Chapter One: CONCEPT OF MATHEMATICS
Introduction
Mathematics is all around us. Almost everything we do involves Mathematics. It is applied in various fields such as science, engineering, technology, social studies, economics and many others. Numerous devices such as mobile phones, computers, television sets, and satellites are designed and manufactured on the basis of mathematical knowledge. In this chapter, you will learn about the meaning of Mathematics, branches of Mathematics, relationship between Mathematics and other subjects, and the importance of studying Mathematics. The competencies developed will enable you to apply Mathematics knowledge and skills to solve daily life problems in different fields such as Agriculture, Entrepreneurship, Botany, Zoology, Music, History, Business, and many other fields.
Meaning of Mathematics
The word 'Mathematics' comes from the Greek word 'mathema', meaning 'which is learnt' or 'science, knowledge or learning'. Numbers, measurements, shapes of physical objects, and equations form a small part of it. Mathematics can be thought of as the science of the structures, orders, patterns, and relations that have evolved from elementary practices of counting, measuring, and describing the shapes of objects. It can also be thought of as a language of science because by using mathematical reasoning, one can develop an understanding and be able to predict nature. Mathematics has the ability to provide precise expression for every concept that can be formulated using mathematical symbols and structures. The knowledge and skills of Mathematics play a crucial role in understanding the concepts of other subjects, both in sciences and arts.
Branches of Mathematics
Mathematics can be categorized into different branches such as Arithmetic, Algebra, and Geometry.
Arithmetic
Arithmetic is a branch of Mathematics that deals with properties and manipulations of numbers. Manipulation of numbers is achieved through the use of basic mathematical operations namely: addition, subtraction, multiplication, and division.
Algebra
Algebra is a branch of Mathematics in which arithmetic operations are applied to symbols rather than specific numbers. The symbols or letters in algebra are called variables which represent quantities with no fixed values.
Geometry
The word geometry was derived from the Greek word 'Geo', which means 'earth' and 'metry', which means 'measurement'. Therefore, Geometry is a branch of Mathematics which deals with the study of the sizes, shapes, positions, angles, and dimensions of different physical objects. Moreover, properties of points, lines, planes, similarities, congruence, and shapes of different regular objects are also studied in Geometry.
Relationship between Mathematics and other subjects
Every aspect of our life makes use of Mathematics in one way or another. Mathematics plays a major role as a tool for effective understanding of other subjects. Numerous concepts in other subjects and fields are described precisely using Mathematics. Learning Mathematics can also benefit students through developing their problem solving and critical thinking skills. Note that, Mathematics formulas which are used to represent and describe concepts and scenarios in other fields are commonly referred to as mathematical models.
Activity 1.1: Discovering the use of Mathematics in daily lives
- Recall your daily activities, identify and record applications of Mathematics in such activities.
- Use a table or any method of your choice to represent the identified activities and its mathematical skills obtained in step 1.
- Explain the mathematical aspects you have observed in step 2.
Mathematics and Agriculture
Agriculture is closely related to Mathematics. For instance, when farmers want to buy seeds, they need to understand the ratio of seeds that is sufficient per piece of land.
Similarly, the determination of the number of bags of fertilizers needed per acre requires some calculations. In these two examples, Mathematics enables farmers to avoid the wastage of financial resources by purchasing only the required amount of inputs. Mathematics is also used in Agriculture to determine suitable amount of water to be used in irrigation and the spacing between seedlings. Similarly, Mathematics is used to determine the investment, expenditure, and savings in cultivating a specific crop, dividing a piece of land, calculating the cost of labour, and so forth.
Mathematics and Biology
There is a direct relationship between Mathematics and Biology. For example, normal animal weights, rate of respiration, nutritive values of food, and transpiration are a few quantities in Biology that can be calculated using mathematical concepts. Mathematics can also be used to estimate the number of blood cells present in the body, measurement of blood pressure, and counting sex chromosomes, among many others.
Mathematics and Chemistry
Mathematics is widely applied in Chemistry to represent and solve various problems. Some Chemistry activities involve measurement of masses, volumes, lengths, temperatures and densities of matter. Mathematics is also used to measure the constituents of mixtures, balancing chemical equations, among many other applications.
Mathematics and Physics
Physics involves the study of laws, principles, and theorems which governs how matter works. Most of the quantities in Physics are expressed mathematically through formulas. In order to understand how to apply the formulas to solve some physics challenges, knowledge of Mathematics is needed. All the quantities in physics are expressed in numbers and units which have to be manipulated using the concepts and skills of Mathematics. For example, to determine the speed of a moving object, one has to find a ratio between the distance covered and the time taken.
Mathematics and Information and Communications Technology
Mathematics plays a big role in the field of Information and Communications Technology (ICT). Computer programs, applications, software, and computer languages make use of mathematical concepts. A common example of the application of numbers is in defining colours in web development languages. For instance, the HTML (hypertext markup language) and CSS (cascading style sheets) uses colour notations such as #000000 which represent Black, #008000 for Green and #800000 for Maroon. Furthermore, the digits 0 and 1 are commonly used in computing systems in computers.
Mathematics and Business Studies
Mathematics is widely applied in daily life activities related to Business Studies. For example, if one wants to determine profit or loss in business, a difference between the selling and buying prices is to be calculated. If the buying price is less than the selling price, the business makes profit. If the buying price is higher than the selling price, the business makes loss. Mathematics is also applied in other areas such as in loan borrowing, determining prices of items, among many other applications.
Mathematics and Geography
Geographers require mathematical calculations to find the distance from one place to another, finding gradients, altitudes of hills and mountains. Through mathematical calculations, geographical locations of different places are determined using latitudes and longitudes, and real-life objects such as buildings are represented on a map through the use of scales.
Mathematics and History
Mathematics helps in describing various historical activities such as duration of events and expressing historical events which happened at different points in time. For example, a simple mathematical operation is used to determine duration of historical events such as World Wars, colonial periods and time spent by leaders in positions, determining dates and ages of fossils by using some mathematical principles such as carbon-14 among many other applications.
Mathematics and Literature
Mastering basic concepts of arithmetic can enable a person to understand better and manage literary works. A kind of writing which can draw attention to readers depend on several factors such as number of words per sentence, the number of sentences per paragraph, and the number of paragraphs per page. For example, 20 words in a sentence are considered as average. Thus, in analysing writing style, one need to find an average number of words in each sentence, which requires Mathematics concepts.
Mathematics and Music
The knowledge of Mathematics plays a vital role in music. For instance, playing one note in piano has a different sound as when three harmonic notes are played together. Music notes are distinguished mathematically by their different values.
A semibreve has four beats, a minim has two beats, crotchet has one beat, quaver has \(\frac{1}{2}\) a bit, semi quaver has \(\frac{1}{4}\) a bit, and so on. Reading and playing music represented by notes in a music sheet depends on the mathematical ability to understand these values.
Mathematics in Sports and Games
Mathematics is important in all forms of sports and games. For example, in many sports and games, winners are those who have accumulated highest number of points. Furthermore, in most of the sports and games, there are specific number of players in the playing ground. For instance, a football team has 11 players while a netball team has 7 players. Moreover, players in some games such as football are given names in terms of numbers. In football, a goalkeeper is normally assigned number 1. In addition, all playing grounds are prepared with specific standards of measurements.
