Adding, subtracting, multiplying and dividing

DECIMAL FRACTIONS

Decimal fractions are added in the same way that the whole numbers are added. Since only like decimal fractions can be added, that is hundredths to hundredths, and tenths to tenths, the addends are arranged in a vertical column with the decimal points directly below one another, all the way down to the answer.

Problem: Find the sum of the following numbers: 2.23, 4.8, 9 and 0.067.

2.230

4.800

Annex zeroes and arrange numbers in columns: 9.000

0.067

_______

16.097 answer.

In subtracting decimal fractions we must write decimal fractions so that the decimal point of the minuend, subtrahend and remainder are below each other. Zeroes should be annexed so that both minuend and subtrahend are carried out to the same number of places. Check the answer the same way that you check the subtraction of whole numbers.

In multiplying a decimal fraction or mixed decimals multiply as you do whole numbers. Then, starting at the right, mark off as many decimal places in the product as there are in the multiplier and multiplicand together.

To divide a number by 10 or any power of ten, move the point in the dividend as many places to the left as there are zeroes in the divisor. Add zeroes when needed.

EXERCISES

I Read the following words paying attention to the pronunciation:

hundredth, tenths, arrange, column, subtrahend, annexed.

II Give all possible derivatives of the following words:

to arrange, to start, product, power.

III Form adverbs of the following words by adding suffixes:

power, use, part, equal.

IV. Make up sentences of your own using the words and expressions given below:

in multiplying, in dividing, in subtracting, in the dividend, in the same way, mark off, since, must be written.

V. Answer the following questions:

1. How are decimal fractions added? 2. How do we write decimal fractions when we want to subtract them? 3. How do we check the answer? 4. How do we multiply (divide) decimal fractions? 5. How do we arrange numbers in adding decimal fractions?

VI. Put 5 questions to the text and answer them.

VII. Translate into Ukrainian:

Decimals are fractions which always have a denominator of 10 or some power of 10, such as 100, 1000, etc. The denominator is usually not written down; but a dot or point called the "decimal point" is placed to the right of the digit in the numerator, which is distant from the extreme left of these digits by the number of zeroes in the denominator.

In addition or subtraction of decimals, the decimal points must be placed in a straight one under the other.

In multiplication of decimals, point off as many places in the product as there are decimal places in the multiplier and the multiplicand.

VIII. Translate into English:

При додаванні десяткових дробів треба дотримуватися наступної послідовності: дроби записують один під одним так, щоб у всіх доданків однакові розряди знаходились один під одним і всі коми стояли в одному й тому ж вертикальному стовбчику; справа від десяткових знаків деяких доданків приписують таке число нулів, щоб усі доданки після коми мали однакову кількість цифр.

TEXT 10

WHAT IS PER CENT?

We have already learned two ways of writing fractional parts: common fractions and decimal fractions. Another method is by using per cents.

Per cent tells the number of parts in every hundred. This number is followed by the per cent sign (%). The word "per cent" and sign % actually refer to the denominator of a fraction expressed as hundredths.

When working with per cent, we do not write the word, but use the sign %, 20 per cent is written 20% and so on.

In working with problems involving percentage¹ we must be able to change per cents to decimals and decimals to per cents.

We can change a per cent to a decimal by dropping the per cent sign and moving the decimal point two places to the left. We can change a per cent to a common fraction with the given number as the numerator and 100 as a denominator.

One hundred per cent of quantity is the entire quantity. To find a per cent of a number, change the per cent to the equivalent decimal fraction or common fraction and multiply the number by the fraction. To find the per cent of one number from a second number, form a fraction in which the first number is the numerator and the second number is the denominator. Divide the numerator into the denominator and change the decimal fraction to a per cent. To find a number when a per cent of it is known, change the per cent to an equivalent decimal fraction or common fraction, divide given number by this fraction.

Note:

¹ problems involving percentage — задачі на визначення процента

EXERCISES

I. Read the following words paying attention to the pronunciation:

entire, original, increase, per cent, percentage, method.

