we
know that when we count we start 1,2 .... . But there are other numbers
like 0, negative numbers and decimals. All these types of numbers are
categorized in different groups like counting numbers, integers,real
numbers, whole numbers and rational and irrational numbers according to
their properties. all this have been covered in this chapter
Natural and Whole Number
Natural
numbers are counting numbers. Counting always starts with 1 and
continues with 2, 3 … (dots means numbers continue with no end).
Therefore Natural or counting numbers starts from 1 and continue to
infinite (no end).
Natural numbers are denoted by N. natural numbers can be represented on a horizontal line called number line as shown above:
The other group of numbers starts from zero and it is called whole numbers denoted by W.
We can represent whole numbers on a number line as shown above:
The Difference between Natural and Whole Numbers
Distinguish between natural and whole numbers
Natural
numbers are either used to count one to one objects or represent the
position of an object in a sequence. They start from one and go on to
infinity.This is why they are sometimes referred to as counting numbers.
The only whole number that cannot be classified as a natural number is
0. Counting numbers can further be classified into perfect numbers,
composite numbers, co-prime/ relatively prime numbers, prime numbers,
even and odd numbers.
Even ,Odd, and Prime Numbers
Identify even ,odd, and prime numbers
Even numbers are
those numbers which are divisible by 2. In other words we can say that
any natural number which can be divided by 2 is an even number. For
example 2, 4, 6, 8, 10, 12, … are even numbers since they are divisible
by 2.
Odd numbers are
those numbers which are not divisible by 2. In other words we can say
that any natural number when divided by 2 and remains 1 including 1 are
called odd numbers. Example 1, 3, 5, 7, 9, 11, 13, 15, … are called odd
numbers.
Prime numbers are
those numbers which are divisible by one and itself excluding one or
any natural number which is divisible by one and itself except one. For
example 2, 3, 5, 7, 11, 13, 17, … are called prime numbers.
Odd and Prime Numbers on Numbers Lines
Show even , odd and prime numbers on number lines
Odd and Prime Numbers
Operations with Whole Numbers
We have four operations which are: addition (+), subtraction (-), multiplication ( X) and division (÷).
Addition of Whole Numbers
Add whole numbers
When
adding numbers we add the corresponding digits in their corresponding
place values and we start adding from the right side i.e. from the place
value of ones to the next.
We can add numbers horizontally or vertically.
Horizontal addition
Example 5
- 972 + 18=
- 23 750 + 250 =
Solution
- 972 + 18 = 990
- 23 750 + 250 = 24 000
Vertical addition
Example 6
Subtraction of Whole Numbers
Subtract whole numbers
Subtraction is denoted by the sign (-). It is sometimes called minus.
Subtraction is the opposite of addition. Subtraction also means reduce a
number from certain number and the answer that is obtained is called difference..
Subtraction
is done in similar way like addition. We subtract the corresponding
digits in their corresponding place value. For example; 505 – 13. We
first subtract ones, which are 5 and 3. Subtract3 from 5 gives 2.
Followed by tens which are 0 and 1. Subtract 1 from 0 is not possible.
In order to make it easy, take 1 from 5 (hundreds). When 1 is added to 0
it has to be changed to be tens since it is added to a place of tens.
So, when 1 comes into a place of tens it becomes 10. So add 10 to 0. We
get 10. Now, subtract 1 from 10. We get 9. We are left with 4 in a place
of hundreds since we took 1. There for our answer will be 492.
Note that similar manner will be used when subtracting.
Example 7
Multiplication of Whole Numbers
Multiply whole numbers
Multiplication
means adding repeatedly depending on the times number given. For
example; 25 6 means add 25, repeat adding 6 times i.e. 25 + 25 + 25 +
25 +25 + 25 = 150. The answer obtained after multiplying two or more
numbers is called product. The number being multiplied is called a multiplicand while the number used in multiplying is called a multiplier. Referring our example, 25 is multiplicand and 6 is multiplier.
Example 8
Division of Whole Numbers
Divide whole numbers
Division is the same as subtraction. You subtractdivisor(the number used to divide another number) fromdividend(the
number which is to be divided), we repeat subtracting divisor to the
answer obtained until we get zero. The answer is how many times you
repeat subtraction.
For
example; 27 ÷9, we take 27 we subtract 9, we get 18. Again we take 18
we subtract 9, we get 9. We take 9 we subtract 9 we get 0. We repeat
subtraction three times. Therefore the answer is 3. The answer obtained
is calledquotient. Referring to our example; 27 is
dividend, 9 is divisor and 3 is quotient. If a number can’t be divided
exactly, what remains or left over is calledremainder.
Example 9
The Four Operations in Solving Word Problems
Use the four operations in solving word problems
Sometimes
you may be given a question with mixed operations +, -, xand ÷ . We do
multiplication and division first then addition and subtraction.
Example 10
- 12 ÷ 4 + 3 x 5
- 14 x2 ÷ 7 – 3 + 6
Solution
- 12 x 4 + 3 x 5 =3 + 15 (do division and multiplication fist) =18
- 14 x 2 ÷ 7 – 3 + 6 =28 ÷ 7 - 3 + 6 (multiply first) =4 – 3 + 6 (then divide) =10 – 3 (add then subtract) =7
We
may use brackets to separate x,÷ , + and – if they are mixed in the
same problem and use what is called BODMAS . BODMAS is the short form of
the following:
B for Brackets O for Open D for Division M for Multiplication A for Addition and S for Subtraction
Therefore,
with mixed operations, we first do the operation inside the brackets;
we say that we open the brackets. Then we do division followed by
multiplication, addition and lastly subtraction.
