FORM ONE MATHEMATICS STUDY NOTES TOPIC 7-9.

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

Algebraic Operations

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

1. 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).

Inequalities

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

Irrational Numbers

Irrational Numbers

Define irrational numbers

An irrational number is areal numberthat cannot be reduced to any ratio between anintegerpand anatural numberq. The union of the set of irrational numbers and the set ofrational numbers forms the set of real numbers. In mathematical expressions, unknown or unspecified irrationals are usually represented byuthroughz. Irrational numbers are primarily of interest to theoreticians. Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.

Examples of irrational numbers are √2 (the square root of 2), the cube root of 3, the circular ratiopi, and the naturallogarithmbasee. The quantities√2andthe cube root of 3are examples ofalgebraic numbers. Pi andeare examples of special irrationals known as atranscendental numbers. The decimal expansion of an irrational number is always nonterminating (it never ends) and nonrepeating (the digits display no repetitive pattern)

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

1. A class has 45 girls and 40 boys. The ratio of number of boys to the number of girls = 40: 45
2. 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

1. 40: 45 = 8: 9
2. 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

1. Increase 6 𝑚 in the ratio 4 ∶ 3
2. 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

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

1. 850, 000/− at 15% per annum for 9 months
2. 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

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