II. Make up sentences of your own using the words and expressions given below:

by using of, to find a per cent of, is followed by, must be able to change, must be able to use, must be able to find, be able to explain. .

III. Answer the following questions:

1. What methods are there for writing fractional parts? 2. Where do we put the per cent sign? 3. What does the sign % actually refer to? 4. How do we change a per cent to a decimal fraction?

IV. Put 5 questions to the text and answer them.

TEXT 11

SCALE DRAWING

When an architect makes plan¹ or blueprint of a house, his drawing is much smaller than the actual house. The plan is reduced in size to fit the paper he is using. This process of reducing in size is called drawing to scale. The reduced drawing is known as a scale drawing.

In a scale drawing each line is a definite fractional part of the line it represents. A line in the scale drawing may be one-half of the line it represents, one-forth of it, one-hundredth, one-thousandth, or any other definite part of it. In a scale drawing the scale may be written: 1´´/4=1´. This means that every ¼-inch length on the scale drawing represents a 1-foot length of the original object.

A scale drawing has the same shape as the original, but not the same size. Thus, we say that scale drawing is similar to the actual object.

The maps printed in history or geography books are also scale drawings with all distances in the same ratio to the corresponding distances in the original. For example, on a map the scale may be given as 1´´/2 = 50 miles. On another map the scale may be 1´´-1000 miles. This means that the actual distance between two points which are 1½ apart a map whose scale reading is 1´´/2=50 miles, is 150 miles.

Note:

¹ makes a plan – креслити план

EXERCISES

I. Read the following words paying attention to the pronunciation:

architect, actual, drawing, scale, blueprint, geography, progress, reduce.

II. Make up sentences of your own using the words and expressions given below:

to take a plan, to make a map, to make a blueprint, to fit, to draw to scale, to be similar to.

III. Put 6 questions to the text and answer them:

IV. Translate into Ukrainian:

In choosing a scale, we always pick one that is convenient to work with—not too large for the paper we are drawing, nor too small to measure. The scale depends upon the size of the original object and how much it must be reduced.

Once a scale is chosen, the same scale must be used in drawing all parts of the same object. Whenever a scale drawing is made, the scale being used must be stated. It is usually written at the bottom of the drawing.

V. Translate into English:

Накреслити план деякого предмета на папері в натуральну величину інколи буває важко. Якщо розміри предмета більше листа паперу, на якому цей предмет потрібно зобразити, то такий предмет креслять у зменшеному вигляді. Якщо архітектор складає план будинку чи креслення «на синьці», то його креслення має бути набагато меншим, ніж справжній будинок. Зменшення розміру називається кресленням у масштабі. В масштабному кресленні кожна лінія — це певна частина тієї лінії, що існує в реальності.

TEXT 12

GRAPHS

A graph represents numerical relationship in visual form. By use of a graph we can show the relation between certain sets of numbers in an interesting, pictorial manner so that they can actually be seen.

The most commonly used graphs are: the pictograph, the bar graph, the line graph and the circle graph. In a pictograph, each picture or symbol represents a definite quantity. In a pictograph we use pictures of objects to represent numbers. The length of bars in a bar graph represents numerical facts. The bars are of varying length but of the same width. They are usually used to show¹ size or amount of different items or size or amount of the same item at different times. The bars of a vertical bar graph are drawn straight up and down, that is at right angles with the horizontal base line of the graph. The bars of a horizontal bar graph are drawn across the page.

The line graph shows the changes in a quantity by the rising or falling of a line. The position of the line with relation to² the horizontal and vertical scales represents numerical facts. The line connects a number of points.

An apportionment or distribution graph shows the relationship of all parts of a particular whole. The whole graph represents 100%. A chart which consists of a circle broken down into subdivision is called a circle graph. A circle graph is used to show how all the parts are related to the whole. The entire circle, which equals 360°, represents the entire thing.