Example 11
Word problems on whole numbers
Example 12
In a school library there are 6 shelves each with 30 books. How many books are there?
Solution
Each shelf has 30 books
6 shelves have 30 × 6 = 180 books.
Therefore, there are 180 books.
Example 13
Juma’s
mother has a garden with Tomatoes, Cabbages and Water Lemons. There are
4 rows of Tomato each with 30 in it. 6 rows of Cabbages with 25 in each
and 3 rows of Water Lemo
Solution
There are 30 Tomatoes in each row
4 rows will have 30 × 4 = 120 Tomatoes
Each row has 25 Cabbages
6 rows have 25 × 6 = 150 Cabbages
Each row has 15 Water Lemons
3 rows have 15 × 3 = 45 Water Lemons
In total there 120 + 150 + 45 = 315 plants.
Therefore in Juma’s mother garden there are 315 plants.
Example 14
A
school shop collects sh 90 000 from customers each day. If sh 380 000
from the collection of 6 days was used to buy books. How much money was
left?
Each day the collection is sh 90 000
6 days collection is sh 90 000 × 6 = sh 540 000
The money left will be = Total collection – Money used
= sh 540 000 – sh 380 000 = sh 160 000
Therefore the money left was sh 160 000
Exercise 1
1. For each of the following numbers write the place value of a digit in a bracket.
- 899 482 (4)
- 1 940 (0)
- 9 123 476
2. Write the numerals for each of the following problems.
- Ten thousand and fifty one.
- Nine hundred thirty millio
Factors And Multiples Of Numbers
Factors of a Number
Find factors of a number
Consider
two numbers 5 and 6, when we multiply these numbers i.e. 5 6 the answer
is 30. The numbers 5 and 6 are called factors or divisors of 30 and
number 30 is called a multiple of 5 and 6. Therefore factors are the
divisors of a number.
Example 15
Find all factors of 12
Note that, when listing the factors we don’t repeat any of it.
Multiples of a Number
Find multiples of a number
Multiples
of a number are the products of a number. For example multiples of 4
are 4, 8, 12, 16, 20, 24, … This means multiply 4 by 1, 2, 3, 4, 5, 6, …
.
Example 16
Example
List all multiples of 6 between 30 and 45.
Solution
The multiples of 6 are: 6×1 = 6; 6×2 = 12; 6×3 = 18; 6×4 = 24; 6×5 = 30; 6×6 = 36; 6×7 = 42; 6×8 = 48 and so on.
Therefore, multiples of 6 between 30 and 45 are 36 and 42.
Factors to Find the Greatest Common Factors(GCF) of Numbers
Use factors to find the greatest common factors(GCF) of numbers
Greatest
Common Factor is sometimes called Highest Common Factor. Its short form
is (GCF) or (HCF) respectively. The Greatest Common Factor is the
largest common divisor of two or more numbers given.
For
example if you are told to find the GCF of 15 and 25. First, list all
factors or divisors of 15 and that of 25. Thus, factors of 15 are 1, 3,
5, 15factors of 25 are 1, 5, 25
The common factors are 1 and 5. Therefore the GCF is 5.
Example 17
Example1
Find the HCF of 72 and 120.
Solution
We have to list factors of our numbers:
Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
The common factors are 1, 2, 3, 4, 6, 8, 12, 24.
Therefore the HCF of 72 and 120 is 24.
Another method which can be used to find the GCF or HCF is prime factorization method.
Example 18
Example
Find the HCF of 36 and 48.
Solution
We have to find prime factors of 36 and 48 first
thus, 36 = 2×2×3×3.
48 = 2×2×2× 2×3.
After
writing the numbers as a product of their prime factors, take only the
common prime factors (prime factors appeared to all numbers). In our
example the common factors are 2×2×3. Therefore the HCF of 36 and 48 is
12.
Lowest Common Multiple (LCM)
Lowest
Common Multiple is also called Least Common Multiple and its short form
is LCM.For example: multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40,
45, 50, 55, …multiples of 3 are 3, 9, 15, 18, 21, 24, 27, 30, 33, ….
When
you look carefully at these multiples of 5 and 3 you notes that 15 had
appeared to both. This multiple which appear to both is called a common
multiple. If there are more than one common multiples which appeared the
smaller common multiple is what is called Lowest Common Multiple.
Example 19
Find the common multiples and then show the Lowest Common Multiple of the numbers 4, 6 and 8.
Solution
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, …
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, …
The common multiples of 4, 6 and 8 are 24, 48, …
Therefore the Lowest Common Multiple of 4, 6, and 8 is 24.
We
can find the Lowest Common Multiple (LCM) of numbers by writing the
numbers as a product of their prime factors. The method is called prime
factorization.
Example 20
Find the LCM of 24 and 36 by prime factorization method.
Solution
Let us find prime factors of each number by dividing the numbers by their prime factors.
Now
take the prime factors which appear to both numbers i.e. 2×2×3 (we take
without repeating). We are left with 2 which is a multiple of 24 and 3
which is a multiple 36. We have also to multiply these prime factors
left i.e. 2×2×2×3×3. This gives 72. Therefore the LCM of 24 and 36 is
2×2×2×3×3= 72.