Activity 1.2: Exploring the use of Mathematics in other fields
Explore various resources such as books and internet to discover examples of mathematical concepts and skills that are used in 3 subjects other than those discussed in this chapter.
Importance of Mathematics
Learning Mathematics and using mathematical skills in daily life enhances problem solving, critical thinking, effective time management among many other importances.
Financial Management
Financial management requires mathematical skills during the preparation of budgets. Calculations are done to ensure the budget prepared correlates with the funds collected.
Problem-solving skills
Problem-solving is one of the most important skills in life. Mathematics is one of the most effective ways to increase analytical and logical thinking, which helps us to become better problem solvers.
Time management
Time management is a key to success for everyone. Therefore, time must be carefully managed. Mathematics helps to determine the time spent on every activity so as to maximize its efficiency.
Activity 1.3: Exploring the importance of Mathematics
Use different sources such as books and internet to explore the importance of Mathematics in daily life.
Exercise 1.1
- What is Mathematics?
- Explain five mathematical skills that are useful in your daily life activities.
- Choose any five subjects of your choice other than Mathematics and explain with vivid examples how each of them is connected to Mathematics.
- Write an essay arguing whether or not learning Mathematics is important in our lives.
Chapter Two: Numbers
Introduction
Numbers play a vital role in everyday life. Different activities are performed with the help of numbers. Numbers are used to quantify and measure quantities. For instance, length, mass, time, volume, and population are some of the quantities represented by numbers. In this chapter, you will learn about rational, irrational, and real numbers. You will also learn about repeating decimals, inequalities and absolute values of real numbers. The competencies developed will enable you to perform daily life activities such as counting things, managing money, distributing items, and interpreting numbers based on contexts, and many other applications.
Concept of numbers
Numbers are classified into different categories. Some categories of numbers that you have already learned include whole numbers, natural numbers, fractions, integers, and decimals. Other major categories of numbers include rational, irrational, and real numbers. Activity 2.1 enables you to identify categories of numbers from daily life activities.
Activity 2.1: Categorising numbers
- Use different measuring tools such as tape measure, ruler, weighing balance, and measuring cylinders to measure lengths, masses, and volumes of different objects, respectively.
- Record the measurements in task 1 and categorise them based on your understanding of categories of numbers.
Rational numbers
A rational number is any number that can be written in the form of \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b \neq 0\). The condition \(b \neq 0\) is essential because division by zero is not defined. The set of rational numbers is denoted by the symbol \(\mathbb{Q}\).
Fractions, integers, whole numbers, terminating and non-terminating decimals form the set of rational numbers. For instance,
\[-\frac{34}{3}, -\frac{20}{3}, -3, -\frac{4}{7}, -\frac{5}{6}, \frac{1}{2}, \frac{1}{3}, \frac{4}{3}, \frac{9}{5}, \frac{100}{5}, 0, \frac{30}{1}, 0.45, 0.3, 0.23\]
and \(1.567\) are rational numbers.
Representation of rational numbers on a number line
Rational numbers can be represented on a number line. Positive rational numbers are represented on the right of zero (the origin) and the negative rational numbers on the left of the origin. The number line helps us to determine other rational numbers between any two rational numbers by increasing the number of divisions. Activity 2.2 enables you to locate numbers on a number line.
Activity 2.2: Locating numbers on a number line
- Prepare a number line (2 to 3 metres long) using sticks and masking tapes or any other materials of your choice.
- Cut manila cards in small sizes and write on them different types of numbers such as whole, integers, and fractions.
- Give the cards to others to post each of the numbers on appropriate position on the number line.
- While sticking the cards, explain to others why you think the positions are appropriate.
The representation of any positive rational number \(\frac{a}{b}\) is done by dividing the unit interval into '\(b\)' equal parts. The '\(a\)' of these parts are taken along the number line to reach the point corresponding to \(\frac{a}{b}\) on the right of zero if the number is positive and to the left of zero if the number is negative.
Example 2.1
Represent \(\frac{1}{4}\) on a number line.
Solution
In order to represent \(\frac{1}{4}\) on a number line, take one unit from 0 towards the right side and then divide that unit into 4 equal parts. Take one part out of the 4 parts to complete a part representing \(\frac{1}{4}\). Therefore, \(\frac{1}{4}\) on a number line is represented as shown in the following number line.
\[\begin{array}{ccc} & & \\ -1 & 0 & \frac{1}{4} \\ \end{array}\]
Example 2.2
Represent \(-2.4\) on a number line.
Solution
\(-2.4 = -2\frac{4}{10} = -2\frac{2}{5}\)
To represent \(-2.4\) on a number line, take 2 units from 0 towards the left side and then divide the third unit into 5 equal parts. Take 2 parts out of the 5 parts to complete a part representing \(-2.4\). Therefore, \(-2.4\) on a number line is represented as shown in the following number line.
\[\begin{array}{ccc} & & \\ -3 & -2.4 & -2 \\ \end{array}\]
Exercise 2.1
- Represent each of the following rational numbers on a number line.
- 5
- \(\frac{7}{3}\)
- \(\frac{1}{5}\)
- 4.5
- \(\frac{3}{4}\)
- \(\frac{3}{2}\)
- 0
- \(3\frac{3}{4}\)
- Show the position of each of the following rational numbers on a number line.
- \(2\frac{2}{3}\)
- \(\frac{6}{1}\)
- \(\frac{16}{8}\)
- \(\frac{0}{3}\)
- \(\frac{25}{6}\)
- \(\frac{25}{50}\)
- \(\frac{5}{5}\)
- \(\frac{50}{25}\)
- Represent each of the following numbers on a number line.
- 3
- \(-\frac{5}{2}\)
- -3
-
- Write all negative and positive rational numbers from the following list of numbers: -5, 0, 12, -3.416520..., 2.85, 7.14, 12.64646464, \(\frac{11}{5}\), \(\frac{3}{5}\), \(-\frac{1}{3}\).
- Which numbers from (a) are not rational numbers? Justify your answer.
- Write each of the following rational numbers in the form of \(\frac{a}{b}\) in its simplest form.
- 0.004
- 4
- -27
- 5.6
- -0.03
- \(\frac{1}{4}\)
- \(-\frac{1}{8}\)
- \(-2\frac{2}{3}\)
Repeating decimals
Decimal numbers are part of rational numbers and are common in our daily life activities. The quantities such as length, height, age, volume and mass can be presented in decimals. Activity 2.3 guides you in expressing various quantities in fractions into decimals.
Activity 2.3: Expressing measurements of quantities in decimals
- Take some fruits or similar objects and divide them into two, three, four, five, six, and seven equal parts.
- Convert each fraction in task 1 into decimals in many decimal places as possible. You can work manually or use a calculator.
- Study carefully the decimal part of the fractions and write down their unique characteristics.
A repeating decimal, also known as recurring decimal is a decimal number with at least one digit in the decimal part that repeats consecutively in a regular order without an end. For example, \(1.6666666 \ldots\) and \(0.639639639639\ldots\) are repeating decimals because 6 and 639 digits from the two decimal numbers, respectively, in the decimal part repeat themselves without an end. The three dots indicate that the repeating digits continue infinitely.