Notes:

¹ are used to show – використовуються для того, щоб показати

² with relation to – по відношенню до

EXERCISES

I. Read the following words paying attention to the pronunciation:

graph, pictograph, circle, straight, right, visual, present, item, interesting, time, entire, page, change, chart.

II. Make up sentences of your own using the words and expressions given below:

can present, visual form, with relation to, is used to show, bar graph, line graph, pictograph.

III. Answer the following questions:

1. What does a graph present? 2. What can we do by using a graph? 3. What are the most commonly used graphs? 4. What is the difference between a pictograph and the bar graph? 5. How are the bars of a vertical (horizontal) graph drawn? 6. What do we call a circle or a line graph?

IV. Put 6 questions to the text.

V. Translate into Ukrainian:

Graphs are used very frequently in newspapers, magazines, textbooks and reference books. Graphs picture facts and figures so clearly that one can understand them at a glance. Graph is the picture of mathematical equation. It is a method of showing on squared paper the changes in value of an expression containing unknown quantities when one of the unknown quantities is given various, definite values. Any other unknown quantity is dependent in some way on the value of the first unknown quantity, which is called the independent value.

VI. Translate into English:

Діаграми, що зустрічаються найчастіше — це піктограми, діаграми у вигляді стовбчиків, діаграми у вигляді кола і лінійні діаграми.

Розподільна діаграма показуєт співвідношення усіх частин одного цілого.

Діаграми звичайно використовуються, щоб показати розмір або кількість різних предметів або розмір і кількість одного й того ж предмета у різний час.

PART II. ALGEBRA

TEXT 1

THE NATURE OF ALGEBRA

Algebra is a generalization of arithmetic. Each statement of arithmetic deals with¹ particular numbers: the statement (20+4)²=20²+2×20×4+4²=576 explains how the square of the sum of the two numbers, 20 and 4, may be computed².It can be shown³ that the same procedure applies if the numbers 20 and 4 are replaced by any two other numbers. In order to state the general rule, we write symbols, ordinarily letters, instead of⁴ particular numbers. Let the number 20 be replaced⁵ by the symbol a, which may denote any number, and the number 4 by the symbol b. Then the statement is true⁶ that the square of the sum of any two numbers a and b can be computed by the rule (a+b)²=a²+2a×b+b².

This is a general rule which remains true no matter what⁷ particular numbers may replace the symbols a and b. A rule of this kind is often called a formula.

Algebra is the system of rules concerning the operations with numbers. These rules can be most easily stated as formulas in terms of letters, like the rule given above for squaring the sum of two numbers.

The outstanding characteristic of algebra is the use of letters to represent numbers. Since the letters used represent numbers, all the laws of arithmetic hold for⁸ operations with letters.

In the same way, all the signs which have been introduced to denote relations between numbers and the operations with them are likewise used with letters.

For convenience⁹ the operation of multiplication is generally denoted by dot as by placing the letters adjacent to each other. For example, a×b is written simply as ab.

The operations of addition, subtraction, multiplication, division, raising to a power and extracting roots are called algebraic expressions.

Algebraic expressions may be given a simpler form by combining similar terms. Two terms are called similar, if they differ only in their numerical factor (called a coefficient).

Algebraic expressions consisting of more than one term are called multinomial’s. In particular, an expression of two terms is a binomial, an expression of three terms is a trinomial. In finding the product of multinomial’s we make use of the distributive law.

Notes:

¹ to deal with – мати справу з

² may be computed – може бути вирахувана

³ it can be shown – можна показати

⁴ instead of – замість

⁵ let the number 20 be replaced – давайте замінимо число 20

⁶ then the statement is true – тоді справедливе твердження

⁷ no matter what – не залежно від того, який

⁸ hold for – годяться для

⁹ for convenience – для зручності

EXERCISES

I. Read the following words paying attention to the pronunciation:

concern, length, letters, generally, mental, check, arithmetic, width, inch, its, division, addition, which, consider, close, total, cost, only.

II. Form nouns and translate them into Russian:

add, divide, multiply, subtract, operate, state, express, represent, introduce.

