© boardworks ltd 2005 1 of 40 a3 formulae ks4 mathematics

© Boardworks Ltd 2005 1 of 40

A3 Formulae

KS4 Mathematics


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Contents

A3.1 Substituting into formulae

A3 Formulae

A3.2 Problems that lead to equations to solve

A3.3 Changing the subject of a formula

A3.4 Manipulating more difficult formulae

A3.5 Generating formulae


Formulae

A formula is a special type of equation that links two or more physical variables.

For example in the formula,

P = 2(l + w)

P represents the perimeter of a rectangle and l and w represent its length and width.

We can use this formula to work out the perimeter of any rectangle given its length and width.

We do this by substituting the values we are given into the formula.


Formulae

Because formulae deal mainly with real-life quantities such as length, mass, temperature or time, the given variables often have units attached.

Units shouldn’t be included in the formula itself.

The units that have to be used are usually defined in the formula. For example,

S = d

t

This formula doesn’t mean much unless we say “S is the average speed in m/s, d is the distance travelled in metres, and t is the time taken in seconds”.


Formulae

Use the formula to find the speed of a car that

Write the distance and the time using the correct units before substituting them into the formula,

S = d

t

travels 2 km in 1 minute and 40 seconds.

2 kilometres = 2000 metres

1 minute and 40 seconds = 100 seconds

Now substitute these numerical values into the formula,

S =2000100

= 20 m/s

We can write the units at the end.


Substituting into formulae

wl

h

The surface area S of a cuboid is given by the formula

S = 2lw + 2lh + 2hwS = 2lw + 2lh + 2hw

where l is the length, w is the width and h is the height.

What is the surface area of a cuboid with a length of 1.5 m, a width of 32 cm and a height of 250 mm?



Before we can use the formula we must write all of the amounts using the same units.

l = 150 cm, w = 32 cm and h = 25 cm

Next, substitute the values into the formula without the units.

S = 2lw + 2lh + 2hw

= (2 × 150 × 32) + (2 × 150 × 25) + (2 × 25 × 32)

= 9600 + 7500 + 1600

= 18,700 cm2

Don’t forget to write the units in at the end.

What is the surface area of a cuboid with a length of 1.5 m, a width of 32 cm and a height of 250 mm?



The distance d, in metres, that an object falls after being dropped is given by the formula,

d = 4.9t2 where t is the time in seconds.

Suppose a boy drops a rock from a 100 metre high cliff.

How far will the rock have fallen after:

a) 2 seconds b) 3 seconds c) 5 seconds?

When t = 2,

d = 4.9 × 22

= 4.9 × 4

= 19.6 metres

When t = 3,

d = 4.9 × 32

= 4.9 × 9

= 44.1 metres

When t = 5,

d = 4.9 × 52

= 4.9 × 25

= 122.5 metres


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A3 Formulae


Problems that lead to equations to solve

Formulae are usually (but not always) arranged so that a single variable is written on the left-hand side of the equals sign. For example, in the formula

v = u + at

v is called the subject of the formula.

If we are given the values of u, a and t, we can find v by substituting these values into the formula.

Suppose instead that we are given to values of v, u and a, and asked to find t.

When these values are substituted we are left with an equation to solve in t.



For example, suppose v = 20, u = 5 and a = 3. Find t.

Substituting these values into v = u + at gives us the equation,

We can then solve this equation as usual.

20 = 5 + 3t

swap both sides: 5 + 3t = 20

subtract 5 from both sides: 3t = 15

divide both sides by 3: t = 5 seconds



What is the height of a trapezium with an area of 40 cm2 and parallel sides of length 7 cm and 9 cm?

The formula used to find the area A of a trapezium with parallel sides a and b and height h is:

A = (a + b)h12

h

a

b



Substituting A = 40, a = 7 and b = 9 into gives

Simplifying,

swap both sides:

h = 5

So the height of the trapezium is 5 cm.

8h = 40

divide by 8:

A = (a + b)h12

40 = (7 + 9)h12

40 = ×16h12

40 = 8h


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A3 Formulae


The subject of a formula

Here is a formula you may know from physics:

V = IRV = IR

where V is voltage, I is current and R is resistance.

V is called the subject of the formula.

The subject of a formula always appears in front of the equals sign without any other numbers or operations.

Sometimes it is useful to rearrange a formula so that one of the other variables is the subject of the formula.

Suppose, for example, that we want to make I the subject of the formula V = IR.


The formula:

can be written as a function diagram:

I × R V

The inverse of this is:

V÷ RI

So:

Changing the subject of the formula

V is the subject of this formula

I is now the subject of this formula

I = VR

V = IR


Matchstick pattern

Look at this pattern made from matchsticks:

PatternNumber, n 1

Number ofMatches, m 3

2

5

3

7

4

9

The formula for the number of matches, m, in pattern number n is given by the formula:

m = 2n + 1

Which pattern number will contain 47 matches?