Repeating decimals can also be represented by using a dot or a bar that is placed on top of a repeating digit.
Example 2.4
Write each of the following repeating decimals by using a dot and a bar.
- \(0.3333 \ldots\)
- \(0.639639639 \ldots\)
- \(0.474747474 \ldots\)
Solution
- \(0.3333 \ldots\) is a repeating decimal which can be written as \(0.\dot{3}\) or \(0.\overline{3}\)
- \(0.639639639 \ldots\) can be written as \(0.\dot{6}3\dot{9}\) or \(0.\overline{639}\)
- \(0.474747474 \ldots\) can be written as \(0.\dot{4}7\) or \(0.\overline{47}\)
Decimals are either terminating or non-terminating. Terminating decimals have a definite number of digits after the decimal point while non-terminating decimals have an endless number of digits after the decimal point. Thus, repeating decimals are non-terminating with one or more repeating digits in the decimal part.
Examples of terminating decimals are 0.5, 1.4, and 7.9 while non-repeating decimals are 3.1415926..., 1.4142135..., and 2.2360679...
Converting repeating decimals into fractions
When working with problems involving repeating decimals, it is important to convert them into simple fractions to maintain accuracy and avoid errors. A repeating decimal can be converted into fraction using the following steps:
- Choose any variable to represent the required fraction.
- Multiply both sides of the equation by a multiple of 10 depending on the number of repeating decimals.
- Subtract the equation in step 1 from the equation in step 2.
- From the equation obtained in step 3, solve for the chosen variable and simplify where necessary.
Example 2.5
Convert each of the following decimals into fractions.
- \(0.\dot{3}\)
- \(0.\dot{8}\dot{3}\)
- \(0.\dot{8}3\dot{5}\)
- \(0.8\dot{3}\)
Solution
- Let \(x = 0.\dot{3}\) (i)
Multiply by 10 both sides of equation (i) to obtain,
\(10x = 3.\dot{3}\) (ii)
Subtract equation (i) from equation (ii):
\(10x - x = 3.\dot{3} - 0.\dot{3}\)
\(9x = 3\)
\(x = \frac{1}{3}\)
Therefore, \(0.\dot{3}\) into fraction is \(\frac{1}{3}\). - Let \(x = 0.\dot{8}\dot{3}\) (i)
Multiply by 100 both sides of equation (i) to obtain
\(100x = 83.\dot{8}\dot{3}\) (ii)
Subtract equation (i) from equation (ii):
\(100x - x = 83.\dot{8}\dot{3} - 0.\dot{8}\dot{3}\)
\(99x = 83\)
\(x = \frac{83}{99}\)
Therefore, \(0.\dot{8}\dot{3}\) into fraction is \(\frac{83}{99}\).
Exercise 2.2
- Convert each of the following decimals into fractions.
- 0.1
- 0.8
- 80.217
- 3.112
- 13.015
- 0.34
- 0.723
- 2.4
- Which of the following rational numbers are repeating decimals?
- \(\frac{1}{9}\)
- \(\frac{22}{7}\)
- \(\frac{2}{3}\)
- \(\frac{20}{11}\)
- If \(x = 2.\dot{6}\) and \(y = 2.\dot{8}\dot{3}\), find the value of \(\frac{x}{y}\).
- If \(x = 0.\dot{3}\) and \(y = 0.\dot{6}\), verify that \(y^2 = x^2 + x\).
Irrational numbers
Irrational numbers are special numbers in our life. A widely known and used irrational number is Pi (\(\pi\)) which appears in formulas for determining circumferences, areas, and volumes of circular shapes. Activity 2.4 allows you to use your experience in decimals to learn about irrational numbers.
Activity 2.4: Differentiating types of decimal numbers
- Identify 10 different fractions which can be converted into terminating or repeating decimals.
- Use a calculator to find answers to the square roots of at most 10 numbers that have no perfect squares and write your answers with at least 10 decimal places.
- Compare the answers in tasks 1 and 2 and write down the differences observed.
An irrational number is a number which can be written as a non-terminating and non-repeating decimal. Also, these numbers cannot be expressed in the form of \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). The set of irrational numbers is denoted by \(\mathbb{Q}'\). Irrational numbers cannot be represented exactly on a number line. However, they can always be approximated to rational numbers.
Example 2.10
Write any 6 irrational numbers.
Solution
The following is a list of some irrational numbers.
- \(\sqrt{2} = 1.414213562373 \ldots\)
- \(e = 2.718281828459 \ldots\)
- \(\pi = 3.141592653589 \ldots\)
- \(-\sqrt{5} = -2.236067977499 \ldots\)
- \(\sqrt{11} = 3.316624790355 \ldots\)
- \(-\sqrt{21} = -4.582575694956 \ldots\)
From these examples, it can be observed that the digits after the decimal point continue infinitely without repeating.
Chapter Three: APPROXIMATIONS
Introduction
Understanding approximations is important in our daily life. Approximations are used in various real life situations like estimating cost of items, distances, and ingredients for a recipe. In this chapter, you will learn about rounding off numbers by place values, number of decimal places and by significant figures. Also, you will learn how to perform approximation in calculations. The competencies developed will help you to perform real-life activities such as estimating number of things, analysing large data, making quick calculations, preparing budgets, and many other applications.
Meaning of approximations
Approximation is a process of finding a number that is closest to the exact value. It involves expressing a number into a higher or lower value which is close to the exact value. Approximation can be termed as estimation and is represented by the symbol '\(\approx\)'.
Activity 3.1: Estimating challenge
- In your surrounding environment (class or home), estimate the number of objects or students available or measurements of objects.
- Record your answers and perform actual counting or measurements and find the difference.
- Compare the differences you have noted between the estimated and actual measurement and comment on the differences.
Rounding off numbers
Rounding off numbers is a process of making a number simpler, but keeping its value closer to the original value. The outcome is less precise but more user-friendly. Rounding off the number can be done by considering place values, number of decimal places, and number of significant figures.
Rounding off numbers by place values
Rounding off numbers by place values is done by considering the given place value for both whole and decimal parts of a number. Thus, for whole numbers, it can be done to the nearest ones, tens, hundreds, thousands or any other place value. For decimal numbers, it can be done to the nearest tenths, hundredths, thousandths and so on. The following are basic procedures for rounding off numbers by place values.
Note: A rounding digit is the digit at a place where a number is supposed to be rounded off.
(a) If the digit to the right of the rounding digit is either 0, 1, 2, 3, or 4, then the digit at the required place value remains unchanged, and each digit to the right of it become zeros.
Example 3.1
Round off each of the following numbers according to the given instruction.
- 591,622 to the nearest hundreds.
- 35.4 to the nearest ones.
- 234 to the nearest hundreds.
Solution
- The digit of 591,622 in the hundreds place is 6 and the digit on the right of 6 is 2, which is less than 5. In this case, maintain 6 and replace each digit to the right of 6 with zero. Thus, 591,622 to the nearest hundreds is 591,600.
- 35.4 to the nearest ones is 35.
- 234 to the nearest hundreds is 200.
Example 3.2
Round off:
- 35.49643 to the nearest thousandths.
