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71. When a number or thing, as an apple or a pear, is divided into two equal parts, one of these parts is called one half. If divided into three equal parts, one of the parts is called one third ; if divided into four equal parts, one of the parts is called one fourth, or one quarter ; if into ten, tenths ; if into a hundred, hundredths, &c.
When a number or thing is divided into equal parts, as halves, thirds, fourths, fifths, &c., these parts are called Fractions. Hence,
72. A FRACTION denotes a part or parts of a number or thing.
An Integer is a whole number.
Note.—The term fraction, is derived from the Latin fractio, which signifies the act of breaking, a broken part or piece. Hence, fractions are sometimes called broken numbers.
73. Fractions are commonly expressed by two numbers, one placed over the other, with a line between them. Thus, one half is written, ž; one third, }; one fourth, 4; three fourths, i ; two fifths, š; nine tenths, *, &c.
The number below the line is called the denominator, and shows into how many parts the number or thing is divided.
Quest.–71. What is meant by one half? What is meant by one third ? What is meant by a fourth? What are fourths sometimes called ? What is meant by fifths ? By sixths ? Eighths? How many sevenths mako a whole one? How many tenths ? What is meant by twentieths ? By hundredths ? 72. What is a Fraction? What is an Integer ? 73. How are fractions commonly expressed ? What is the number below the line called ? What does it show?
The number above the line is called the numerator, and shows how many parts are expressed by the fraction. Thus in the fraction }, the denominator 3, shows that the number is divided into three equal parts ; the numerator 2, shows that two of those parts are expressed by the fraction.
The numerator and denominator taken together, are called the terms of the fraction.
74. A proper fraction is a fraction whose numerator is less than its denominator; as, }, }, $.
An improper fraction is one whose numerator is equal to, or greater than its denominator, as 3, 7.
A simple fraction is a fraction which has but one numerator and one denominator, and may
proper, or improper ; as, , .
A compound fraction is a fraction of a fraction; as, f of of 5. A complex fraction is one which has a fraction in its
21 4 21 numerator, or denominator, or in both; as
5'53'8' A mixed number is a whole number and a fraction written together; as, 43, 2571.
76. The value of a fraction is the quotient of the numerator divided by the denominator. Thus, the value of
is two; of 4 is one ; of } is one third ; &c. Read the following fractions, and name the kind of each: 1. ; ; ; ; ; ; 14; 18; 25. 2. of $; of of 1,2; 1 of 1 of of 75.
434 3. 25; 143; 8514; 9935+;
16 8' 21' 7
QUEST.-What is the number above the line called ? What does it show ? What are the denominator and nunerator, taken together, called ? 74. What is a proper fraction ? An improper fraction ? A simple fraction ? A conto pound fraction ? A complex fraction ? A mixed number? 76. What is the value of a fraction ?
To find a fractional part of a given number. Ex. 1. If a loaf of bread costs 4 cents, what will half a loaf cost ?
Analysis.-If a whole loaf costs 4 cents, 1 half a loaf • will cost 1 half of 4 cents; and i half of 4 cents is 2 cents, Half a loaf of bread will therefore cost 2 cents.
2. If a pound of sugar costs 12 cents, what will 1 third of a pound cost ?
Analysis.-Reasoning as before, if a whole pound costs 12 cents, 1 third of a pound will cost 1 third of 12 cents; and i third of 12 cents is 4 cents.
Ans. 4 cents. 77. From these examples the learner will perceive that
A half of a number is equal to as many units, as 2 is contained times in that number.
A third of a number is equal to as many units as 3 is contained times in that number.
A fourth of a number is equal to as many units, as 4 is contained times in that number, &c. Hence,
78. To find a Half of a number, divide it by 2. To find a THIRD of a number, divide it by 3. To find a FOURTH of a number, divide it by 4, &c. Note.-For mental exercises in Fractions, see Mental Arithmetic, Section VII.
3. What is half of 257 ?
Dividing 257 by 2, the quotient is 128 Operation. and 1 over. Placing the 1 over the 2
2)257 and annexing it to the quotient, we have
1284 Ans. 1284, wbich is the answer required.
4. What is a third of 21 ? 33? 48 ? 78 ? 151 !
QUXST.-78. How do you find
half of a number? A third ? A four. 3
8. What is a seventh of 88? 133 ? 175 ? 250 ? 9. What is a ninth of 126 ? 163 ? 270 ? 316 ? 79. To find what part one given number is of another.
Make the number called the part, the numerator, and the other given number the denominator. The fraction thus formed will be the answer required.
1. What part of 3 is 2? Ans. j. 2. What part of 4 is 1 ? Is 2 ? Is 3? Is 5 ? 3. What part of 7 is 2 ? Is 4? Is 5 ? Is 6 ? 4. What part of 9 is 1 ? Is 2? Is 4? Is 5? 5. 5 is what part of 11? Of 12 ? Of 13? 6. 8 is what part of 17 ? Of 19? Of 45 ? 7. 15 is what part of 38 ? Of 57 ? Of 85 ? 80. A part of a number being given to find the whole.
Multiply the given part by the number of parts into which the whole is divided, and the product will be the answer required.
1. 27 is 1 ninth of what number?
Suggestion.—Since 27 is 1 ninth, 9 ninths, or the whole, must be 9 times 27; and 27X9=243. Ans.
The given part is 27, and the Operation. number of parts into which the 27=1 ninth. whole is divided, is 9 ninths;
9=no. parts. we therefore multiply 27 by 9. Ans, 243=the whole,
2. 18 is 1 fifth of what number?
QUEST.-19. How do you find what pa one number is of another 1
When a part of a number is given, how do you find the whole ?
Multiplying a whole number by a fraction. 81. We have seen that multiplying by a whole number is taking the multiplicand as many times as there are units in the multiplier. (Art. 36.) On the other hand,
If the multiplier is only a part of a unit, it is plain we must iake only a part of the multiplicand. Hence,
82. Multiplying by a fraction is taking a certain PORTION of the multiplicand as many times as there are like portions of a unit in the multiplier. That is,
Multiplying by 1, is taking 1 half of the multiplicand once. Thus, 6xI=3. Multiplying by }, is taking 1 third of the multiplicand
Thus, 6x1=2. Multiplying by $, is taking 1 third of the multiplicand twice. Thus, 6Xž=4.
Obs. If the multiplier is a unit or 1, the product is equal to the multiplicand; if the multiplier is greater than a unit, the product is greater than the multiplicand; (Art. 36;) and if the multiplier is less than a unit, the product is less than the multiplicand. Hence,
83. To multiply a whole number by a fraction.
Divide the given number by the denominator, and multiply the quotient by the numerator.
Obs. 1. The result will be the same if we first multiply the given number by the numerator, then divide this product by the denomi
2. When the numerator is 1, it is unnecessary to multiply by it; for, multiplying by 1 does not alter the value of a number. (Art. 82. Obs.)
Quest.–81. What is meant by multiplying by a whole number? 82. By a fraction? What is meant by multiplying by ?? By 4? By j? By *? Obs. If the multiplier is a unit or 1, what is the product equal to? When the multiplier is greater than 1, how is the product compared with the multiplicand ? When less, how? 83. How do you multiply a whole number by a fraction ? Obs. What other method is mentioned ? When the numerator is 1, is It necessary to multiply by it! Why not?