The formula:

m = 2n + 1 m = 2n + 1

can be written as a function diagram:

n + 1× 2 m

The inverse of this is:

m÷ 2 – 1n

or

m is the subject of this formula

n is the subject of this formula


n =m – 1

2



To find out which pattern will contain 47 matches, substitute 47 into the rearranged formula.

n =m – 1

2

n =47 – 1

2

n =462

n = 23

So, the 23rd pattern will contain 47 matches.



We can also change the subject by performing the same operations on both sides of the equals sign. For example, to make C the subject of

subtract 32: F – 32 =9C5

multiply by 5: 5(F – 32) = 9C

divide by 9:5(F – 32)

9= C

F = + 329C5

5(F – 32)9

C =


Change the subject of the formula 1


Find the equivalent formulae


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A3 Formulae


Formulae where the subject appears twice

Sometimes the variable that we are making the subject of a formula appears twice. For example,

S = 2lw + 2lh + 2hw

where S is the surface area of a cuboid, l is its length, w is its width and h is its height.

Make w the subject of the formula.

To do this we must collect all terms containing w on the same side of the equals sign.

We can then isolate w by factorizing.


Formulae where the subject appears twice

S = 2lw + 2lh + 2hw

Let’s start by swapping the left-hand side and the right-hand side so that the terms with w’s are on the left.

2lw + 2lh + 2hw = S

subtract 2lh from both sides: 2lw + 2hw = S – 2lh

factorize: w(2l + 2h) = S – 2lh

divide by 2l + 2h: w =S – 2lh

2l + 2h


Formulae involving fractions

When a formula involves fractions we usually remove these by multiplying before changing the subject.

Make R the subject of the formula

For example, if two resistors with a resistance a and b ohms respectively, are arranged in parallel their total resistance R ohms can be found using the formula,

1R

=1a

+1b

aΩ bΩ


Formulae involving fractions

multiply through by Rab: = +RabR

Raba

Rabb

simplify: ab = Rb + Ra

factorize: ab = R(b + a)

1R

=1a

+1b

divide both sides by a + b: = Rab

a + b

R =ab

a + b


Formulae involving powers and roots

The length c of the hypotenuse of a right-angled triangle is given by

where a and b are the lengths of the shorter sides.

c = √a2 + b2

Make a the subject of the formula

square both sides: c2 = a2 + b2

subtract b2 from both sides: c2 – b2 = a2

a = √c2 – b2

square root both sides: √c2 – b2 = a



The time T needed for a pendulum to make a complete swing is

T = 2π lg

where l is the length of the pendulum and g is acceleration due to gravity.

Make l the subject of the formula

When the variable that we wish to make the subject appears under a square root sign, we should isolate it on one side of the equation and then square both sides.



T = 2π lg

divide both sides by 2π:T2π

= lg

square both sides:T2

4π2= l

g

multiply both sides by g:T2g4π2

= l

l =T2g4π2


Change the subject of the formula 2


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A3 Formulae


Writing formulae

Write a formula to work out,

1) the cost, c, of b boxes of crisps at £3 each

c = 3bc = 3b

2) the distance left, d, of a 500 km journey after travelling k km

d = 500 – kd = 500 – k

3) the cost per person, c, if a meal costing m pounds is shared between p people

c =mp


4) the number of seats in a theatre, n, with 25 seats in each row, r

n = 25rn = 25r

5) the age of a boy Andy, a, if he is 5 years older than his sister Betty, b

a = b + 5 a = b + 5

6) the average weight, w, of Alex who weighs a kg, Bob who weighs b kg and Claire who weighs c kg.

w =a + b + c

3

Writing formulae


C = 7n + 10C = 7n + 10

Using this formula, how much would it cost to clean all 105 windows of Formula Mansion?

We substitute the value into the formula,

C = 7 × 105 + 10

= 735 + 10

= 745

It will cost £745.

Writing formulae

A window cleaner charges a £10 call-out fee plus £7 for every window that he cleans. Write a formula to find the total cost C when n windows are cleaned.


C = 7n + 10

At another house the window cleaner made £94.How many windows did he have to clean?

We substitute this value into the formula to give an equation,

He cleaned 12 windows.

Writing formulae

94 = 7n + 10

swap both sides: 7n + 10 = 94

subtract 10 from both sides: 7n = 84

n = 12


Using formulae to write mathematical rules

When conducting a mathematical investigation, it is usually necessary to use formulae to write rules and generalizations.

For example, Sophie is investigating patterns of shaded squares on a numbered grid.

She starts by looking at arrangements of numbers in two by two squares on a 100 square grid. For example,

34 35

44 45

She notices that the sum of the numbers in a two by two square is always equal to four times the number in the top left-hand square plus 22. Sophie writes this as a formula.


Number grid patterns

© boardworks ltd 2005 1 of 40 a3 formulae ks4 mathematics

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