formulas 5 5.1sequences 5.2introduction to functions 5.3simple algebraic fractions chapter summary...

Formulas5

5.1 Sequences

5.2 Introduction to Functions

5.3 Simple Algebraic Fractions

Chapter Summary

Case Study

5.4 Formulas and Substitution

5.5 Change of Subject

P. 2

Case Study

The following table shows the range of the BMI and the corresponding body condition.

Body Mass Index (BMI) is frequently used to check whether a person’s body weight and body height are in an appropriate proportion. It can be calculated by using the following formula:

)m(in heightBody

kg)(in t Body weigh (BMI)Index MassBody

22

For example, if a person’s weight and height are 50 kg and 1.6 m respectively, then the BMI 50 1.62 19.5.

BMI ConditionLess than 18.5 Underweight

18.5 – 22.9 Ideal23 – 24.9 Overweight25 – 29.9 Obese30 or over Severely obese

Let us check your body mass index and see whether you are normal or not.

Is my weight normal?

P. 3

We call such a list of numbers a sequence.

A. Introduction to Sequences

John recorded the weights of 10 classmates. He listed their weights (in kg) in the order of their class numbers.

Each number in a sequence is called a term.

We usually denote the first term as T1, the second term as T2, and so on.

In the above sequence, T1 46, T2 42, T3 50, ... , etc.

5.1 Sequences

P. 4

5.1 Sequences

B. Some Common Sequences

Some sequences have certain patterns, but some do not.

1. Consider the sequence 1, 5, 9, 13, ... .

When the difference between any 2 consecutive terms is a constant, such a sequence is called an arithmetic sequence, and the difference is called a common difference.

We can guess the subsequent terms: 17, 21, 25, ...

2. Consider the sequence 8, 16, 32, 64, ... .

When the ratio of each term (except the first term) to the preceding term is a constant, such a sequence is called a geometric sequence, and the ratio is called a common ratio.

We can guess the subsequent terms: 128, 256, 512, ...

+4 +4

2

2

P. 5

B. Some Common Sequences

5.1 Sequences

3. We can arrange some dots to form some squares.

4. We can arrange some dots to form some triangles.

The number of dots used in each squareis called a square number.

The number of dots used in each triangleis called a triangular number.

5. Consider the sequence 1, 1, 2, 3, 5, 8, ... .In this sequence, starting from the third term, each term is the sum of the 2 preceding terms. This sequence is called the Fibonacci sequence.

P. 6

5.1 Sequences

C. General Terms

For a sequence that shows a certain pattern, we can use Tn to represent the nth term.

It is a common practice to write the general term of a sequence as an algebraic expression in terms of n.

Tn is called the general term of the sequence.

For example, since the sequence 2, 4, 6, 8, 10, ... has consecutive even numbers, we can deduce that the general term of this sequenceis Tn 2n.Once the general term of a sequence is obtained, we can use it to describe any term in a sequence.

P. 7

(a) The sequence 7, 14, 21, 28, ... can be written as7(1), 7(2), 7(3), 7(4), ...

5.1 Sequences

C. General Terms

Example 5.1T

... ,16

1 ,

8

1 ,

4

1 ,

2

1

Find the general terms of the following sequences.(a) 7, 14, 21, 28, ... (b) 1, 2, 4, ...

(c) (d) 15, 14, 13, 12, ...

Solution:

(b) The sequence 1, 2, 4, ... can be written as21 1, 22 1, 23 1 ...

∴ The general term of the sequence is 7n.

∴ The general term of the sequence is 2n 1.

P. 8

5.1 Sequences

C. General Terms

Example 5.1T

Solution:

... ,16

1 ,

8

1 ,

4

1 ,

2

1

Find the general terms of the following sequences.(a) 7, 14, 21, 28, ... (b) 1, 2, 4, ...

(c) (d) 15, 14, 13, 12, ...

... ,21 ,

21 ,

21 ,

21 4321

(c) The sequence can be written as... ,16

1 ,

8

1 ,

4

1 ,

2

1

n

2

1∴ The general term of the sequence is .

(d) The sequence 15, 14, 13, 12, ... can be written as16 1, 16 2, 16 3, 16 4, ...

∴ The general term of the sequence is 16 n.

P. 9

5.1 Sequences

C. General Terms

Example 5.2TFind the general term and the 9th term of each of the following sequences.(a) 1, 2, 5, 8, ... (b) 1, 4, 9, 16, ...

Solution:Rewrite the terms of the sequence as expressions in which the order of the terms can be observed.

