Adding, subtracting, multiplying and dividing the whole numbers

The result of additions of numbers is called the sum or total of the numbers. The numbers to be added¹ are called the addends. In adding a series of numbers, begin with the column at the right. If the sum of a column of digits is ten or larger, carry the tens digits and add it to the sum of the digits in the next column to the left. Careless mistakes are sometimes made because the work was not checked. It is always wise therefore to check your answer.

In subtracting whole numbers, the number which is to be made smaller², or diminished is called the minuend; the number "taken away" or subtracted is called subtrahend. The answer is the difference between the minuend and the subtrahend and it is called the remainder, or difference. In checking a subtraction example, add the remainder and the subtrahend. If your answer is correct, the result obtained by addition equals the minuend.

In multiplication, the number by which you multiply is called the multiplier, the number being multiplied is called multiplicand. The number resulting from the multiplication is called the product. Multiplication can be checked by interchanging the multiplier and multiplicand and multiplying again. Remember that the product of any number multiplied by zero is zero. The product of any number multiplied by one is the same number. The order in which numbers are multiplied does not change the product.

In division, the number that is to be divided is called the dividend. The number by which the dividend is to be divided is called the divisor. The answer is called the quotient. The remainder is what is left over after the dividend has been divided into equal parts. If there is a remainder, it may be written over the divisor and expressed as a fraction in the quotient.

Notes:

¹ the numbers to be added - числа, которые надо сложить

² is to be made smaller- должно быть уменьшено

EXERCISES

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

to add, addends, adding, to subtract, subtrahend, minuend, remainder, to multiply, product, dividend, divisor, quotient.

II. Give all possible derivatives of the following verbs:

to differ, to check, to answer, to change, to obtain.

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

the numbers to be added, the exercise to be checked, the work to be done, the number to be divided, can be made, can be divided, can be checked.

IV. Answer the following questions:

1. How is the result of addition called? 2. What do we do while adding a series of numbers? 3. Why do we sometimes make mistakes in adding numbers? 4. What is the result of subtracting whole numbers called? 5. How do we check a subtraction example? 6. What is the result of multiplication called? 7. What is the result of division called?

V. Make up 6 questions to the text and answer them.

VI. Translate into Russian:

Signs of Operations Used in Arithmetic The signs most used in arithmetic to indicate operations with numbers are plus (+), minus (—), multiplication (X), and division ( : ) signs. When either of these is placed between any two numbers it indicates respectively that the sum, difference, product, or quotient of the two numbers is to be found. The equality sign (=) shows that any indicated operation or combination of numbers written before it (on the left) produces the result or number written after it.

VII. Learn by heart:

Five times five are twenty five; five times six are thirty; five times seven are thirty five; five times eight are forty; five times nine are forty five; five times ten are fifty; five times eleven are fifty five; five times twelve are sixty; six times nine are fifty four; six times ten are sixty; seven times nine are sixty three, seven times ten are seventy; eight times nine are seventy two; eight times ten are eighty; nine times nine are eighty one; nine times ten are ninety.

VIII. Translate into English:

Числа, которые нужно сложить, называются слагаемыми, а результат сложения, т.е. число, получающееся от сложения, называется суммой.

Вычитанием называется действие, посредством которого (by means of which) по данной сумме и одному данному слагаемому отыскивается другое слагаемое.

Число, которое умножают, называется множимым; число, на которое умножают, называется множителем. Результат действия, т.е. число, полученное при умножении, называется произведением.

Число, которое делят, называется делимым; число, на которое делят, называется делителем; число, которое получается в результате деления, называется частным.

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FRACTIONS AND THEIR MEANING

A fraction represents a part of one whole thing. A fraction indicates that something has been cut or divided into a number of equal parts. For example, a pie has been divided into four equal parts. If you eat one piece of the pie, you have taken one part out of four parts. This part of the pie can be represented by the fraction 1/4. The remaining portion of the pie, which consists of¹ three of the four equal parts of the pie, is represented by the fraction² 3/4.

In a fraction the upper and lower numbers are called the terms of the fraction. The horizontal line separating the two numbers in each fraction is called the fraction line. The top term of a fraction or the term above the fraction line is called numerator; the bottom term or the term below the fraction line is called the denominator.