Integers
Identify integers
Consider a number line below
The
numbers from 0 to the right are called positive numbers and the numbers
from 0 to the left with minus (-) sign are called negative numbers.
Therefore all numbers with positive (+) or negative (-) sign are called
integers and they are denoted by Ζ. Numbers with positive sign are
written without showing the positive sign. For example +1, +2, +3, …
they are written simply as 1, 2, 3, … . But negative numbers must carry
negative sign (-). Therefore integers are all positive and negative
numbers including zero (0). Zero is neither positive nor negative
number. It is neutral.
The
numbers from zero to the right increases their values as the increase.
While the numbers from zero to the left decrease their values as they
increase. Consider a number line below.
If
you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater
than 2. We use the symbol ‘>’ to show that the number is greater than
i. e. 3 >2(three is greater than two). And since 2 is to the left of
3, we say that 2 is smaller than 3 i.e. 2<3. The symbol ‘<’ is
use to show that the number is less than.
Consider
numbers to the left of 0. For example if you take -5 and -3. -5 is to
the left of -3, therefore -5 is smaller than -3. -3 is to the right of
-5, therefore -3 is greater than -5.
Generally, the number which is to the right of the other number is greater than the number which is to the left of it.
If
two numbers are not equal to each to each other, we use the symbol ‘≠’
to show that the two numbers are not equal. The not equal to ‘≠’ is the
opposite of is equal to ‘=’.
Example 21
Represent the following integers Ζ on a number line
- 0 is greater than Ζ and Ζ is greater than -4
- -2 is less than Ζ and Ζ is less than or equal to 1.
Solution
a.
0 is greater than Ζ means the integers to the left of zero and Ζ is
greater than -4 means integers to the left of -4. These numbers are -1,
-2 and -3. Consider number line below
b.
-2 is less than Ζ means integers to the right of -2 and Ζ is less than
or equal to 1 means integers to the left of 1 including 1. These
integers are -1, 0 and 1. Consider the number line below
Example 22
Put the signs ‘is greater than’ (>), ‘is less than’ (<), ‘is equal to’ (=) to make a true statement.
Addition of Integers
Add integers
Example 23
2 + 3
Show a picture of 2 and 3 on a number line.
When
drawing integers on a number line, the arrows for the positive numbers
goes to the right while the arrows for the negative numbers goes to the
left. Consider an illustration bellow.
The
distance from 0 to 3 is the same as the distance from 0 to -3, only the
directions of their arrows differ. The arrow for positive 3 goes to the
right while the arrow for the negative 3 goes to the left.
Example 24
-3 + 6
Solution
Subtraction of Integers
Subtract integers
Since
subtraction is the opposite of addition, if for example you are given
5-4 is the same as 5 + (-4). So if we have to subtract 4 from 5 we can
use a number line in the same way as we did in addition. Therefore 5-4
on a number line will be:
Take five steps from 0 to the right and then four steps to the left from 5. The result is 1.
Multiplication of Integers
Multiply integers
Example 25
2×6 is the same as add 2 six times i.e. 2×6 = 2 + 2 + 2 + 2 + 2 +2 = 12. On a number line will be:
Multiplication
of a negative integer by a negative integer cannot be shown on a number
line but the product of these two negative integers is a positive
integer.
From
the above examples we note that multiplication of two positive integers
is a positive integer. And multiplication of a positive integer by a
negative integer is a negative integer. In summary:
- (+)×,(+) = (+)
- (-)×,(-) = (+)
- (+)×,(-) = (-)
- (-)×,(+) = (-)
Division of Integers
Divide integers
Example 26
6÷3 is the same as saying that, which number when you multiply it by 3 you will get 6, that number is 2, so, 6÷3 = 2.
Therefore
division is the opposite of multiplication. From our example 2×3 = 6
and 6÷3 = 2. Thus multiplication and division are opposite to each
other.
Dividing
two integers which are both positive the quotient (answer) is a
positive integer. If they are both negative also the quotient is
positive. If one of the integer is positive and the other is negative
then the quotient is negative. In summary:
- (+)÷(+) = (+)
- (-)÷(-) = (+)
- (+)÷(-) = (-)
- (-)÷(+) = (-)
Mixed Operations on Integers
Perform mixed operations on integers
You
may be given more than one operation on the same problem. Do
multiplication and division first and then the rest of the signs. If
there are brackets, we first open the brackets and then we do division
followed by multiplication, addition and lastly subtraction. In short we
call it BODMAS. The same as the one we did on operations on whole
numbers.
Example 27
9÷3 + 3×2 -1 =
Solution
9÷3 + 3×2 -1
=3 + 6 -1 (first divide and multiply)
=8 (add and then subtract)
Example 28
(12÷4 -2) + 4 – 7=
Solution
(12÷4 -2) + 4 – 7
=1 + 4 – 7 (do operations inside the brackets and divide first)
=5 – 7 (add)
=2
TOPIC 2: FRACTIONS
A fraction is a number which is expressed in the form of a/b where a - is the top number called numerator and b- is the bottom number called denominator.
A Fraction
Describe a fraction
A fraction is a number which is expressed in the form of a/b where a - is the top number called numerator and b- is the bottom number called denominator.