- 274 to the nearest tens.
- 3.2743 to the nearest hundredths.
- 856,145 to the nearest thousands.
- 42.245468 to the nearest thousandths.
Solution
- 35.49643 rounded off to the nearest thousandths is 35.496.
- 274 rounded off to the nearest tens is 270.
- 3.2743 rounded off to the nearest hundredths is 3.27.
- 856,145 rounded off to the nearest thousands is 856,000.
- 42.245468 rounded off to the nearest thousandths is 42.245.
(b) If the digit to the right of a rounding digit is greater than 5, then 1 is added to the rounding digit and each digit to the right is replaced by zero.
Example 3.3
Round off:
- 0.267 to nearest tenths.
- 17.82 to the nearest ones.
- 63,504 to the nearest thousands.
Solution
- 0.267 to the nearest tenths is 0.3.
- 17.82 rounded off to the nearest ones is 18.
- 63,504 rounded off to the nearest thousands is 64,000.
Example 3.4
The population of a certain country in 1967 census showed that there were 5,834,875 men and 6,111,188 women. Round off the figures to the nearest millions.
Solution
5,834,875 men is rounded off to the nearest millions as 6,000,000 men (1 is added to 5 because the digit on the right of 5 is 8).
6,111,188 women is rounded off to the nearest millions as 6,000,000 women (6 remain unchanged since the digit on the right of 6 is 1 which is less than 5 and each digit on the right of 6 is replaced by zero).
(c) If the digit to the right of a rounding digit is 5, then the rounding digit is treated in three main ways:
- Add 1 to the rounding digit if it is odd.
- Add 1 to the rounding digit (whether even or odd) if there are other digits that follow 5 which are all not zero.
- The digit remains unchanged if all the digits following 5 are zeros and the rounding digit is even.
Example 3.5
Round off each of the following numbers according to the given instructions.
- 2.635 to the nearest hundredths.
- 42,243,568 to the nearest thousands.
- 8,115,684 to the nearest ten thousands.
Solution
- 2.635 rounded off to the nearest hundredths is 2.64 since the rounding digit is odd and the digit to its immediate right is 5.
- 42,243,568 rounded off to the nearest thousands is 42,244,000 since digits after 5 (which is on the right of 3 as a rounding digit) are all not zero.
- 8,115,684 rounded to the nearest ten thousands is 8,120,000 since digits after 5 (which is on the right of 1 as a rounding digit) are all not zero.
Exercise 3.1
- Round off each of the following numbers to the nearest thousands.
- 8,259
- 12,222
- 13,709
- 100,998
- 17,501
- 2,349,673
- 60,500
- 9,999
- 1,234,567
- Round off each of the following numbers to the nearest ones.
- 41.4
- 0.49
- 2.613
- 0.8
- 0.379
- 2.55
- Round off each of the following numbers to the nearest tens.
- 25.12
- 16.15
- 28.929
- 10.0089
- 50.408
- 33.456
- 20.17
- 311.114
- The total mass of cotton harvested in a certain district was 17,816,273 kg. Round off the mass to the nearest:
- Millions of kilograms
- Thousands of kilograms
- In 1983, the number of primary school pupils in a certain region was 237,268. Round off the number of pupils to the nearest thousands.
Rounding off by number of decimal places
Rounding off decimals involves a similar process of rounding numbers by place values. When rounding off decimals, the digits to the right of the given decimal place are dropped rather than being replaced by zeros.
In the process of rounding off, the following steps are used:
(a) If the digit after the rounding digit is either 0, 1, 2, 3, or 4, then the rounding digit remain unchanged, and all the other digits to the right are dropped.
Example 3.8
Round off 0.2464 to 3 decimal places.
Solution
The digit in the third decimal place is 6 and the nearest digit to the right is 4 which is less than 5. In this case, 6 will remain unchanged and 4 is dropped. Therefore, \(0.2464 \approx 0.246\) to 3 decimal places.
(b) If the digit to the right of a rounding digit is greater than 5, then 1 is added to the rounding digit and other digits to the right are dropped.
Example 3.9
Round off 0.97381 to 3 decimal places.
Solution
The digit in the third decimal place is 3 and the nearest digit to the right is 8 which is greater than 5. In this case, 1 is added to the rounding digit and each digit to the right is dropped. Therefore, \(0.97381 \approx 0.974\) to 3 decimal places.
(c) If the digit to the right of a rounding digit is 5, then the rounding digit is considered as follows:
- Add 1 to the rounding digit if it is odd.
- Add 1 to the rounding digit (whether even or odd) if there are other digits that follow 5 which are all not zero.
- The digit remains unchanged if all the digits following 5 are zeros and the rounding digit is even.
Example 3.10
Round off the following numbers.
- 6.2358 to two decimal places.
- 3.4655 to two decimal places.
- 0.2475 to three decimal places.
Solution
- 6.2358 rounded off to two decimal places is 6.24 since the rounding digit is odd and the digit to the right of the rounding digit is 5.
- 3.4655 is rounded off as 3.47 to two decimal places since the rounding digit is followed by 5 and the digit after 5 are not all zero.
- 0.2475 is rounded off as 0.248 to three decimal places since the rounding digit is odd.
Significant figures
A significant figure is a digit in a number that gives information about the precision and accuracy of the measured value.
Rules for significant figures
- Any digit from 1 to 9 appearing in a number is a significant figure.
- Each zero appearing in a number between digits 1 to 9 is a significant figure. For example, 0 in 602 is a significant figure. That is, 602 has 3 significant figures.
- When zeros are written to the right of the last non-zero digit, the zeros are not significant figures. For example, 72,000 has only 2 significant figures.
- In decimals, any zero to the left of the first non-zero digit is not a significant figure. For example, the zeros in 0.025 are not significant figures. Thus, 0.025 has 2 significant figures.
- All zeros that are on the right of a decimal point are significant only if a non-zero digit does not follow them. For example, 0.040 has two significant figures while 0.6700 has four significant figures, and 10.000 has five significant figures.
- When zero is written at the end of an approximated number, including decimals, it is a significant figure. For example, \(2.73 \approx 3.0\), and in this case, 0 is a significant figure. Also, \(45,961 \approx 46,000\), the zero at the third-place value is a significant figure.
- All the zeros that are on the right of the non-zero digit are significant if they come from measurements. For instance, 20 m has two significant figures and 300 g has 3 significant figures.
Note:
- Significant figures are also known as significant digits.
- Significant figures are always counted from left towards the right of a number.
Rounding off numbers by significant figures
Rounding off numbers to a specified number of significant figures is one of the methods of approximations. Similar rules for rounding off numbers are used in writing numbers to a given number of significant figures.
Example 3.14
Determine the number of significant figures in each of the following numbers.
- 2.3004
- 0.0804
- 0.00002
- 4,002,000,000
- 0.00165000
- 3,000 km
Solution
- 2.3004 has 5 significant figures.
- 0.0804 has 3 significant figures.
- 0.00002 has 1 significant figure.
- 4,002,000,000 has 4 significant figures.
- 0.00165000 has 6 significant figures.
- 3,000 km has 4 significant figures.
Example 3.15
Write 38,176 correct to 1 significant figure.
