Addition, subtraction, multiplication and division of fractions
To add fractions having the same denominator (like fractions) add their numerators and write the sum over the common denominator (do not add the denominators). Reduce the Resulting fraction to lowest terms.
To add fractions having different denominators (unlike fractions) the fractions must be changed1 to equivalent fractions which have the same or a common denominator. The least number which will be a common denominator, for example, of 2/3 and 3/5 is 15. 15 is the least common denominator, or lowest common denominator of 2/3 and 3/5. The least common denominator is sometimes denoted by the letters L.C.D.
To subtract fractions having the same denominator subtract the numerators and write the difference over the common denominator (do not subtract denominators).
To subtract fractions having different denominators first change the fractions to equivalent fractions having a common denominator. To subtract the fractions when they have a common denominator, subtract the numerators and write the difference over the denominator.
To multiply a mixed number and a fraction: 1) reduce the fraction to its lowest terms; 2) change the mixed number to an improper fraction; 3) multiply the two numerators to obtain the numerator of the answer; 4) multiply the denominators to obtain the denominator of the answer; 5) reduce the fraction obtained when possible. Reduction can be done by dividing a numerator and denominator by the same number. The numbers that are divided are crossed out, and the quotients are written as the new numerator and the new denominator.
To divide a whole number by a fraction, multiply the whole number by the denominator of the fraction and divide the result by the numerator of the fraction.
Note:
1 must be changed - должны быть превращены.
EXERCISES
I. Read the following words paying attention to the pronunciations:
subtract, unlike, multiply, result, change, cross, equivalent, quotient.
II. Form verbs of the following nouns:
subtraction, multiplication, division, addition, difference, equality.
III. Make up sentences of your own using the words and expressions given below:
to change to, to cross out, must be changed, can be done, divide the result by, write over, write under.
IV. Answer the following questions:
1. What should one do in order to add fractions having the same denominator? (different denominators?) 2. What should one do in order to subtract fractions having the same denominator (different denominators?) 3. How do you multiply fractions having the same (different) denominators? 4. How do you multiply a mixed number and a fraction?
V. Put 6 questions to the text and answer them.
VI. Translate into Russian:
When fractions have a common denominator, they can be added by simply adding the numerators and writing the sum over the same denominator. Any fractions with a common denominator are subtracted by subtracting the numerator of the subtrahend fraction from that of the minuend fraction, and writing the remainder over the common denominator to form the remainder fraction. Thus to add or subtract fractions, first change them into ones with the L.C.D., and then add or subtract the numerators, writing the result as the numerator of a fraction with the common denominator. This fraction is the desired sum or difference respectively.
To multiply a fraction by a whole number, multiply the numerator by that number, and write the product as the numerator of a new fraction with the same denominator. This fraction is the desired product. In order to divide a fraction by any number, multiply the denominator by that number.
VII. Translate into English:
Чтобы сложить дроби с одинаковыми знаменателями, надо сложить их числители и оставить тот же знаменатель.
Чтобы сложить дроби с разными знаменателями, нужно предварительно привести их к наименьшему общему знаменателю, сложить их числители и написать общий знаменатель.
Чтобы вычесть дробь из дроби, нужно предварительно привести дроби к наименьшему общему знаменателю, затем из числителя уменьшенной дроби вычесть числитель вычитаемой дроби и под полученной разностью написать общий знаменатель.
Чтобы умножить дробь на целое число, нужно умножить на это целое число числитель и оставить тот же знаменатель.
Чтобы разделить дробь на целое число, нужно умножить на это число знаменатель, а числитель оставить тот же.
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CHANGING FRACTIONS
The numerator and denominator of a fraction may be multiplied by the same number without changing the value of the fraction. The resulting equivalent fraction is actually the same fraction expressed in higher terms.
To change a mixed number to an improper fraction we must: 1) multiply the denominator of the fraction by the whole number; 2) add the numerator of the fraction to the product of the multiplication; 3) write the result over the denominator.
To change an improper fraction to a whole or a mixed number we must divide the numerator by the denominator. If there should be a remainder, write it over the denominator. The resulting fraction should then be reduced to its lowest terms.
To change a whole number to an improper fraction with a specific denominator: 1) multiply the specific denominator and whole number; 2) write the result over the specific denominator.
