© boardworks ltd 2004 1 of 33 a3 formulae ks3 mathematics

© Boardworks Ltd 20041 of 33

A3 Formulae

KS3 Mathematics


A3.1 Introducing formulae

Contents

A3 Formulae

A3.2 Using formulae

A3.4 Deriving formulae

A3.3 Changing the subject of a formula


Formulae

What is a formula?

In maths a formula is a rule for working something out.

The plural of formula is formulae.

For example,

How can we work out the number of days in a given number of weeks?

number of days = 7 × number of weeks


What formula would we write to work out the number of table legs in the classroom?

× 4

We can now work out the number of table legs for any given number of tables.

For example, if there are 16 tables in the classroom:

Number of table legs = 16 × 4 = 64

Writing formulae in words

Number of table legs = number of tables


Writing formulae in words

Mark is 5 years older than his sister, Kate.

What formula would we write to work out Mark’s age if we are given Kate’s age?

+ 5

For example, if Kate is 8 years old:

Mark’s age = 8 + 5 = 13

If Kate is 49 years old:

Mark’s age = 49 + 5 = 54

Mark’s age = Kate’s age


Writing formulae in symbols

Formulae are normally written using letters instead of words.

Each letter in a formula represents a numerical amount.

For example,

The perimeter of a square = 4 × the length of one of the sides.

s

s

s

s

We can write this as,

Where p is the perimeter of the square and s is the length of a side.

p = 4s


Sally earns a basic salary of £200 a week working in an office. She also earns £7 an hour for each hour of overtime she does.

What formula could we use to work out Sally’s weekly income?

Weekly income, in pounds = 7 × number of extra hours + 200

We can write this in symbols as:

How much does Sally earn if she works 9 hours overtime?

W = 7 × 9 + 200 = £263

Writing formulae in symbols

W = 7h + 200


Examples of formulae

Remember, a formula is a special type of equation that links physical variables.

Here are some examples of well-known formulae from maths.

The area, A, of rectangle of length, l, and width, w, is given by the formula:

A = lw

The perimeter, P, of a rectangle is given by the formula:

P = 2(l + w)


Writing formulae

Write a formula to work out:

The cost, c, of b boxes of crisps at £3 each.

c = 3b

The distance left, d, of a 500 km journey after travelling k km.

d = 500 – k

The cost per person, c, if a meal costing m pounds is shared between p people.

c =mp

© Boardworks Ltd 200410 of 331 of 20

The number of seats in a theatre, n, with 25 seats in each row, r.

n = 25r

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

a = b + 5

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


Castle entrance prices

Adult £3

Stony Castle

What formula could we use to work out the total cost in pounds, T, for a number of adults, A and a

number of children, C, to visit the castle?

3A + 2C

Child £2

Entrance fee

T =


T = 3A + 2C

Using this formula, how much would it cost for 4 adults and a class of 32 children to visit the castle?

We substitute the values into the formula:

T = 3 × 4 + 2 × 32

= 12 + 64

= 76

It will cost £76.

Castle entrance prices


Newspaper advert

To place an advert in a local newspaper costs £15. There is then an additional charge of £2 for each word used.

Write a formula to work out the total cost in pounds, C, to place an advert containing n words.

C = 15 + 2n

How much would it cost to place an advert containing 27 words?

C = 15 + 2 × 27

= 15 + 54

= 69 It will cost £69


A3.2 Using formulae

Contents

A3 Formulae





Substituting into formulae

wl

h

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

S = 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 × 32 × 25)

= 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


Substituting into formulae and solving equations

What is the length of a rectangle with a perimeter of 20 cm and a width of 4 cm?

l

w

The formula used to find the perimeter P of a rectangle with length l and width w is

P = 2l + 2w



Substitute P = 20 and w = 4 into the formula:

Solve this equation

Simplifying:

Subtracting 8:

20 = 2l + 8

6 = l

So the length of the rectangle is 6 cm.

P = 2l + 2w

20 = 2l + (2 × 4)

12 = 2l

Dividing by 2:

l = 6



The area A of a triangle with base b and perpendicular height h is given by the formula

A = 12

bh

What is the length of the base of a triangle with an area of 48 cm2 and a perpendicular height of 12 cm?

b

h



Substitute A = 48 and h = 12 into the formula:

48 = 1

2b × 12

Solve this equation

A = 12

bh

Dividing by 12: 4 = 1

2b

Multiplying by 2: 8 = b

b = 8

So the base of the triangle measures 8 cm.



Contents

A3 Formulae

A3.2 Using formulae




Using inverse operations

Andy is 5 years older than his brother, Brian. Their ages are linked by the formula:

where A is Andy’s age in years and B is Brian’s age in years.

Using this formula it is easy to find Andy’s age given Brian’s age.

Suppose we want to find Brian’s age given Andy’s age.

Using inverse operations, we can write this formula as:

A = B + 5

B = A – 5


The subject of a formula

Look at the formula,V = 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:

I × R V

The inverse of this is:

V÷ RI

or

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

can be written as:

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.



To make C the subject of the formula

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 =


Equivalent formulae



Contents

A3 Formulae

A3.2 Using formulae




Connecting Dots


Dotty patterns

© boardworks ltd 2004 1 of 33 a3 formulae ks3 mathematics

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