Solution
Counting from left to right, the given number has 5 significant figures. The first digit, 3 is increased by one to become 4 (since 8 is greater than 5), and each of the right-side digits become zero. Thus, \(38,176 \approx 40,000\) correct to 1 significant figure.
Approximations in calculations
When working out some calculations involving numbers, sometimes it is important to find an estimate of the answer. To find an estimated answer, take a suitable approximation by rounding off the numbers involved. A suitable estimation can be to round off a number to the nearest whole number or in simple decimals which enables you to estimate the given problem easily and quickly.
Example 3.18
Estimate the value of \(38 \times 71\).
Solution
For easy estimation, round off 38 and 71 to the nearest tens.
That is, \(38 \times 71 \approx 40 \times 70 = 2,800\).
Therefore, \(38 \times 71 \approx 2,800\).
Example 3.19
Estimate the value of \(256.5 \div 63.5\).
Solution
Round off both numbers to the nearest ones.
\(256.5 \approx 256\) to the nearest ones and \(63.5 \approx 64\) to the nearest ones. It follows that,
\(256.5 \div 63.5 \approx 256 \div 64\)
\(\approx 4\)
Therefore, \(256.5 \div 63.5 \approx 4\).
Exercise 3.4
In question 1 to 10, estimate the value of each of the given expressions.
- \(43 \times 28\)
- \(2,912 \times 32\)
- \(82 \times 61\)
- \(868 \times 31\)
- \(2.94 \times 248\)
- \(171,220 \div 79\)
- \(35,164 \times 23.04\)
- \(1.029 \div 0.021\)
- \(4,981 \div 6,438\)
- \(9,110,218,800 \div 4,081\)
Chapter summary
- Approximating or rounding off a number is a process of writing a number close enough to the exact number.
- During approximation, if the digit to the right is 5, the rounding digit is:
- increased by 1 if it is odd.
- left unchanged if it is even and has zero digits after 5 or has no digit after 5.
- Any digit from 1 to 9 appearing in a number is a significant figure.
- When zeros are written to the right of the last non-zero digit of an exact number, the zeros are not significant figures.
- In decimals, any zero written to the left of the first non-zero digit is not a significant figure.
- All zeros that are on the right of the decimal point are significant, only if a non-zero digit does not follow them.
- When a zero is written at the end of an approximated decimal or number, it is a significant figure.
- All the zeros that are on the right of the last non-zero digit are significant if they come from measurements.
Chapter Four: Ratios and Proportions
Introduction
Ratios and proportions play an important role in our everyday life. They are used in various daily life activities such as in cooking, financial activities, medicine prescription, construction, arts and design, and many more applications. Understanding proportions helps us manage various daily life activities such as balancing diets, distributing things, and planning of events. In this chapter, you will learn about ratios and proportions. The competencies developed will enable you to ensure equal and proportional sharing and distribution of things among other applications.
Ratios
Sharing things is a common practice in our life. For example, people can share portions of land, wealth, and many other things. A required portion of water, fruit paste and sugar can result into a nice juice. Activity 4.1 introduces you with some everyday uses of ratios.
Activity 4.1: Ratios in daily life
- Use various resources to explore carefully the relationship that exists between:
- Human body weight and height
- Circumference of the neck and waist
- Represent these facts mathematically and provide your interpretation.
- Identify 3 more similar facts in everyday life and describe them mathematically.
A ratio is a quantitative comparison between two or more quantities of the same units, normally expressed as a fraction. It shows the number of times one value contains the other. A ratio between the numbers \( p \) and \( q \) is written as \( p : q \), which is equivalent to \( p + q \) or \( \frac{p}{q} \) and is read as \( p \) to \( q \).
Note: Multiplying or dividing both terms of a ratio by the same number does not change the ratio. In other words, 2:4 = 4:8 (multiplying each term of the ratio by 2). The ratio can be reduced to its lowest terms by expressing the ratio as a fraction and simplify it to its lowest term.
Example 4.1
Simplify 48:36.
Solution
Given 48:36
The given ratio can be written in fraction as \( \frac{48}{36} \)
Dividing the numerator and denominator by 12 gives,
\( \frac{48}{36} = \frac{4}{3} \)
Therefore, 48:36 in its simplified form is 4:3.
Example 4.2
Given \( a:b = 1:2 \) and \( b:c = 3:5 \), find \( a:b:c \) and \( a:c \).
Solution
Given \( a:b = 1:2 \) and \( b:c = 3:5 \). \( b \) is in both ratios, so make the two ratios equivalent by multiplying \( a:b = 1:2 \) by 3 and \( b:c = 3:5 \) by 2 as follows:
\( a:b = (1:2) \times 3 = 3:6 \)
\( b:c = (3:5) \times 2 = 6:10 \)
Therefore, \( a:b:c = 3:6:10 \) and \( a:c = 3:10 \).
Example 4.3
Jaha and Siwema shared 40,000 Tanzanian shillings. Jaha received 15,000 Tanzanian shillings and Siwema received 25,000 Tanzanian shillings. Find the ratio of the amount of money each of them received.
Solution
The ratio is written as 15,000 to 25,000 or
\( 15,000 \div 25,000 = \frac{15,000}{25,000} = \frac{3}{5} \)
Therefore, the ratio of the amount of money received by Jaha and Siwema is 3:5.
Example 4.4
A man's monthly income is 1,274,000 Tanzanian shillings. He spends 1,078,000 Tanzanian shillings every month. Find the ratio between his income to expenditure and savings to income.
Solution
Income to expenditure ratio:
Given, income = Tshs 1,274,000 and expenditure = Tshs 1,078,000
Hence, income : expenditure = \( \frac{1,274,000}{1,078,000} = \frac{13}{11} = 13:11 \)
Therefore, the ratio of income to expenditure is 13:11.
Savings to income ratio:
Savings = Tshs (1,274,000 - 1,078,000) = Tshs 196,000
Thus, savings: income = \( \frac{196,000}{1,274,000} = 2:13 \)
Therefore, the ratio of savings to income is 2:13.
Example 4.5
A Mathematics club has 21 members of which 13 are females and the rest are males. Find the ratio of males to all club members.
Solution
Given, total number of club members = 21, number of females = 13
Thus, number of males = total number of club members - number of females = 21 - 13 = 8
Therefore, the ratio of males to all club members is 8:21.
Example 4.6
Express the ratio 200 cm:1m in its lowest form.
Solution
Using comparison of metric units, it is known that 1 m = 100 cm. Since the units must be the same, then it follows that,
200 cm:1 m = 200 cm:100 cm = 2:1
Therefore, the ratio 200 cm:1m in its lowest form is 2:1.
Example 4.7
A special cereal mixture contains rice, wheat, and corn in the ratio 2:3:5, respectively. If a bag of the mixture contains 3 kilograms of rice, how much corn does it contain?
Solution
Let \( x \) be the amount of corn in kilograms that a special cereal mixture contains.
The ratio of rice to corn is given by \( 2:5 = 3 \text{ kg} : x \text{ kg} \)
The ratio can be written in fraction as \( \frac{\text{rice}}{\text{corn}} = \frac{2}{5} = \frac{3}{x} \)
Thus, \( \frac{2}{5} = \frac{3}{x} \)
Cross multiplication gives, \( 2x = 3 \times 5 \)
\( x = \frac{3 \times 5}{2} = 7.5 \)
Therefore, the special cereal mixture contains 7.5 kilograms of corn.