A fraction may stand for³ part of a group. There is a group 5 apples. Each is-1/5 (one fifth) of the group. If we take away 2 apples, we say that we are removing 2/5 of the number of apples present. If we take away 3 apples, we are removing 3/5 of the apples present. In this instance, a fraction is being used to stand for a part of a group.

A fraction also indicates division. For example: one apple was divided into eight parts and the man has eaten one part. Therefore he has eaten 1/8 of the apple. How much of the apple is left? How many eighths are in the whole apple?

Principle to Remember. If in any fraction the numerator and denominator are equal, the fraction is equal to 1.

Notes:

¹ to consist of — состоять из

² is represented by (the fraction) — представлено (дробью)

³ may stand for- может означать

EXERCISES

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

sign, piece, upper, numerator, denominator, number, fraction.

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

remaining portion, equal, to apply, to consist of, represented by, may stand for, is being used, to indicate.

III. Answer the following questions:

1. What does a fraction represent? 2. What do we call "the terms of fractions"? 3. What is the numerator? (denominator?) 4. What does a fraction indicate? 5. When is the fraction equal to 1?

IV. Translate into Russian:

The horizontal line separating the two numbers in each fraction is called the fraction line. The number above the fraction line is the numerator and that below is the denominator of the fraction. The denominator names the fractional unit and the numerator indicates the number of those units contained in the fraction.

V. Translate into English:

Дробь представляет собой часть целого. Число, стоящее над чертой, называется числителем дроби. Число, стоящее под чертой, называется знаменателем дроби. Числитель и знаменатель называется членами дроби.

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TYPES OF FRACTIONS

Common Fraction. A common fraction is a number that has the numerator and the denominator represented by numbers placed the one above and the other below a horizontal line. 3/7 is a common fraction.

Proper Fraction. If the numerator of a fraction is less than denominator the fraction is called a proper fraction. The value of a proper fraction is always less than 1. 6/7, 1/5 and 9/10 are proper fractions.

Improper Fraction. If the numerator of a fraction is equal to¹ or larger than the denominator, the fraction is called a improper fraction. The value of an improper fraction is equal to or larger than 1. 5/3, 3/2, 8/8 are improper fractions.

Mixed number. A number which consists of a whole number and a fraction is called a mixed number.2¼, 5¾, 9¼ are mixed numbers.

Reducing a Fraction to Lower Terms. For convenience and clarity a fraction must always be expressed in its simplest form. That is, it must be reduced to its lowest terms. To reduce a fraction to its lowest terms², divide the numerator and the denominator by the largest number that will divide into both of them evenly.

The process of crossing all common factors out of numerator and denominator is called the redaction of a fraction to its lowest terms. The greatest (largest) quantity which is a common divisor of two or more quantities is called a greatest common divisor of these quantities. It is written G.C.D.

Notes:

1 is equal to – равно

2 to reduce a fraction to its lowest terms —здесь, для того, чтобы сократить дробь

EXERCISES

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

reduce, value, both, other, mixed, proper, improper.

II. Give all possible derivatives of the following words:

value, convenience, to represent, to express, to divide.

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

evenly, to reduce to, for the convenience, expressed in, is equal to.

IV. Answer the following questions:

1. What is a common fraction called? 2. What is a proper fraction called? 3. Is the value of a proper fraction more or less than 1? 4. What do we call mixed numbers? 5. How do you reduce a fraction to its lower terms?

V. Put 6 questions to the text.

VI. Translate into Russian:

Fractions indicate division, the numerator being a dividend, the denominator a divisor, and the value of the fraction the quotient.

A fraction can be reduced to lower terms if the numerator and the denominator are divisible by a single number, that is if they have a common divisor. In order to reduce a fraction to its lowest terms, therefore, it is seen at once that the greatest common divisor must be used.

VII. Translate Into English:

Дробь, у которой числитель меньше знаменателя, называется правильной дробью. Правильная дробь меньше единицы.

Дробь, у которой числитель равен знаменателю или больше его, называется неправильной дробью. Таким образом, неправильная дробь или равна единице, или больше ее.

Числа, которые состоят из целого числа и дроби, называются смешанными числами.

Сокращением дроби называется замена ее другой, равной ей дробью с меньшими членами, путем деления числителем и знаменателем на одно и тоже число. Это число называется наибольшим общим делителем.

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