Comparing Fractions. Fractions can be compared. To compare unlike fractions we must change them to equivalent fractions so that all have like denominators.
When fractions have different numerators but the same denominator, the fraction having the largest numerator has the greatest value.
When fractions have different denominators but the same numerator, the fraction having the largest denominator has the smallest value.
EXERCISES
I. Read the following words paying attention to the pronunciation:
express, high, whole, compare, specific, actually.
II. Give as many derivatives as you can of the following words:
to express, actually, result, high, to remain.
III. Make up sentences of your own using the words and expressions given below:
expressed in higher terms, mixed number, to change a fraction, to change a whole number, can be compared, can be multiplied, without changing, without dividing, without comparing.
IV. Answer the following questions:
1. What is an equivalent fraction? 2. How do you change a mixed number to an improper fraction? 3. How do you change an improper fraction to a whole number or mixed number? 4. How do you change a whole number to an improper fraction with a specific denominator? 5. What must you do to compare unlike fractions? 6. How do you compare fractions?
V. Put 6 questions to the text and answer them.
VI. Translate into Russian:
When denominators and numerators of different fractions are both different, the values of the fractions cannot be compared until they are converted so as to have the same denominators.
Since fractions indicate division, all changes in the terms of a fraction (numerator and denominator) will affect its value (quotient) according to the general principles of division. These relations constitute the general principles of fractions.
VII Translate into English:
Чтобы обратить смешанное число в неправильную дробь, нужно целое число умножить на знаменатель дроби, к произведению прибавить числитель и сделать эту сумму числителем искомой (sought for) дроби, а знаменатель оставить прежним.
Чтобы обратить неправильную дробь в смешанное число, нужно числитель дроби разделить на знаменатель и найти остаток. Частное покажет число целых единиц; остаток взять в качестве числителя, а знаменатель оставить прежним.
Если числитель дроби уменьшить в несколько раз, не изменяя знаменателя, то дробь уменьшится во столько же раз.
Если числитель и знаменатель дроби увеличить в одинаковое число раз, то дробь не изменится.
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DECIMAL FRACTIONS
A decimal fraction is a special type of fraction written without a denominator (which is 10 or a power of 10) but in which the number of figures on the right-hand side of a dot, called the decimal point, indicates whether the denominator is 10 or a higher power of 10; e.g. 2/10 is written as a decimal in the form 0.2, 23/100 as 0.23, and 23/1000 as 0.023.
If any figure of the number is moved one place to the left, the value of the number is multiplied by 10.
To round off a decimal to a particular, inspect the figure to the right of the required place: if it is 5 or over, change the last required digit to the next higher figure and drop all decimals to the right of the required figures, if it is less than 5, drop all decimals to the right of the required figure.
To compare decimal fractions, annex zeroes so that the decimals have the same number of places.
A decimal fraction may be changed to a common fraction by: leaving out the decimal point, writing the decimal number as the numerator and the number shown by the name of the last decimal place as the denominator.
EXERCISES
I. Read the following words paying attention to the pronunciation:
special, place, particular, decimal, indicate, without, whether, higher, power, leave, greatest, figure, require, determine.
II. Form nouns of the following verbs:
to determine, to place, to indicate, to move, to inspect, to require, to compare.
III. Answer the following questions:
1. What is the decimal fraction? 2. How do we write decimal fractions? 3. How do you round off a decimal to a particular place? 4. How do you compare decimal fractions? 5. How do you change decimal fractions?
IV. Put 4 questions to the text and answer them.
V. Translate into Russian:
Since fractional parts of thing may be written either as common fractions or as decimal fractions, we should be able to change a decimal fraction to a common fraction.
To change a decimal fraction to a common fraction take out the decimal point and write the decimal number as the numerator of the fraction. For the denominator write the number as shown by the name of the last decimal place. Reduce the common fraction to lowest terms.
VI. Translate into English:
Дроби, знаменателями которых являются числа, выраженные единицей с последующими нулями (одним или несколькими), называются десятичными. Из двух десятичных дробей та больше, у которой число целых больше; при равенстве целых та дробь больше, у которой число десятых больше и т. д.
Перенесение запятой на один знак вправо увеличивает число в десять раз. Чтобы увеличить его в сто раз, нужно перенести запятую на два знака вправо и т. д.
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