Consider the diagram below
The shaded part in the diagram above is 1 out of 8, hence mathematically it is written as 1/8
Example 1
(a) 3 out of 5 ( three-fifths) = 3/5
Example 2
(b) 7 0ut of 8 ( i.e seven-eighths) = 7/8
Example 3
- 5/12=(5 X 3)/(12 x 3) =15/36
- 3/8 =(3 x 2)/(8 X 2) = 6/16
Dividing the numerator and denominator by the same number (This method is used to simplify the fraction)
Difference between Proper, Improper Fractions and Mixed Numbers
Distinguish proper, improper fractions and mixed numbers
Proper fraction -is a fraction in which the numerator is less than denominator
Example 4
4/5, 1/2, 11/13
Improper fraction -is a fraction whose numerator is greater than the denominator
Example 5
12/7, 4/3, 65/56
Mixed fraction -is a fraction which consist of a whole number and a proper fraction
Example 6
(a) To convert mixed fractions into improper fractions, use the formula below
(b)To convert improper fractions into mixed fractions, divide the numerator by the denominator
Example 7
Convert the following mixed numbers into improper fractions
Comparison of Fractions
In
order to find which fraction is greater than the other, put them over a
common denominator, and then the greater fraction is the one with
greater numerator.
A Fraction to its Lowest Terms
Simplify a fraction to its lowest terms
Example 8
For the pair of fractions below, find which is greater
Solution
Equivalent Fractions
Identify equivalent fractions
Equivalent Fraction
- Are equal fractions written with different denominators
- They are obtained by two methods
<!-- [if !supportLists]-->(a) <!--[endif]-->Multiplying the numerator and denominator by the same number
<!-- [if !supportLists]-->(a) <!--[endif]-->Dividing the numerator and denominator by the same number (This method is used to simplify the fraction
NOTE: The fraction which cannot be simplified more is said to be in its lowest form
Example 9
Simplify the following fractions to their lowest terms
Solution
Fractions in Order of Size
Arrange fractions in order of size
Example 10
Arrange in order of size, starting with the smallest, the fraction
Solution
Put them over the same denominator, that is find the L.C.M of 3, 7, 8 and 9
Addition of Fractions
Add fractions
Operations on fractions involves addition, subtraction, multiplication and division
- Addition and subtraction of fractions is done by putting both fractions under the same denominator and then add or subtract
- Multiplication of fractions is done by multiplying the numerator of the first fraction with the numerator of the second fraction, and the denominator of the first fraction with the denominator the second fraction.
- For mixed fractions, convert them first into improper fractions and then multiply
- Division of fractions is done by taking the first fraction and then multiply with the reciprocal of the second fraction
- For mixed fractions, convert them first into improper fractions and then divide
Example 11
Find
Solution
Subtraction of Fractions
Subtract fractions
Example 12
Evaluate
Solution
Multiplication of Fractions
Multiply fractions
Example 13
Division of Fractions
Divide fractions
Example 14
Mixed Operations on Fractions
Perform mixed operations on fractions
Example 15
Example 16
Word Problems Involving Fractions
Solve word problems involving fractions
Example 17
- Musa is years old. His father is 3¾times as old as he is. How old is his father?
- 1¾of a material are needed to make suit. How many suits can be made from
TOPIC 3: DECIMAL AND PERCENTAGE
The Concept of Decimals
Explain the concept of decimals
A
decimal- is defined as a number which consist of two parts separated by
a point.The parts are whole number part and fractional part
Example 1
Example 2
Conversion of Fractions to Terminating Decimals and Vice Versa
Convert fractions to terminating decimals and vice versa
The first place after the decimal point is called tenths.The second place after the decimal point is called hundredths e.t.c
Consider the decimal number 8.152
NOTE
- To convert a fraction into decimal, divide the numerator by denominator
- To convert a decimal into fraction, write the digits after the decimal point as tenths, or hundredths or thousandths depending on the number of decimal places.
Example 3
Convert the following fractions into decimals
Solution
Divide the numerator by denominator
Operations and Decimals
Addition of Decimals
Add decimals
Example 4
Evaluate
Solution
Subtraction of Decimals
Subtract decimals
Example 5
Solution
Multiplication of Decimals
Multiply decimals
Example 6
Solution
Mixed Operations with Decimals
Perform mixed operations with decimals
Example 7
Solution
Word Problems Involving Decimals
Solve word problems involving decimals
Example 8
If 58 out of 100 students in a school are boys, then write a decimal for the part of the school that consists of boys.
Solution
We can write a fraction and a decimal for the part of the school that consists of boys.
| fraction | decimal |
| 58/100 | 0.58 |
Expressing a Quantity as a Percentage
Express a quantity as a percentage
The
percentage of a quantity is found by converting the percentage to a
fraction or decimal and then multiply it by the quantity.
Example 9
NOTE:The
concept of percentage of a quantity can be used to solve the problems
involving percentage increase and decrease as shown in the below
examples:-
A Fractions into Percentage and Vice Versa
Convert a fraction into percentage and vice versa
To change a fraction or a decimal into a percentage, multiply it by 100%
Example 10
Convert the following fractions into percentages
A Decimal into Percentage and Vice Versa
Convert a decimal into percentage and vice versa
To change a percentage into a fraction or a decimal, divide it by 100%
Example 11
Convert the following percentages into decimals
Percentages in Daily Life
Apply percentages in daily life
Example 12
In an assignment, Regina scored 9 marks out of 12. Express this as a percentage
Solution
Example 13
A school has 400 students of which 250 are girls. What percentage of the students are not girls?