III. Form adverbs of the following words by adding the suffix -ly and translate them into Russian:

general, ordinary, particular, simple, similar, different.

IV. Make up sentences of your own using the words and expressions given below:

to deal with, concerning, it can be shown, may be computed, remains true, for convenience, to square, in particular, to extract a root, in terms of letters.

V. Answer the following questions:

1. What is the relationship between arithmetic and algebra? 2. In what operations in arithmetic do we use numbers? 3. What do we use in algebra to represent numbers? 4. What may a formula be considered? 5. What examples of the close relationship between arithmetic and algebra can you give?

VI. Translate into Ukrainian:

Algebra is used in a many walks of life, from that of the philosopher to that of the manual labor. The skilled worker may use algebra to determine the location of the centre or the size of holes he must drill. Doctors, engineers, and scientists use algebra in their research.

But the use of algebra we can reduce complex problems to simple formulas. We can find the answer to problems about the universe, and problems of sewing, building, cooking, measuring, buying and selling as well.

VII. Translate into English:

Алгебра – це система правил, що стосуються дій з числами. В алгебрі числа позначаються буквами, а не цифрами. Оскільки букви позначають числа, всі закони арифметики є придатними для дій з буквами. Знаки, які означають дії з цифрами, також вживаються для букв.

TEXT 2

SIGNS USED IN ALGEBRA

In algebra, the signs plus (+) and minus (-) have their ordinary meaning, indicating addition and subtraction and also serve to distinguish¹ between opposite kinds of numbers, positive (+) and negative (-). In such an operation as +10-10=0, the minus sign means that the minus 10 is combined with the plus 10 to give a zero result² or that 10 is subtracted from 10 to give a zero remainder.

The so-called “double sign” (±), which is read “plus-or-minus”, is sometimes used. It means that the number or symbol which it precedes may be “either plus or minus”³ or “both plus and minus”⁴.

As in arithmetic, the equality sign (=) means “equals” or “is equal to”.

The multiplication sign (×) has the same meaning as in arithmetic. In many cases, however, it is omitted. A dot (·) placed between any two numbers a little above the line (to distinguish it from a decimal point) is sometimes used as a sign of multiplication.

The division sign (÷) has the same meaning as in arithmetic. It is frequently replaced by the fraction line; thus 6/3 means the same as 6÷3 and in both cases the result or quotient is 2. The two dots above and below the line in the division sign (÷) indicate the position of the numerator and denominator in a fraction, or the dividend and divisor in division.

Parentheses ( ), brackets [ ], braces { }, and other inclosing signs are used to indicate that everything between the two signs is to be treated as⁵ a single quantity and any sign placed before it refers to everything inside as a whole and to every part of the complete expression inside.

Another sign which is sometimes useful is the sign which means “greater than” or “less than”. The sign (>) means “greater than” and the sign (<) means “less than”. Thus, a>b means that “a is greater than b”, and 3<5 means “3 is less than 5”.

The sign .·., three dots at the corners of a triangle, means “hence” or “therefore”.

Notes:

¹ serve to distinguish – слугують для того, щоб розрізняти

² to give a zero result – дати у результаті нуль

³ either plus and minus – чи плюс, чи мінус

⁴ both plus and minus – як плюс, так і мінус

⁵ is to be treated as – слід розглядати як

EXERCISES

I. Read the following words paying attention to the pronunciation:

algebra, also, double, triangle, product, quotient, quantity, frequently, sign, minus, combine, twice, inside, sum, number, meaning, between, complete, parentheses, arithmetic, fraction, subtraction, operation.

II. Form nouns of the following words:

to indicate, to add, to operate, to subtract, to mean, to express, to divide, to place, to differ.

III. Make up sentences of your own using the words and expressions given below:

serve to distinguish, to give a zero result, to give a zero remainder, combine with, both plus and minus, either plus or minus.

