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215.

VULGAR FRACTIONS.

89

rates the parts of the denominator signified by the fraction. As there are no limits to the number of parts into which a thing, or whole, may be divided, it is evident that it is possible for every number to be a numerator, or a denominator of a fraction. Hence the variety of fractions must be unlimited.

2. Fractions are of two kinds, Vulgar and Decimal, which differ in the form of expression, and the modes of operation.

3. A Vulgar Fraction is expressed by two numbers, called the numerator and denominator, written the former over the latter, with a line between, as, the former before the latter, as 3-8-3.

4. A Decimal Fraction, or a Decimal, is a fraction which denotes parts of a unit which become ten times smaller by each successive division (113), and is expressed by writing down the numerator only. (See Part H. Sect. III). A decimal is read in the same manner as a vulgar fraction; thus 0.5 is read 5 tenths, 0.25 25 hundredths, and it is put into the form of a vulgar fraction by drawing a line under it, and writing as many ciphers under the line as there are figures in the decimal, with a 1 at the left hand; thus, 0.5 ⚫ becomes 15, 0.25, and 0.005, 1000.

215.

VULGAR FRACTIONS

1. A proper fraction is one whose numerator is less than its denominator; as, 1, 3, &c. (23).

2. An improper fraction is one whose numerator is greater than its denominator; as,,,, &c. (24).

3. The numerator and denominator of a fraction are called its terms (30).

4. A compound fraction is a fraction of a fraction; as, of 1. 5. A mixed number is a whole number and a fraction written together, as 124, and 63 (23).

6. A common divisor, or common measure, of two or more numbers, is a number which will divide each of them without a remainder.

7. The greatest common divisor of two or more numbers, is the greatest number which will divide those numbers severally without a remainder.

8. Two or more fractions are said to have a common denomi nator, when the denominator of each is the same number (25)

9. A common multiple of two or more numbers is a number, which may be divided by each of those numbers without a remainder. The least common multiple is the least number, which may be divided as above.

10. A prime number is one which can be divided without a remainder, only by itself, or a unit.

11. An aliquot part of any number, is such part of it, as being taken a certain number of times, will exactly make that number.

12. A perfect number is one which is just equal to the sum of all its aliquot parts.

The smallest perfect number is 6, whose aliquot parts are 3, 2, and 1, and 3+2+1=6; the next is 28, the next 496, and the next 8128. Only ten perfect numbers are yet known.

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1. Change 16 to a whole or

mixed number. 3)76

As the denominator de

1. Change 25 to an improper fraction.

denotes the di

notes the number of parts 25x376 vision of 1 by 3,

25 into which the whole, or

3

(129); if now we multiply 25 by 3, and add the product to 1, making (25×3+1) 76, and then write the 76 over 3, thus,

unit, is divided, and the numerator shows how many of those parts are contained in the fraction (22), there are evidently as many wholes, as the number of times the, we evidently both multiply and

numerator contains the denominator; or, otherwise, since every fraction denotes the division of the numerator by the denominator (129), where the numerator is greater than the denominator, we have only to perform the division which is denoted.

217. To change an improper fraction to an equivalent whole or mixed number.

'RULE.-Divide the numerator by the denominator, and the quotient will be the whole, or mixed number required.

divide 25 by 3; but as the multiplication is actually performed, and the division only denoted, the expression becomes an improper frac

tion.

A whole number is changed to an improper fraction, by writing 1 under it, with a line between.

218. To change a whole or mixed number to an equivalent improper fraction.

RULE.-Multiply the whole number by the denominator of the fraction, add the numerator to the product, and write the I sum over the denominator for the required fraction.

QUESTIONS FOR PRACTICE

2. Change 26 to a mixed number.

3. Change 24 to a mixed number.

4. In 2368. shillings, how many shillings?

5. In 2 of a week, how many weeks?

per

2. Change 84 to an improfraction.

3. Change 273 to an im proper fraction.

4. In 1988. how many 12ths?

5. In 33 weeks, how many 7ths?

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1. Henry had of a peck of plums, which were twice the quan tity James had; how many had

James?

1. James had of a peck of plums, and Henry had twice as many; how many had Henry? Here we have evidently to multiply by 2; but two times is; Here we have evidently to divide hence, to multiply by 2, we multi-into 2 equal parts; but 2 divided ply the numerator 2 by 2, and write into 2 parts, one of them is ; then the product, 4, over 8, the denom- to divide by 2, we must divide inator; or, otherwise, if we divide 8, the numerator by 2, and write the the denominator, by 2, and write quotient, 1, over 4, the denominator; the quotient, 4, under 2, the nu- or, otherwise, if we multiply 4, the merator, thus,, the fraction be- denominator, by 2, and write the comes multipled; for while the numproduct, 8, under 2, the numerator, ber of parts signified remains the thus,, the fraction becomes dividsame, the division has rendered those ed by 2; for while the number of parts twice as great; and these re- parts remains the same, the multiplisults, and, are evidently the cation has rendered the parts only same in value, though differing in half as great; and these results, the magnitude of the terms. Therefore

220. To multiply a fraction by a whole number.

RULE.-Multiply the nume-rator, or divide the denominator, of the fraction by the whole number; the result will be the product required.

and, are evidently the same in value, though expressed in different terms. Hence

221. To divide a fraction by a whole number.

RULE. Divide the numerator, or multiply the denominator, of the fraction by the whole number; the result will be the required quotient.