Solution
TOPIC 7: ALGEBRA
An
algebraic expression – is a collection of numbers, variables, operators
and grouping symbols.Variables - are letters used to represent one or
more numbers
Symbols to form Algebraic Expressions
Use symbols to form algebraic expressions
The parts of an expression collected together are called terms
Example
- x + 2x – are called like terms because they have the same variables
- 5x +9y – are called unlike terms because they have different variables
An algebraic expression can be evaluated by replacing or substituting the numbers in the variables
Example 1
Evaluate the expressions below, given that x = 2 and y = 3
Example 2
Evaluate the expressions below, given that m = 1 and n = - 2
An expression can also be made from word problems by using letters and numbers
Example 3
A rectangle is 5 cm long and w cm wide. What is its area?
Solution
Let the area be A.
Then
A = length× widith
A = 5w cm2
Simplifying Algebraic Expressions
Simplify algebraic expressions
Algebraic expressions can be simplified by addition, subtraction, multiplication and division
Addition and subtraction of algebraic expression is done by adding or subtracting the coefficients of the like terms or letters
Coefficient of the letter – is the number multiplying the letter
<!--[endif]-->Multiplication and division of algebraic expression is done on the coefficients of both like and unlike terms or letters
Example 4
Simplify the expressions below
Solution
Equations with One Unknown
An equation – is a statement that two expressions are equal
An Equation with One Unknown
Solve an equation with one unknown
An equation can have one variable (unknown) on one side or two variables on both sides.
When you shift a number or term from one side of equation to another, its sign changes
- If it is positive, it becomes negative
- If it is negative, it becomes positive
Example 5
Solve the following equations
Solution
An Equation from Word Problems
Form and solve an equation from word problems
Some word problems can be solved by using equations as shown in the below examples
Example 6
Naomi is 5 years young than Mariana. The total of their ages 33 years. How old is Mariana?
Solution
Mariana is 19 years
Equations with Two Unknowns
Simultaneous Equations
Solve simultaneous equations
Simultaneous equations – are groups of equations containing multiple variables
Example 7
Examples of simultaneous equation
A simultaneous equation can be solved by using two methods:
- Elimination method
- Substitution method
ELIMINATION METHOD
STEPS
- Choose a variable to eliminatee.g x or y
- Make sure that the letter to be eliminated has the same coefficient in both equations and if not, multiply the equations with appropriate numbers that will give the letter to be eliminated the same coefficient in both equations
- If the signs of the letter to be eliminated are the same, subtract the equations
- If the signs of the letter to be eliminated are different, add the equations
Example 8
Solve the following simultaneous equations by elimination method
Solution
- Eliminate y
To find y put x = 2 in either equation (i) or (ii)
From equation (i)
(b)Eliminate x
In order to find y, put x = 2 in either equation (i) or (ii)
From equation (ii)
(c) Given
To find g put r = 3 in either equation (i) or (ii)
From equation (i)
(d) Given
To find x, put y = - 1 in either equation(i) or (ii)
From equation (ii)
BY SUBSTITUTION
STEPS
- Make the subject one letter in one of the two equation given
- Substitute the letter in the remaining equation and proceed as in case of elimination
Example 9
Solve the following simultaneous equations by substitution method
Solution
Linear Simultaneous Equations from Practical Situations
Solve linear simultaneous equations from practical situations
<!--[endif]-->Simultaneous equations can be used to solve problems in real life involving two variables
Example 10
If
3 Mathematics books and 4 English books weighs 24 kg and 5 Mathematics
books and 2 English books weighs 20 kg, find the weight of one
Mathematics book and one English book.
Solution
Let the weight of one Mathematics book = x and
Let the weight of one English book = y
To find y, put x = 2.29 in either equation (i) or (ii)
From equation(i).
An
inequality – is a mathematical statement containing two expressions
which are not equal. One expression may be less or greater than the
other.The expressions are connected by the inequality symbols<,>,≤
or≥.Where< = less than,> = greater than,≤ = less or equal and ≥ =
greater or equal.
Linear Inequalities with One Unknown
Solve linear inequalities in one unknown
An
inequality can be solved by collecting like terms on one side.Addition
and subtraction of the terms in the inequality does not change the
direction of the inequality.Multiplication and division of the sides of
the inequality by a positive number does not change the direction of the
inequality.But multiplication and division of the sides of the
inequality by a negative number changes the direction of the inequality
Example 11
Solve the following inequalities
Solution
Linear Inequalities from Practical Situations
Form linear inequalities from practical situations
To represent an inequality on a number line, the following are important to be considered:
- The endpoint which is not included is marked with an empty circle
- The endpoint which is included is marked with a solid circle
Example 12
Compound statement – is a statement made up of two or more inequalities
Example 13
Solve the following compound inequalities and represent the answer on the number line
Solution
TOPIC 8: NUMBERS
A Rational Number
Define a rational number
ARational Numberis a real number that can be written as a simple fraction (i.e. as aratio). Most numbers we use in everyday life are Rational Numbers.
| Number | As a Fraction | Rational? |
|---|---|---|
| 5 | 5/1 | Yes |
| 1.75 | 7/4 | Yes |
| .001 | 1/1000 | Yes |
| -0.1 | -1/10 | Yes |
| 0.111... | 1/9 | Yes |
| √2(square root of 2) | ? | NO ! |
The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they arenot rationalthey are calledIrrational.