(a) T1 1T2 2T3 5T4 8∴ Tn 3n 4

3 4 6 4 9 4 12 4

(b) T1 1T2 4T3 9T4

16 ∴ Tn n2

12 22

32

42

T9 3(9) 4 23

T9 92

81

+3 +3 +3

3(1) 4 3(2) 4 3(3) 4 3(4) 4

P. 10


Consider a sequence with the general term Tn 5n 2.

The above figure shows an ‘input-process-output’ relationship, which is called a function.

In this example, we call Tn a function of n.

In the previous section, we learnt how to find the values of the terms in a sequence from the general term by substituting different values of n in the general term.

For each input value of n, there is exactly one output value of Tn.

P. 11

For every value of x, there is only one corresponding value of y.

Suppose that each can of cola costs $5.

Let x be the number of cans of cola, and $y be the corresponding total cost.

Since the total cost of x cans of cola is $5x, the equation y 5x represents the relationship between x and y.The following table shows some values of x and the correspondingvalues of y.


The idea of function is common in our daily lives.

x 1 2 3 4 5

y 5 10 15 20 25

We say that y is a function of x.

P. 12


Example 5.3T

5

2

If p is a function of q such that p 4q 5, find the values of p for the following values of q.(a) 6 (b) 5 (c)

Solution:

Substitute q 6 into the expression 4q 5.

(a) When q 6, p 4(6) 5

(b) When q 5, p 4(5) 5 25

(c) When q , p 5

25

5

24

55

8

5

17

19

P. 13

we call such an expression an algebraic fraction.

,)2)(1(

6or

12

5 ,

4

3 2

xx

x

b

b

b

A. Simplification


b

aWe learnt at primary level that numbers in the form , where

a and b are integers and b 0, are called fractions.

When both the numerator and the denominator of a fractional expression are polynomials, where the denominator is not a constant, such as:

Note that is

not an algebraic fraction because the denominator is a constant.

5

43 yx

P. 14

For example:

3

2312

212

36

24

For algebraic fractions, we can simplify them in a similar way, when the common factors are numbers, variables or polynomials.

2

23

3

2)3(2

)2(2

6

4

a

aa

a

a

a

A. Simplification


We can simplify a numerical fraction, whose numerator and denominator both have common factors, by cancelling the common factors.

For example:

P. 15

ucua

ucua

42

105 (a)

y

yy

21

714 (b)

2

A. Simplification


Example 5.4T

ucua

ucua

42

105

y

yy

21

714 2

Simplify the following algebraic fractions.

(a) (b)

First factorize the numerator and the denominator. Then cancel out the common factors.

Solution:

2

5

)2(2

)2(5

cau

cau

3

12 y)3(7

)12(7

y

yy

3

21or

y

We cannot cancel common factors from the terms 7y and 21y only, i.e., the fraction cannot be simplified as

.3

14

21

714 22 y

y

yy

3

P. 16

hk

knkmnhmh

2

422 (a)

2)26(

3 (b)

x

x

A. Simplification


Example 5.5T

hk

knkmnhmh

2

422Simplify the following algebraic fractions.

(a) (b)2)26(

3

x

x

Solution:

nm 2

hk

nmknmh

2

)2(2)2(

hk

nmkh

2

)2)(2(

2)]3(2[

3

x

x

)3(4

1

x

2)3(4

3

x

x

First factorize the numerator. You may check if 2k + h (the denominator) is a factor of the numerator.

P. 17

B. Multiplication and Division


For example:

We can perform the multiplication or division of algebraic fractions in a similar way.

When multiplying or dividing a fraction, we usually try to cancel out all common factors before multiplying the numerator and the denominator separately to get the final result.

For example:

6

5)3(3

)5(5

)5(2

3

9

25

10

3

a

b

aa

bb

b

a

a

b

b

a

6

5

)3(3

)5(5

)5(2

3

9

25

10

3

2

2

2

3

P. 18

32

2 8

864

2 (a)

k

t

tt

k

205

24

4

6 (b)

2

n

q

n

q

B. Multiplication and Division


Example 5.6T

Solution:

32

2 8

864

2

k

t

tt

k

Simplify the following algebraic fractions.

(a) (b)205

24

4

6 2

n

q

n

q

3

2 8

)8(8

2

k

t

tt

k

tk4

1

224

)4(5

4

6

q

n

n

q

q4

5

P. 19

a

aaa

9

79

52

9

5

9

2

a

aaa

18

718

)2(2)3(1

9

2

6

1

C. Addition and Subtraction


For example:

The method used for the addition and subtraction of algebraicfractions is similar to that for numerical fractions.

