 A
2r A
Remove the square root sign by squaring both sides.
(2r)2 A
4r2 4r 2 A
Multiply both sides by and then cancel.
A 4r 2
Write the new subject, A, on the left-hand side.
Example 12
Make c the subject of a 3c 4(bc 2d)
Solution 12
a 3c 4(bc 2d )
There are c terms on both sides.
a 3c 4bc 8d
Expand the brackets.
a 3c 4bc 8d
Subtract 4bc from both sides to get all c terms on one side.
3c 4bc 8d a
Subtract a from both sides.
c(3 4b) 8d a
8d a
c 3 4b
There is more than one term involving c. Factorise the left-hand side.
Divide both sides by (3 4b).
Example 13
The radius of a circle is r cm.
The length of a rectangle is (x 2) cm and its width is x cm.
The area of the circle and the area of the rectangle are equal.
Express r in terms of x and .
Solution 13
Area of circle r2
r2 x(x 2)
x(x 2)
r 2 x(x 2)
r (x 2) cm
r cm
x cm
Area of rectangle x(x 2)
Area of the circle area of the rectangle.
Divide both sides by .
Take the square root of both sides.
291
Formulae
CHAPTER 18
Exercise 18E
1 Make x the subject of y 2x 1 3x.
yx
2 Rearrange 2(x 1) to make x the subject.
3
3 The diagram shows a right-angled triangle
and a rectangle.
The area of the triangle is equal to the area
of the rectangle.
Express h in terms of b and x.
4 Make m the subject of the formula
W a(3 m) 5m
(x 3) cm
h cm
x cm
b cm
5 Make c the subject of a bc 5 3c
6 Rearrange 2p 3q r(q 4) to make q the subject.
7 The radius of a circle is R cm.
The length of a side of a square is x cm.
The area of the circle and the area of the square are equal.
a Express R in terms of x and .
b Hence find the radius of the circle when the perimeter
of the square is 40 cm.
Give your answer in terms of .
8 Rearrange 5(x 3) y(7 2x)
to make x the subject.
x cm
R cm
3 2w
9 Make w the subject of the formula y w
1x
10 Make x the subject of y
1x
x
11 Make x the subject of 1 bx
a
12 Make A the subject of the formula
13 Make a l b g
P
A 5
A
l
the subject of the formula T 2 g
Chapter summary
You should now know:
292
that a formula can be written using algebraic expressions to describe a rule or a relationship.
For example, the area of a rectangle can be written as an algebraic formula A lw, where
A area, l length and w width.
that in a formula, for example, A lw, A is called the subject of the formula.
the meanings of expression, identity, equation and formula.
You should also be able to:
find the value of the subject of a formula by substituting given values for the letters on the
right-hand side.
change the subject of a simple formula by rearranging the terms in the formula.
change the subject of a formula which may involve square roots, or the required subject
occurring twice.
Chapter 18 review questions
CHAPTER 18
Chapter 18 review questions
Q(P 2)
2 M 6
P 4, Q 30
Work out the value of M.
1 D 3s 7t
s 4, t 2
Work out the value of D.
3 P Q2 2Q
Find the value of P when Q 3
(1388 March 2004)
4 The number of diagonals, D, of a polygon with n sides is given by the formula
n2 3n
D 2
A polygon has 20 sides.
Work out the number of diagonals of this polygon.
(1388 March 2005)
5 A bicycle has 2 wheels.
A tricycle has 3 wheels.
In a shop there are x bicycles and y tricycles.
The total number of wheels on the bicycles and the tricycles in the shop is given by W.
Write down a formula for W, in terms of x and y.
6 The cost, in pounds, of hiring a car can be worked out using this rule.
Add 3 to the number of days’ hire
Multiply your answer by 10
The cost of hiring a car for n days is C pounds.
Write down a formula for C in terms of n.
(1387 June 2005)
7 A ruler costs 45 pence. A pen costs 30 pence.
Louisa buys x rulers and y pens. The total cost is C pence.
a Write down a formula for C in terms of x and y.
C 240, x 2
b Work out the value of y.
8 Pat plays a game with red cards and green cards.
Red cards are worth 5 points each.
Green cards are worth 3 points each.
Pat has r red cards and g green cards.
His total number of points is N.
Write down, in terms of r and g, a formula for N.
9 Which of the following is an identity?
C 2r
4a 3 8a 4
2(c 1) 2c 2
(1388 March 2003)
4m 3
293
Formulae
CHAPTER 18
10 Copy the table and write either
expression or equation or
identity or formula in each space.
The first has been done for you.
A 12 bh
formula
x2 5 86
7m 5n
3p 2q p p p q q
(1388 March 2005)
11 Make k the subject of the formula p 9k 20
12 Jo uses the formula F 95 C 32 to change degrees Celsius (C ) to degrees Fahrenheit (F).
a Use this formula to find
ii the value of C when F 50
i the value of F when C 40
b Rearrange the formula to make C the subject of the formula.
13 Make p the subject of the formula m 3n 2p
14 a Solve 4(x 3) 6
b Make t the subject of the formula
v u 5t
(1387 June 2005)
16 Use a calculator for this question.
(1388 March 2005)
15 Make a the subject of the formula
a
s 8u
4
(1388 March 2005)
P r 2r 2a
P 84 r 6.7
a Work out the value of a.
Give your answer correct to three significant figures.
b Make r the subject of the formula P r 2r 2a
(1387 June 2005)
17 Make m the subject of the formula 2(2p m) 3 5m
(1388 January 2003)
18 Make a the subject of the formula 2(3a c) 5c 1
(1388 January 2004)
19 The diagram shows a shape made up of a right-angled triangle and a semicircle.
The area of the triangle is equal to the area of the semicircle.
a Express h in terms of and d.
b Express d in terms of and h.
h
d
(1388 January 2004)
20
21
22
23
w(1 y)
Make y the subject of the formula T y
x
Make x the subject of the formula y (1385 June 2002)
ax
x4
a Simplify (3xy3)4.
b Rearrange 2y to give x in terms of y.
(1385 June 2001)
5
The fraction, p, of an adult’s dose of medicine which should
3w 20
p be given to a child who weighs w kg is given by the formula
200
a Use the formula to find the weight of a child whose dose is the same as an adult’s dose.
b Make w the subject of the formula.
c Express A in terms of w in the following formula.
A
3w 20
200
A 12
294
(1387 November 2003)