IV. Answer the following questions:

1. What signs are used in algebra? 2. What do signs (+) and (-) indicate? 3. How is the sign (±) read? 4. What is the equality sign? 5 What is the meaning of the multiplication sign? 6. What is the meaning of the division sign? 7. What does the expression (a+b) mean?

V. Translate into Ukrainian:

ab means the same as a×b and 2×c means the same as 2c, twice c. We cannot write 23, however, for 2×3 as 23 has another meaning, namely, the number twenty three. Therefore, in general, the multiplication sign (×) may be omitted between algebraic symbols or between an algebraic symbol and an ordinary arithmetical number, but not between two arithmetical numbers.

Another sign which is sometimes used is the inclined fraction line (/); thus 6/3 means the same as 6:3. This form has the advantage of being compact and also allowing both dividend and divisor (or numerator and denominator) to be written or printed on the same line.

VI. Translate into English:

В алгебрі ми вживаємо наступні знаки: плюс, мінус, знак рівності, знак множення, знак ділення, дужки круглі, квадратні і фігурні, знак «більше, ніж», знак «меньше, ніж» та інші. Знак три крапки по кутам трикутника означає «отже» або «тому».

TEXT 3

EQUATIONS

An equation is a statement of equality. The statement may be true for¹ all values of the letters.

The value of the letters for which the equation is true is the root or solution of the equation.

When a statement of equality of this kind is given, our interest is in finding² the value of the letter for which it is true. The following rules aid in finding the root.

1. The roots of an equation remain the same if the same expression is added to or subtracted from both sides of the equation.

2. The roots of an equation remain the same if both sides of the equation are multiplied or divided by the same expression other than zero and not involving the letter whose value is in question³.

The equation 2x = 4 where x is the unknown, is true for x=2. To illustrate the first of the above two rules, add 5x to both sides of the equation 2x = 4. We get 2x+5x = 4+5x which, like equation 2x = 4 is true for only x = 2. To illustrate the importance of the restriction in the second of the above two laws, multiply both sides of the equation by x and get (2x)x = (4)x which is true not only for

x = 2 but also for x = 0.

It is always a good plan to check the accuracy⁴ of one's work by substituting the result in the original equation to see whether the equation is true for this value.

Rule 1 is applied very frequently. It is, therefore, desirable to state it in a way which mechanizes its application.

If the equation 4x = 28-3x is given, in applying Rule 1, 3x may be added to both sides of the equation, yielding 4x+3x = 28-3x+3x = 28.

The result of the operation consists in omitting⁵ the term +3x to the left side. We call this operation transposition of the term 3x. This operation is an application of Rule 1 and may be explained in the following way:

Any term of one side of an equation may be transposed to the other side if its sign is changed.

Example. Find the value of x which satisfies 3x+ 7(4-x)+6x = 15. Clearing of parentheses and combining terms:

3x + 28 – 7x+ 6x = 15,

2x+28 =15.

Transposing +28 from the left side:

2x = 15 - 28,

2x = - 13.

Dividing each side by 2, according to Rule 2:

2x/2=-13/2; x=-13/2.

An equation which can be reduced to the form ax+b =0 (a≠0), is called a linear equation in x.

To solve an equation containing fractions, first reduce each fraction to its lowest terms. Then multiply each side of the equation by the least common denominator of all the denominators. This process is called clearing of fractions.

A quadratic equation is one which can be reduced to the form ax²+bx+c = 0 (a≠0) where a, b and c are known and x is unknown.

Notes:

¹ may be true for – може бути дійсним для, правильним, справедливим

² our interest is in finding – нам цікаво знайти

³ in question – невідоме шукане

⁴ to check the accuracy – щоб перевірити точність

⁵ consists in omitting – полягає в опускнні

EXERCISES

I. Read the following words paying attention to the pronunciation:

subtract, illustrate, result, where, which, parentheses, other, whether, always, way, statement, remain, same, equation.

II. Give the Infinitives of the following words:

is, given, finding, got, saw, known.

III. Make up sentences of your own using the words and expressions given below:

may be true for, in finding, in question, by substituting, to consist in omitting, a linear equation, the least common denominator.