When the denominators of algebraic fractions are not equal, first we have to find the lowest common multiple (L.C.M.) of the denominators.

For example:

P. 20

hh 8

3

2

7



Example 5.7T

hh 8

3

2

7 Simplify .

Solution:

hh 8

3

8

)4(7

h8

328

h8

25

P. 21

uvvu

3

3

62

5



Example 5.8T

Solution:

Simplify .uvvu

3

3

62

5

)3(2

1

uv

)3(

3

)3(2

5

vuvu

)3(2

)2(35

vu

)3(2

1

vu

For any number x 0,

xxx

111

P. 22

)3(3

275

pp

p



Example 5.9T

pp

3

93

4

Simplify .

Solution:

Since 3(p 3) and p have no common factors other than 1, the L.C.M. of 3(p 3) and p is 3p(p 3).

pp

3

93

4 pp

3

)3(3

4

)3(3

)3(334

pp

pp

)3(3

2794

pp

pp

P. 23


Consider the volume (V) of a cuboid:

V lwh where l is the length, w is the widthand h is the height.

If the values of the variables l, w and h are already known, we can find V by the method of substitution.

Similarly, if the values of V, l and w are known, we can find h.

Actually, in any given formula, we can find any one of the variables by the method of substitution if all the others are known.

P. 24

22h


Example 5.10T

4

32 hhrT Consider the formula . Find the value of h if T 121

and r 2.5.

Solution:

4

34

3

2

2

rhT

hhrT

4

35.2121 2h

4

3

2

5121

2

h

4

3

4

25121 h

2

11121

h

11

2121h

Factorize the expression.

P. 25


2

bhA

Given the area (A) of a triangle:

where b is the base length and h is the height.

In the formula, A is the only variable on the left-hand side.

We call A the subject of the formula.

This formula can be used if we want to find A, when b and h are known.

b

Ah

2

In another case, if we need to find h when A and b are known, it is more

convenient to use another formula , with h being the subject.

The process of obtaining the formula from is called the change of subject. 2

bhA

b

Ah

2

We can use the method of solving equations to change the subject of a formula.

P. 26

ukuk

543543

5

43 ku


Example 5.11TMake u the subject of the formula 3k 4 5u.

Solution:

First move the other terms to one side such that only the variable u remains on the other side.

435 ku Rewrite the formula such that only thevariable u is on the L.H.S.

4 is transposed to the L.H.S. to become 4.

P. 27

prm

rmp132

321

rp

prm

3

2

)3(2 rpmpr


Example 5.12T

rmp

321 Make m the subject of the formula .

Solution:

pr

rp

m

32

prrpm 2)3(

Simplify the fractions.The L.C.M. of r and p is pr.

Multiply both sides by mpr.

P. 28

mhmkm

mhk

2)23(23

2

)1(2

3

k

hkm


Example 5.13T

m

mhk

23

2

Make m the subject of the formula .

Solution:

First move all the terms that involve m to one side.

)1(23 kmhk

mhmkk 223 Remove the brackets

mkmhk 223 Take out the common factor m.

hkkm 3)1(2

P. 29

)24(9

2Tt 6

)27(9

2

∴ It takes 6 hours for the temperature of the food to become 3C.


Example 5.14T

2

924

tT

A bag of food is put into a refrigerator. The temperature T (in C) of

the food after time t (in hours) is given by the formula .

(a) Make t the subject of the formula.(b) How long will it take for the temperature of the food to become

–3C?

Solution:

Tt

tT

242

92

924(a) (b) )]3(24[

9

2 t

P. 30

Chapter Summary

5.1 Sequences

A list of numbers arranged in an order is called a sequence.

Each number in a sequence is called a term.

For a sequence with a certain pattern, we can represent the sequence by its general term.

P. 31


Chapter Summary

A function describes an ‘input-process-output’ relationship between 2 variables. Each input gives only one output.

P. 32


Chapter Summary

The manipulations of algebraic fractions are similar to those of numerical fractions.

P. 33


Chapter Summary

A formula is any equation that describes the relationship of 2 or more variables.

By the method of substitution, we can find the value of a variable in a formula when the other variables are known.

P. 34


Chapter Summary

When there is only one variable on one side of a formula, this variable is called the subject of the formula.

formulas 5 5.1sequences 5.2introduction to functions 5.3simple algebraic fractions chapter summary...

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