QUESTIONS FOR PRACTICE.

2. What is the product of 2. How many times 24 in by 24?-of by 32?-of 72?-32 in 1§2?—36 in 198? by 36?-of by 42?-of-42 in 136 ?-9 in 27? by 3?

3. How many are 5 times ?-3 times ?—14 times -7 times?

4. If 1lb. of rice cost of a dollar, what will 5 lb. cost? 5. If a bushel of wheat cost of a dollar, what will -6 bushels cost?

3. How many times 5 in 14 in 11?—7

3? 3 in

in §, or 5?

4. If 5 lb. of rice cost of a dollar, what will 1 lb. cost?

5. If 6 bushels of wheat cost of a dollar, what is it a bushel?

7

MULTIPLICATION BY FRACTIONS.

ANALYSIS.

222. If a load of hay be worth $12, what are of it worth?

Here 12 and are evidently two factors, which, multiplied together, will give the price; and since the result is the same, whichever is made the muluplier (86), we may make the multiplicand, and proceed (220) thus, X 12-24-8 dollars. Ans. Otherwise, since in the multiplication by a whole number, the multiplicand is repeated as many times as the multiplier contains units, if therefore the multiplier be 1, the multiplicand will be repeated one time, and the product will be just equal to the multiplicand; if the multipher be, the multiplicand will be repeated half a time, and the product will be half the multiplicand; if the multiplier be, it will be repeated one third of a time, and the product will be one third of the multiplicand, and generally, multiplying by a fraction is taking out such a part of the multiplicand as the fraction is part of a unit. Hence the product of 12 by , is of 12; and to find of 12, we must first find 3 of 12, by dividing 12 by 3, and then multiply this third by 2; thus, 12-3-4, and 4×2-8; $8 then are of $12, or the product of $12 by 3, as by the former method. Therefore,

223. To multiply a whole number by a fraction.

RULE.-Divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator, or multiply the whole number by the numerator, and divide the product by the denominator.

QUESTIONS FOR PRACTICE.

2. What is the product of 4 multiplied by ?-of 7 multiplied by ?- of 9 by of 17 by +?

3. If a barrel of rum cost $24, what cost of it? Ans. $18.

4. What cost 18 bushels of corn, at of a dollar a bushel ? Ans. $6.

224.

MULTIPLICATION

5. If a bushel of pears cost 75 cents, what cost of them? Ans. 15 cts.

6. What is the product of 16 by?-256 by †?—of 12 by &?

NOTE. It will be observed from the above examples, that multiplica tion by a proper fraction gives a product which is less than the multiplicand (121).

OF ONE FRACTIONAL QUANTITY BY ANOTHER.

son owning of a gristmill, sold of his share; what part of the whole mill did he sell?

Here we wish to take out of, which has been shown (222) to be the same as multiplying by; but to multiply by a fraction, we must divide the multiplicand by the denominator, and multiply the quotient by the nu merator; is divided by 3, by multiplying the denominator 4 by 3 (12,

225,

226.

VULGAR FRACTIONS.

93

and the quotient is; and is multiplied by 2, by multiplying the numerator, 3, by 2 (220), and the product is equal to the part of the mill sold. Hence,

To multiply a fraction by a fraction, or to change a compound fraction to a single one.

RULE.-Multiply the numerators together for a new nume rator, and the denominators together for a new denominator.

QUESTIONS FOR PRACTICE (56).

2. A man owning of a 4. What part of a mile is farm, sold of his share; of of a mile? what part of the farm did he sell?

of

Ans.

3. What part of a foot is

of a foot?

Ans.

Ans.

of of of

to a single fraction.
Aus. 403

5. Change

of

6. Multiply 48 by 27

225. DIVISION BY FRACTIONS.

1. In 6 dollars, how many times of a dollar? Here we wish to divide 86 into parts, each of which shall be of a dok lar, or in other words, divide 6 by. Now in order to find how many times in 6, we reduce 6 to 4ths, by multiplying it by 4, the denominator of the fraction, thus: 4 times 6 are 24; 6 dollars, then, are 24 fourths, or quarters of a dollar; and dividing 24 fourths by 3 fourths (the numerator), the quo tient, 8, is evidently the number of times of a dollar may be had in 2 or 6 dollars. Hence,

RULE

226. To divide a whole number by a fraction.

Multiply the number to be divided by the denomina tor of the fraction, and divide the product by the numerator.

QUESTIONS FOR PRACTICE.

2. In 7 shillings, how many

times

6. In a pound of tobacco, how many quids, each weighof an ounce ?

of a shilling?
Ans. 28.

ing

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3. In 17 bushels of wheat,

how many times

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Ans. 384-1014

How many are 7 8÷? 2÷32?

NOTE. Here it will be seen that division by a fraction, gives a quo tient larger than the dividend.

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