IV. Answer the following questions:

1. What is an equation? 2. What are the expressions on either side of the sign of equality called? 3. What should be done to keep the balance of the equation? 4. How do we check an equation? 5. What operations must one do when solving an equation by the combination of rules?

V. Translate into Ukrainian:

If the same number is added to each side of an equation, the equality of the two sides is not altered. If the same number is subtracted from each side of an equation, the equality of the two sides is not altered. If both sides of an equation are multiplied by the same number, the equality of the two sides is not altered. If both sides of an equation are divided by the same number, the equality of the two sides is not altered.

VI. Translate into English:

Корінь рівняння залишається таким же, якщо до обох частин рівняння додати або від обох частин рівняння відняти один і той же вираз. Корінь рівняння залишається таким же, якщо обидві частини рівняння помножити чи поділити на один і той же вираз.

Розв'язати рівняння – значить знайти ті (those) значения невідомого, при яких обидві частини рівняння дорівнюють одному й тому ж числу (іншими словами, всі ті значения невідомого, при яких рівність буде вірною). Кажуть, що ці значення невідомого задовольняють рівнянню. Значення невідомого, які задовольняють рівнянню, називаються коренями або розв'язками рівняння.

TEXT 4

MONOMIAL AND POLYNOMIAL

Algebraic expressions are divided into two groups according to the last algebraic operation indicated.

A monomial is an algebraic expression whose last operation in point of order is neither addition nor¹ subtraction.

Consequently, a monomial is either a separate number represented by a letter or by a figure, e.g. -a, +10, or a product, e.g. ab, (a+b)c, or a quotient, e.g. (a-b)/c, or a power, e.g. b2, but it must never be either a sum or a difference.

If a monomial is a quotient, it is called a fractional monomial; all the other monomials are called integral monomials. Thus, (a-b)/c is a fractional monomial, while (x—y)ab; a(x+y)² are integral monomials.

An algebraic expression which consists of several monomials connected by the + and - signs, is known as a polynomial². Such is for instance, the expression

ab-a+b-10+(a-b)/c.

Terms of a polynomial are separate expressions which form the polynomial by the aid of the + and — signs. Usually, the terms of a polynomial are taken with the signs prefixed to them; for instance, we say: term -a, term +62, and so on. When there is no sign before the first term it is ab or +ab.

A binomial is an algebraic expression of two terms; a trinomial is an expression of three terms and so on.

Notes:

¹ neither ...nor – ні…ні

² is known as a polynomial – відомий як багаточлен, називається багаточленом

EXERCISES

I. Read the following words paying attention to the pronunciation:

fractional, integral, binomial, trinomial, monomial, polynomial, divided, indicated, represented, connected.

II. Give words of the same root as:

Model : operate v; operation n; operative a

serve, express, indicate, divide, represent, connect.

III. Point out the nouns, adjectives and adverbs and write them down in three columns:

algebraic, integral, addition, last, while, point, several, separate, sign, easily, fractional, difference, term.

IV. Make up sentences of your own using the words and expressions given below:

neither addition nor subtraction, neither sum nor difference, either monomial or polynomial, either multiplication or division.

V. Answer the following questions:

1. Into how many groups are algebraic expressions divided? 2. What is a monomial algebraic expression? 3. By what is a monomial represented? 4. What algebraic expression is called polynomial? 5. What are terms of a polynomial?

VI. Translate into Ukrainian:

An algebraic expression of one term is called a monomial or simple expression. An algebraic expression of more than one term is called a polynomial; a polynomial of two terms is called a binomial.

3a+2b and x2—y2 are binomials. a+b+c is a trinomial.

VII. Translate into English:

Алгебраїчний вираз, що містить тільки дії множення, ділення та піднесення у ступінь, називається одночленом.

Алгебраїчна сума кількох одночленів називається багаточленом. Двочлен – це алгебраїчний вираз, що складається з двох членів, тричлен – алгебраїчний вираз, що складається з трьох членів.

TEXT 5