The Basic Operations on Rational Numbers
Perform the basic operations on rational numbers
Addition of Rational Numbers:
To
add two or morerational numbers, the denominator of all the rational
numbers should be the same. If the denominators of all rational numbers
are same, then you can simply add all the numerators and the denominator
value will the same. If all the denominator values are not the same,
then you have to make the denominator value as same, by multiplying the
numerator and denominator value by a common factor.
Example 1
1⁄3+4⁄3=5⁄3
1⁄3 +1⁄5=5⁄15 +3⁄15 =8⁄15
Subtraction of Rational Numbers:
To
subtract two or more rational numbers, the denominator of all the
rational numbers should be the same. If the denominators of all rational
numbers are same, then you can simply subtract the numerators and the
denominator value will the same. If all the denominator values are not
the same, then you have to make the denominator value as same by
multiplying the numerator and denominator value by a common factor.
Example 2
4⁄3 -2⁄3 =2⁄3
1⁄3-1⁄5=5⁄15-3⁄15=2⁄15
Multiplication of Rational Numbers:
Multiplication
of rational numbers is very easy. You should simply multiply all the
numerators and it will be the resulting numerator and multiply all the
denominators and it will be the resulting denominator.
Example 3
4⁄3x2⁄3=8⁄9
Division of Rational Numbers:
Division
of rational numbers requires multiplication of rational numbers. If you
are dividing two rational numbers, then take the reciprocal of the
second rational number and multiply it with the first rational number.
Example 4
4⁄3÷2⁄5=4⁄3x5⁄2=20⁄6=10⁄3
Real Numbers
Real Numbers
Define real numbers
he
type of number we normally use, such as 1, 15.82, −0.1, 3/4,
etc.Positive or negative, large or small, whole numbers or decimal
numbers are all Real Numbers.
They are called "Real Numbers" because they are not Imaginary Numbers.
Absolute Value of Real Numbers
Find absolute value of real numbers
The
absolute value of a number is the magnitude of the number without
regard to its sign. For example, the absolute value of 𝑥 𝑜𝑟 𝑥
written as 𝑥 . The sign before 𝑥 is ignored. This is because the
distance represented is the same whether positive or negative. For
example, a student walking 5 steps forward or 5 steps backwards will be
considered to have moved the same distance from where she originally
was, regardless of the direction.
The 5 steps forward (+5) and 5 steps backward (-5) have an absolute value of 5
Thus |𝑥| = 𝑥 when 𝑥 is positive (𝑥 ≥ 0), but |𝑥| = −𝑥 when 𝑥 is negative (𝑥 ≤ 0).
For example, |3| = 3 since 3 is positive (3 ≥ 0) And −3 = (−3) =3 since −3 is negative (3 ≤ 0)
Related Practical Problems
Solve related practical problems
Example 5
Solve for 𝑥 𝑖𝑓 |𝑥| = 5
Solution
For any number 𝑥, |𝑥| = 5, there are two possible values. Either 𝑥,= +5 𝑜𝑟 𝑥 = 5
Example 6
Solve for 𝑥, given that |𝑥 + 2| =4
Solution
TOPIC 9: RATIO, PROFIT AND LOSS.
Ratio
A ratio – is a way of comparing quantities measured in the same units
Examples of ratios
- A class has 45 girls and 40 boys. The ratio of number of boys to the number of girls = 40: 45
- A football ground 100 𝑚 long and 50 𝑚 wide. The ratio of length to the width = 100: 50
NOTE: Ratios can be simplified like fractions
- 40: 45 = 8: 9
- 100: 50 = 2: 1
A Ratio in its Simplest Form
Express a ratio in its simplest form
Example 1
Simplify the following ratios, giving answers as whole numbers
Solution
A Given Quantity into Proportional Parts
Divide a given quantity into proportional parts
Example 2
Express the following ratios in the form of
Solution
To increase or decrease a certain quantity in a given ratio, multiply the quantity with that ratio
Example 3
- Increase 6 𝑚 in the ratio 4 ∶ 3
- Decrease 800 /− in the ratio 4 ∶ 5
Solution
Profits and Loss
Profit or Loss
Find profit or loss
If
you buy something and then sell it at a higher price, then you have a
profit which is given by: Profit = selling price − buying price
If
you buy something and then sell it at a lower price, then you have a
loss which is given by: Loss = buying price − selling price
The profit or loss can also be expressed as a percentage of buying price as follows:
Percentage Profit and Percentage Loss
Calculate percentage profit and percentage profit and percentage loss
Example 4
Mr. Richard bought a car for 3, 000, 000/− and sold for 3, 500, 000/−. What is the profit and percentage profit obtained?
Solution
Profit= selling price − buying price = 3,500,000-3,000,000=500,000
Therefore the profit obtained is 500,000/-
Example 5
Eradia bought a laptop for
Solution
But buying price = 780, 000/− and loss = buying price − selling price = 780, 000 − 720, 000 = 60, 000/−
Simple Interest
Calculate simple interest
The amount of money charged when a person borrows money e. g from a bank is called interest (I)
The amount of money borrowed is called principle (P)
To calculate interest, we use interest rate (R) given as a percentage and is usually taken per year or per annum (p.a)
Example 6
Calculate the simple interest charged on the following
- 850, 000/− at 15% per annum for 9 months
- 200, 000/− at 8% per annum for 2 years
Solution
Real Life Problems Related to Simple Interest
Solve real life problems related to simple interest
Example 7
Mrs.