Exercise 4.1
- Express each of the following ratios in its lowest form.
- 8:12
- 40:50
- 100:50
- 77:35
- 48:36
- 160:96
- 144:81
- \( \frac{3}{2} : \frac{1}{4} \)
- \( \frac{14}{2} : \frac{1}{2} \)
- \( \frac{1}{2} : \frac{2}{2} \)
- \( \frac{1}{4} : \frac{1}{7} \)
- \( \frac{2}{5} : \frac{3}{5} \)
- 0.17:1
- 3.5:0.07
- 7:0.007
- 120:105
- Tshs 136 to Tshs 4
- 50 s to 90 s
- 0.75 km to 0.5 km
- 120 km to 105 km
- Complete each of the following ratios.
- 25:____ = 5:6
- 15:20 = 18:____
- 24:8 = ____:7
- 30:6 = ____:9
- 4:7 = 20:____
- 4:9 = ____:63
- 12:____ = 3:7
- ____:8 = 21:24
- A herd of 52 horses consists of white and black horses. If it contains 12 white horses, find the ratio of white horses to black horses.
- The ratio of girls to boys in a swimming club was 2:4. If 14 members were girls, how many members were there in the club?
- If 4M = 5N = 6P, find M:N:P.
Proportions
Proportion is essential in solving real-life problems involving scaling, ratios, and percentages. Such problems involve comparing and relating different quantities. Thus, proportion is a part, a share, or a number considered in comparison to a whole. Engage in Activity 4.2 to recall some real-life activities on proportions.
Activity 4.2: Relating quantities
- In your daily life, recall things or events for which the occurrence of one affects the occurrence of another. Some examples include number of litres of fuel used by a car per kilometre and number of people and time used to complete a task.
- Write your own three examples and show mathematically how the two things or events relate to each other.
The concept of equivalent fractions describes correctly what it means by a proportion. For example, \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent, and hence the two fractions are proportional. Thus, a proportion is a statement which shows that two ratios are equal. That means, if two ratios are proportional, the corresponding parts of each ratio have the same relationship.
Proportions can be written in two ways:
- As two equal fractions; \( \frac{a}{b} = \frac{c}{d} \)
- Using colons; \( a : b = c : d \)
When two ratios are equal, that is \( a : b = c : d \), it follows that \( \frac{a}{b} = \frac{c}{d} \). Thus, \( a \times d = b \times c \).
When three numbers \( a \), \( b \), and \( c \) are in continued proportion, then \( c \) is called the third proportional. The third proportional of two numbers \( a \) and \( b \) is defined as \( a : b = b : c \), that is, \( \frac{a}{b} = \frac{b}{c} \).
Example 4.9
Find the proportional parts of 156 in the ratio \( 3 : 4 : 5 \).
Solution
Given the total number of parts = 156, ratio = \( 3 : 4 : 5 \). The sum of the terms of the ratio is \( 3 + 4 + 5 = 12 \). The required proportional parts are:
\( \frac{3}{12} \times 156 = 39 \)
\( \frac{4}{12} \times 156 = 52 \)
\( \frac{5}{12} \times 156 = 65 \)
Therefore, the required proportional parts are 39, 52, and 65.
Example 4.10
The first, second, third, and fourth terms are proportional. If the first, second, and third terms are 42, 36, and 35, respectively, find the fourth term.
Solution
Let \( x \) be the fourth term.
Thus, 42, 36, 35, \( x \) are in proportion.
Writing the numbers in ratios gives, \( \frac{42}{36} = \frac{35}{x} \)
Cross multiplication gives, \( 42 \times x = 35 \times 36 \)
\( x = \frac{35 \times 36}{42} = 30 \)
Therefore, the fourth term is 30.
Example 4.11
The ratio of the number of girls to that of boys in a school is 7:12. If the number of boys is 1,380, find:
- the number of girls.
- the total number of students in the school.
Solution
(a) Given number of girls: number of boys = 7:12
Number of boys = 1,380
Let \( x \) be the number of girls. It follows that 7:12 = \( x : 1380 \).
Writing in fraction gives, \( \frac{7}{12} = \frac{x}{1380} \)
Cross multiplication gives, \( 12x = 7 \times 1380 \)
\( x = \frac{7 \times 1380}{12} = 805 \)
Therefore, the number of girls is 805.
(b) The total number of students = number of girls + number of boys = 805 + 1380 = 2,185.
Therefore, the total number of students in the school is 2,185.
Example 4.12
In a road safety campaign, it was revealed that 1 out of every 10 drivers in a certain city attended safety driving seminars. If there were 400 drivers in the city, how many drivers have attended the seminars?
Solution
Let \( y \) be the number of drivers who are likely to have attended driving seminars.
The ratio of drivers who have attended safety driving seminars = 1:10
The ratio of the drivers who are likely to have attended the seminar = \( y : 400 \)
According to proportions, the two ratios are equal. Thus, \( \frac{1}{10} = \frac{y}{400} \)
Cross-multiplication gives, \( 10y = 1 \times 400 \)
\( y = \frac{1 \times 400}{10} = 40 \)
Therefore, 40 drivers out of 400 have attended the driving safety seminar.
Exercise 4.2
- Divide each of the following in the given ratios.
- 100 in ratio 7:3
- 75 in the ratio 3:2
- 16 in the ratio 3:3:2
- \( \frac{1}{2} \) in the ratio 4:1
- If 800 kilograms of rice are shared between two families in the ratio 3:2, how much does each family get?
- Divide 60,000 Tanzanian shillings among Juma, Ali, and Aisha in the ratio 5:3:2.
- Musa, Josephine, Mary, and Kato have 300, 100, 500, and 600 shares in a cooperative shop, respectively. Divide 150,000 Tanzanian shillings among them in the ratio of their shares.
- Magnesium combines with oxygen in the ratio 3:2 by mass to form a new substance. What mass of magnesium will be needed to combine with 1.4 kg of oxygen?
- Divide 28.6 kg of meat among four families in the ratio 4:5:6:7.
- A powdery mixture is made up of powders A and B in the ratio 5:4. If 72 kg of this mixture are required, how much of each type should be used?
- An alloy is made up of metals \( x \) and \( y \) in the ratio 2.4:1 by mass. How much mass of \( y \) is required if 8 kg of \( x \) is used to make the alloy?
- If the ratio of lead to tin in a solder is 1.8:1, how much mass of tin is needed when 90 kg of lead is used?
- A line segment AB, which is 24 cm long, is divided at a point P (between A and B) in the ratio 7:5. Find the lengths of \( \overline{AP} \) and \( \overline{PB} \).
Chapter summary
- A ratio is a comparison between two or more quantities which are in the same unit and can be simplified as a fraction.
- A ratio can be expressed as a fraction, using colon or in words.
- A proportion is a statement that two ratios are equal.
- In general, a proportion describes a part, a share or a number considered in comparison to a whole.
Project: Planning for a catering of an event
You have been appointed in a catering committee for preparation of a graduation ceremony. A total number of 250 participants are expected to attend the graduation.