Mihambo deposited money in CRDB bank for 3 years and 4 months. A t the
end of this time she earned a simple interest of 87, 750/− at 4.5% per
annum. How much had she deposited in the bank?
Solution
Given I = 87, 750/− R = 4.5% % T = 3 years and 4 months
Change months to years
TOPIC 10: COORDINATE OF A POINT
Read the coordinates of a point
Coordinates
of a points – are the values of 𝑥 and 𝑦 enclosed by the bracket which
are used to describe the position of a point in the plane
The
plane used is called 𝑥𝑦 − plane and it has two axis; horizontal axis
known as 𝑥 − axis and; vertical axis known as 𝑦 − axis
A Point Given its Coordinates
Plot a point given its coordinates
Suppose
you were told to locate (5, 2) on the plane. Where would you look? To
understand the meaning of (5, 2), you have to know the following rule:
Thex-coordinate (alwayscomes first. The first number (the first coordinate) isalwayson the horizontal axis.
A Point on the Coordinates
Locate a point on the coordinates
The location of (2,5) is shown on the coordinate grid below. Thex-coordinate is 2. They-coordinate is 5. To locate (2,5), move 2 units to the right on thex-axis and 5 units up on they-axis.
The order in which you writex- andy-coordinates in an ordered pair is very important. Th ex-coordinate always comes first, followed by they-coordinate. As you can see in the coordinate grid below, the ordered pairs (3,4) and (4,3) refer to two different points!
Gradient (Slope) of a Line
The Gradient of a Line Given Two Points
Calculate the gradient of a line given two points
Gradient
or slope of a line – is defined as the measure of steepness of the
line. When using coordinates, gradient is defined as change in 𝑦 to the
change in 𝑥.
Consider two points 𝐴 (𝑥1, 𝑦1)and (𝐵 𝑥2, 𝑦2), the slope between the two points is given by:
Example 1
Find the gradient of the lines joining:
- (5, 1) and (2,−2)
- (4,−2) and (−1, 0)
- (−2,−3) and (−4,−7)
Solution
Example 2
- The line joining (2,−3) and (𝑘, 5) has gradient −2. Find 𝑘
- Find the value of 𝑚 if the line joining the points (−5,−3) and (6,𝑚) has a slope of½
Solution
Equation of a Line
The Equations of a Line Given the Coordinates of Two Points on a Line
Find the equations of a line given the coordinates of two points on a line
The equation of a straight line can be determined if one of the following is given:-
- The gradient and the 𝑦 − intercept (at x = 0) or 𝑥 − intercept ( at y=0)
- The gradient and a point on the line
- Since only one point is given, then
- Two points on the line
Example 3
Find the equation of the line with the following
- Gradient 2 and 𝑦 − intercept −4
- Gradient −2⁄3and passing through the point (2, 4)
- Passing through the points (3, 4) and (4, 5)
Solution
Divide by the negative sign, (−), throughout the equation
∴The equation of the line is 2𝑥 + 3𝑦 − 16 = 0
The equation of a line can be expressed in two forms
- 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and
- 𝑦 = 𝑚𝑥 + 𝑐
Consider the equation of the form 𝑦 = 𝑚𝑥 + 𝑐
𝑚 = Gradient of the line
Example 4
Find the gradient of the following lines
- 2𝑦 = 5𝑥 + 1
- 2𝑥 + 3𝑦 = 5
- 𝑥 + 𝑦 = 3
Solution
Intercepts
The line of the form 𝑦 = 𝑚𝑥 + 𝑐, crosses the 𝑦 − 𝑎𝑥𝑖𝑠 when 𝑥 = 0 and also crosses 𝑥 − 𝑎𝑥𝑖𝑠 when 𝑦 = 0
See the figure below
Therefore
- to get 𝑥 − intercept, let 𝑦 = 0 and
- to get 𝑦 − intercept, let 𝑥 = 0
From the line, 𝑦 = 𝑚𝑥 + 𝑐
𝑦 − intercept, let 𝑥 = 0
𝑦 = 𝑚 0 + 𝑐 = 0 + 𝑐 = 𝑐
𝑦 − intercept = c
Therefore, in the equation of the form 𝑦 = 𝑚𝑥 + 𝑐, 𝑚 is the gradient and 𝑐 is the 𝑦 − intercept
Example 5
Find the 𝑦 − intercepts of the following lines
Solution
Graphs of Linear Equations
The Table of Value
Form the table of value
The graph of a straight line can be drawn by using two methods:
- By using intercepts
- By using the table of values
Example 6
Sketch the graph of 𝑦 = 2𝑥 − 1
Solution
The Graph of a Linear Equation
Draw the graph of a linear equation
By using the table of values
Simultaneous Equations
Linear Simultaneous Equations Graphically
Solve linear simultaneous equations graphically
Use
the intercepts to plot the straight lines of the simultaneous
equations. The point where the two lines cross each other is the