- Plan for the amount of different varieties of food needed for the ceremony.
- Show all the necessary mathematical calculations in determining the total amount of each type of variety of food for the meal.
Chapter Five: Algebra
Introduction
Algebra is a branch of Mathematics that helps to solve real-life problems by using letters, numbers, and symbols. It unifies all branches of Mathematics, and thus, it is a tool used in daily life activities. Algebra is used to understand and work with unknown quantities in various situations. For instance, it helps to calculate expenditure when shopping, plan travel routes, cooking or preparing recipes for different servings. In this chapter, you will learn about algebraic expressions, equations with one unknown, equations with two unknowns, and inequalities with one unknown. The competencies developed will enable you to solve daily life problems such as planning for recipes, scheduling routes, estimating profit and loss, currency conversions, among many other applications.
Algebraic expressions
In algebra, letters or symbols are used to represent numbers. For instance, it is common to represent the length of an object by the letter \( l \), radius by letter \( r \), and diameter by letter \( d \). In addition, letters such as \( x, y, z, \alpha \) and \( \beta \) can be used to represent numbers.
An algebraic expression is a mathematical statement which contains numbers, letters and mathematical operations such as \( +, -, \times, \) and \( \div \). An algebraic expression is made up of terms.
A term can be a number, a variable, a number multiplied by a variable, or a variable multiplied by a variable. For example, in the algebraic expression \( 5x - 3y + 8 \), terms are \( 5x, -3y \) and \( 8 \). Terms with the same variables and exponents are called like terms. For example, \( 8y \) and \( 9y \) or \( 8xy \) and \( 3xy \) or \( 6x^2y \) and \( 2x^2y \) are like terms. Terms with different variables or exponents are called unlike terms. For example, \( 6x \) and \( 6z \) or \( 3mn \) and \( 4vy \) or \( 6u^2y \) and \( 2uy^2 \) are unlike terms.
Simplification of algebraic expressions
Simplification of algebraic expressions involves combining like terms by adding or subtracting their coefficients in order to obtain a simple term. It may also involve multiplying and dividing terms as well as working with fractions and brackets.
Example 5.1
Multiply \( 5a + 2b - 3c \) by 4.
Solution
\( (5a + 2b - 3c) \times 4 = 4(5a + 2b - 3c) \)
\( = 4 \times 5a + 4 \times 2b + 4 \times (-3c) \)
\( = 20a + 8b - 12c \)
Therefore, \( 5a + 2b - 3c \) multiplied by 4 gives \( 20a + 8b - 12c \).
Example 5.2
Multiply \( 2x - 3y \) by \( -2a \).
Solution
\( (2x - 3y) \times (-2a) = (-2a)(2x) + (-2a)(-3y) \)
\( = -4ax + 6ay \)
Therefore, when \( 2x - 3y \) is multiplied by \( -2a \), the result is \( -4ax + 6ay \).
Example 5.3
Re-write \( -5a(m + 2n - 2) \) without brackets.
Solution
\( -5a(m + 2n - 2) = -5am + (-5a \times 2n) + (-5a \times -2) \)
\( = -5am - 10an + 10a \)
Therefore, \( -5a(m + 2n - 2) = -5am - 10an + 10a \).
Example 5.4
Divide \( 4ax + 6ay - 10az \) by \( 2a \).
Solution
\( (4ax + 6ay - 10az) \div 2a = \frac{4ax + 6ay - 10az}{2a} \)
\( = \frac{4ax}{2a} + \frac{6ay}{2a} - \frac{10az}{2a} \)
\( = 2x + 3y - 5z \)
Therefore, \( 4ax + 6ay - 10az \) divided by \( 2a \) is \( 2x + 3y - 5z \).
Exercise 5.1
- Simplify each of the following algebraic expressions and then state the number of terms and its coefficients.
- \( n+n+n+n+n+k+k+k+x+x \)
- \( 3x+4y-7z+3x-7y+2z \)
- \( \frac{1}{2}x+7x-\frac{1}{4}x \)
- Simplify each of the following algebraic expressions.
- \( 12m+13m \)
- \( -5hk(2r-3d-1) \)
- \( 9z+6z-8z+5z \)
- \( 0.5m(3a-2b) \)
- \( 9a-6b+3a \)
- \( 15n-9n \)
- \( x+4-5x \)
- \( 4k-k+3k \)
- \( r+2r+3r+4 \)
- \( 8y-3-7y+48y-3 \)
- \( 5(5x+6m+4y) \)
- \( -2m(-13x+y-6z) \)
Activity 5.1: Using algebraic expressions in daily life experiences
- Explore different resources to discover various daily life activities whose processes can be represented by algebraic expressions.
- Express algebraically the scenarios explored in task 1 and provide justifications for your answers.
Algebraic equations
An algebraic equation is a mathematical statement connecting two algebraic expressions with an equal sign (=). It expresses a statement or a problem in a clear and short way. For example, \( 2x = 16 \) and \( 3x + 4y = 8 \) are algebraic equations. Therefore, an equation has two equal sides, the left-hand side and the right-hand side.
Linear equations
A linear equation is an algebraic equation in which each variable has an exponent of one. A linear equation can have one or more unknown variables. For example, \( 3x - 2 = 7 \) and \( 2z + 1 = 0 \) are linear equations with one variable \( x \) and \( z \), respectively, whereas \( 5x + 2y = 9 \) and \( 2x - 3y = 1 \) are linear equations with two variables \( x \) and \( y \).
Formulation of linear equations
A linear equation can be formulated from a real-life activity or from a given word problem. Formulation of a linear equation from a real-life scenario or word problem involves identifying the unknown, assigning the unknown with a variable, and finally writing the equation based on the given conditions.
The following table shows some common words used in word problems which involve mathematical operations.
| Words | Mathematical symbol |
|---|---|
| Addition, sum, increased by, plus, total, more than, exceed | + |
| Difference, subtract, decreased, reduced by, minus, less than | – |
| Multiplication, times, product, multiplied by | × |
| Division, divided by, quotient, ratio | ÷ |
| Equals, result is, is equal to, is, gives, the same as | = |
Example 5.12
The sum of two numbers is 20. If one of the numbers is 8, form a linear equation connecting the two numbers.
Solution
Let \( x \) be the unknown number. Since the known number is 8, then the algebraic equation connecting the two numbers is \( 8 + x = 20 \).
Solving linear equations with one unknown
Solving an equation with one unknown means finding the value of the unknown variable which satisfy the equation. It involves several steps of simplification to obtain the value of the variable which satisfies the given equation.
Note: Whatever operation is done on the left-hand side of the equation must be done on the right-hand side.
Example 5.16
Find the solution of \( x+5=8 \).
Solution
Given \( x+5=8 \).
Collect the like terms by subtracting 5 from both sides. That is,
\( x+5-5=8-5 \)
\( x=3 \)
Therefore, \( x = 3 \).
Example 5.17
Find the value of \( x \) that satisfies the equation \( x-8=16-2x \).
Solution
Given \( x-8=16-2x \).