solution to the simultaneous equations
Example 7
Solve the following simultaneous equations by graphical method
Solution
Consider: 𝑥 + 𝑦 = 4
If 𝑥 = 0, 0 + 𝑦 = 4 𝑦 = 4
If 𝑦 = 0, 𝑥 + 0 = 4 𝑥 = 4
Draw a straight line through the points 0, 4 and 4, 0 on the 𝑥𝑦 − plane
Consider: 2𝑥 − 𝑦 = 2
If 𝑥 = 0, 0 − 𝑦 = 2 𝑦 = −2
If 𝑦 = 0, 2𝑥 − 0 = 2 𝑥 = 1
Draw a straight line through the points (0,−2) and (1, 0) on the 𝑥𝑦 − plane
From the graph above the two lines meet at the point 2, 2 , therefore 𝑥 = 2 𝑎𝑛𝑑 𝑦 = 2
TOPIC 11: PERIMETERS AND AREAS
Perimeters of Triangles and Quadrilaterals
The Perimeters of Triangles and Quadrilaterals
Find the perimeters of triangles and quadrilaterals
Perimeter
– is defined as the total length of a closed shape. It is obtained by
adding the lengths of the sides inclosing the shape. Perimeter can be
measured in 𝑚𝑚 , 𝑐𝑚 ,𝑑𝑚 ,𝑚,𝑘𝑚 e. t. c
Examples
Example 1
Find the perimeters of the following shapes
Solution
- Perimeter = 7𝑚 + 7𝑚 + 3𝑚 + 3𝑚 = 20 𝑚
- Perimeter = 2𝑚 + 4𝑚 + 5𝑚 = 11 𝑚
- Perimeter = 3𝑐𝑚 + 6𝑐𝑚 + 4𝑐𝑚 + 5𝑐𝑚 + 5 𝑐𝑚 + 4𝑐𝑚 = 27 𝑐𝑚
Circumference of a Circle
The Value of Pi ( Π)
Estimate the value of Pi ( Π)
The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as3.14159.
It has been represented by the Greek letter "π" since the mid 18th
century, though it is also sometimes spelled out as "pi" (/paɪ/).
The
perimeter of a circle is the length of its circumference 𝑖. 𝑒
𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒. Experiments show that
the ratio of the circumference to the diameter is the same for all
circles
The Circumference of a Circle
Calculate the circumference of a circle
Example 2
Find the circumferences of the circles with the following measurements. Take 𝜋 = 3.14
- diameter 9 𝑐𝑚
- radius 3½𝑚
- diameter 4.5 𝑑𝑚
- radius 8 𝑘𝑚
Solution
Example 3
The circumference of a car wheel is 150 𝑐𝑚. What is the radius of the wheel?
Solution
Given circumference, 𝐶 = 150 𝑐𝑚
∴ The radius of the wheel is 23.89 𝑐𝑚
Areas of Rectangles and Triangles
The Area of a Rectangle
Calculate the area of a rectangle
Area
– can be defined as the total surface covered by a shape. The shape can
be rectangle, square, trapezium e. t. c. Area is measured in mm!,
cm!,dm!,m! e. t. c
Consider a rectangle of length 𝑙 and width 𝑤
Consider a square of side 𝑙
Consider a triangle with a height, ℎ and a base, 𝑏
Areas of Trapezium and Parallelogram
The Area of a Parallelogram
Calculate area of a parallelogram
A parallelogram consists of two triangles inside. Consider the figure below:
The Area of a Trapezium
Calculate the area of a trapezium
Consider a trapezium of height, ℎ and parallel sides 𝑎 and 𝑏
Example 4
The area of a trapezium is120 𝑚!. Its height is 10 𝑚 and one of the parallel sides is 4 𝑚. What is the other parallel side?
Solution
Given area, 𝐴 = 120 𝑚2, height, ℎ = 10 𝑚, one parallel side, 𝑎 = 4 𝑚. Let other parallel side be, 𝑏
Then
Areas of Circle
Calculate areas of circle
Consider a circle of radius r;
Example 5
Find the areas of the following figures
Solution
Example 6
A circle has a circumference of 30 𝑚. What is its area?
Solution
Given circumference, 𝐶 = 30 𝑚
C = 2𝜋𝑟
Ndugu mwalimu/mwanafunzi kama unapata changamoto yoyote ile
kuhusu upatikanaji wa NOTES HIZI kwenye website hii, tafadhari wasiliana nasi kwa
simu no/hotline. 0759-104804 Paschal kabonge
barua pepe. paschalkabonge2@gmail.com
au andika maoni yako kwenye sehemu ya comment hapa chini ya post hii.
Kumbuka: mitihani yote inapatikana bureeee kabisa.
Asante.
ELIMU NI HAKI KWA WOTE.
8 Comments
This is very helpful
ReplyDeleteThank you for visiting our website.
DeleteMathematics Form One Full Notes >>>>> Download Now
Delete>>>>> Download Full
Mathematics Form One Full Notes >>>>> Download LINK
>>>>> Download Now
Mathematics Form One Full Notes >>>>> Download Full
>>>>> Download LINK z3
Nice notes
ReplyDeleteHuda Mustaphe
ReplyDeleteGood
ReplyDeleteThank you for visiting our website.
DeleteMathematics Form One Full Notes >>>>> Download Now
ReplyDelete>>>>> Download Full
Mathematics Form One Full Notes >>>>> Download LINK
>>>>> Download Now
Mathematics Form One Full Notes >>>>> Download Full
>>>>> Download LINK px