Collect the like terms by performing the following:
\( x-8+8=16-2x+8 \) (Add 8 on both sides)
\( x = 16 - 2x + 8 \)
\( x + 2x = 24 - 2x + 2x \) (Add \( 2x \) on both sides)
\( 3x = 24 \)
Divide by 3 on both sides of the equation to get,
\( x = \frac{24}{3} = 8 \)
Therefore, \( x = 8 \).
Linear simultaneous equations
Linear simultaneous equations, also referred to as a system of linear equations, are two or more algebraic equations which share the variables and the solutions. They are called simultaneous equations because they are solved at once (simultaneously) and have common solution.
Consider the following linear equation:
Equation (1) involves two unknowns \( x \) and \( y \). Different values of \( x \) and \( y \) can be used to make the equation true. The values of \( x \) and \( y \) that satisfy equation (1) in ordered pairs \( (x, y) \) are \( (1,19) \), \( (2,18) \), \( (3, 17) \), \( (10,10) \), \( (13.4, 6.6) \) and so on. Therefore, in order to get unique values of \( x \) and \( y \) in equation (1), another linear equation is needed. For instance, if it is also known that
It is possible to use equation (1) and (2) to find the unique values of \( x \) and \( y \) which satisfy both equations. These equations are such that the value of \( x \) in equation (1) is the same as the value of \( x \) in equation (2) and the same applies to \( y \), that is, the equations have common solution.
Solving linear simultaneous equations
There are several methods of solving simultaneous equations. This section discusses how solutions of linear simultaneous equations are obtained by elimination and substitution methods.
Solving linear simultaneous equations by elimination method
Elimination method involves performing suitable mathematical operations to reduce the two equations into a single equation with one unknown. The following are the steps for solving linear simultaneous equations by elimination method:
- Choose the easiest variable to be eliminated. Multiply the given equations by suitable numbers so as to make the coefficients of the chosen variable in both equations equal.
- Add the new equations if the coefficients of the chosen variable to be eliminated are opposite in signs, otherwise subtract them.
- Solve the obtained equations. This gives the value of one of the unknowns.
- Repeat steps 1 to 3 for the value of the second unknown or substitute the solution obtained in step 3 into one of the equations to get the value of the other unknown.
Example 5.30
Solve the following linear simultaneous equations by elimination method.
\( 3x + y = 9 \) (ii)
Solution
Choose the easiest variable to be eliminated. In this case, \( y \) can be eliminated easily.
The coefficient of \( y \) in equation (i) is 1 and in equation (ii) is 1.
Since the signs of the variable \( y \) in (i) and (ii) are the same, subtract equation (ii) from (i) as follows:
\( - (3x + y = 9) \)
\( 3x = 6 \)
It follows that, \( 3x = 6 \)
\( x = 2 \)
Substitute the value of \( x \) into one of the equations to obtain the value of \( y \).
Substitute \( x = 2 \) into the equation (ii) to obtain,
\( 3(2) + y = 9 \)
\( 6 + y = 9 \)
\( y = 3 \)
Therefore, \( x = 2 \) and \( y = 3 \) is the solution of the given simultaneous equations.
Solving linear simultaneous equations by substitution method
Substitution method involves choosing one equation and transposing one of its variables by making it the subject of the other. The resulting equation is substituted into the other to obtain a single equation with one unknown. The following are the steps for solving simultaneous equations of two unknowns by substitution method:
- Express one of the variables in terms of the other from one of the equations.
- Substitute the equation obtained in Step 1 into the second equation to obtain an equation in one unknown.
- Solve the equation in one unknown obtained in Step 2.
- Substitute the value of the variable obtained in Step 3 to one of the equations to get the value of the other unknown.
Example 5.35
Solve the following simultaneous equations by substitution method.
\( y + 2x = 11 \) (ii)
Solution
Using equation (ii), express \( y \) in terms of \( x \) as follows.
From \( y + 2x = 11 \), it implies that,
\( y = 11 - 2x \) (iii)
Substituting equation (iii) into equation (i) gives,
\( 3(11 - 2x) - 2x = -7 \)
Expanding and collecting like terms gives,
\( 33 - 6x - 2x = -7 \)
\( -8x = -40 \)
\( x = \frac{40}{8} = 5 \)
Substitute \( x = 5 \) into equation (iii), that is,
\( y = 11 - (2 \times 5) = 11 - 10 = 1 \)
Therefore, \( x = 5 \) and \( y = 1 \).
Solution of inequalities with one unknown
Solutions of inequalities with one variable are determined by using similar approach as that used in solving linear equations. However, there are some specific aspects to consider when solving inequalities. In most cases, solutions in inequalities are presented in range or as a set. Thus, solving linear inequalities requires adhering to the following rules:
- Adding or subtracting an equal value to both sides does not change the inequality sign.
- Multiplying or dividing on both sides by a positive number does not change the inequality sign.
- Multiplying or dividing on both sides by a negative number changes the inequality sign.
Example 5.41
Solve for the values of \( x \) if \( x - 3 < \frac{1}{2} \).
Solution
Given \( x - 3 < \frac{1}{2} \).
Add 3 to both sides of the inequality to obtain
\( x - 3 + 3 < \frac{1}{2} + 3 \)
\( x < 3 \frac{1}{2} \)
Therefore, \( x < 3 \frac{1}{2} \).
Example 5.42
Solve for the value of \( x \) if \( 5x + 2 > 1 \).
Solution
Given \( 5x + 2 > 1 \).
Subtract 2 from both sides of the inequality. That is,
\( 5x + 2 - 2 > 1 - 2 \)
\( 5x > -1 \)
Divide by 5 on both sides
\( \frac{5x}{5} > -\frac{1}{5} \)
\( x > -\frac{1}{5} \)
Therefore, \( x > -\frac{1}{5} \).
Exercise 5.8
- List the numbers which satisfy each of the following conditions.
- \( x < 6 \) if \( x \) is a counting number.
- \( x \leq 4 \) if \( x \) is a whole number.
- \( x > 3 \) if \( x \) is an odd number.
- \( x > -3 \) and \( x \leq 4 \) if \( x \) is an integer.
- Solve for \( x \) in each of the following inequalities.
- \( 5x > 12 \)
- \( 4 - x < 10 \)
- \( 2 - x > 8 \)
- \( 2x - 5 \leq \frac{1}{2} \)
- \( \frac{x}{2} - \frac{1}{2} < \frac{1}{2} \)
- \( 6x + 4 \geq 2 \)
- \( 4x - \frac{3}{4} > 2x + \frac{1}{4} \)
Chapter summary
- Algebra uses letters or symbols to represent numbers in mathematical statements called algebraic expressions and equations.
- An algebraic expression is formed by one or more terms.
- A term of an algebraic expression can be a single letter, or letters multiplied by numbers, or a constant separated by addition or subtraction.
- The number which multiplies a variable is called the coefficient of the variable.
- Simplification of an algebraic expression is a process of combining like terms to obtain a single term as possible.
- An equation is a statement which connects two expressions with an equal sign (=).
- Simultaneous equations are a set of two or more related equations with multiple unknowns.
- Inequalities are used to compare two values or expressions.
- Common inequalities are less than (<), greater than (>), less than or equal to (\( \leq \)), greater than or equal to (\( \geq \)) and not equal (\( \neq \)).
No comments
Post a Comment