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OPERATION.

9 14 30 | 35 47.
9

2 | 6
3

47 X 35 X 6 X 3 29610 Ans. 4. What is the least common multiple of 6, 8, 10, 18, 20, and 24 ?

5. What is the least common multiple of 14, 19, 38, and 57 ?

Ans. 798. 6. What is the least common multiple of 20, 36, 48, and 50 ?

Ans. 3600. 7. What is the least common multiple of 15, 25, 35, 45, and 100 ?

Ans. 6300. 8. What is the least common multiple of 100, 200, 300, 400, and 575 ?

Ans. 27600. 9. The least common multiple of 1, 2, 3, 4, 5, 6, 8, 9, and one other number prime to them, is 2520. What is that other number?

Ans. 7. 10. What is the least common multiple of 18, 24, 36, 126, 20, and 48?

11. I have four different measures; the first contains 4 quarts, the second 6 quarts, the third 10 quarts, and the fourth 12 quarts. How large is a vessel, that may be filled by each one of these, taken a certain number of times full ?

Ans. 60 quarts. 12. What is the smallest sum of money with which I can purchase a number of oxen at $ 50 each, cows at $ 40 each, or horses at $ 75 each ?

Ans. $ 600.

MISCELLANEOUS EXAMPLES.

1. How many times does 7 occur as a factor of 6174 ?

Ans. 3 times. 2. Required the largest prime factor of 5775. 3. Required the largest composite factor of 19929.

Ans. 6643. 4. Required the quotients of 2338 divided by its two prime factors next larger than 1.

Ans. 1169; 334. 5. Required all the prime numbers that will divide 17385 without a remainder.

6. A farmer has 3000 bushels of grain ; which are the threo smallest-sized bags, and the three largest-sized bins, holding an exact number of bushels, that will each measure the same without a remainder ? Ans. Bags of 1, 2, or 3 bushels each ; and bins of 1500,

1000, or 750 bushels each. 7. A teacher having a school consisting of 152 ladies and 136 gentlemen, divided it in such a manner that each class of ladies equalled each class of gentlemen, and the classes were the largest the school would admit of, and have them all of the same size.

Required the number of classes, and the number in each class. Ans. 19 classes of ladies, 17 classes of gentlemen, and 8

pupils in a class. 8. At noon the second, minute, and hour hand of a clock are together; how long after will they be again, for the first time, in the same position ?

9. J. Porter has a four-sided garden, the first side of which is 348 feet in length; the second, 372 feet; the third, 444 feet; and the fourth, 492 feet. Required the length of the longest rails that can be used in fencing it, allowing the end of each rail to lap by the other 9 inches, and all the panels to be of equal length ; also, the number of rails, if 5 rails be allowed to each panel.

Ans. Length 12ft. 9in.; and 690 rails. 10. L. Ford has 5 pieces of land, the first containing 3A. 2R. 1p. ; the second, 5A. 3R. 15p. ; the third, 8A. 29p.; the fourth, 12A. 3R. 17p. ; and the fifth, 15A. 31p. Required the largest sized house-lots, containing each an exact number of square rods, into which the whole can be divided.

Ans. 1A. 27p. each. 11. What three numbers between 30 and 140 have 12 for their greatest common divisor, and 2772 for their least common multiple.

Ans. 36, 84, and 132. 12. Four men, A, B, C, and D, are engaged in making regular excursions into the country, between which each stays at home just 1 day; and A is always absent exactly 3 days, B 5 days, and C and D 7 days. Provided they all start off on the same day, how many days must elapse before they can all be at home again on the same day?

Ans. 23 days.

COMMON FRACTIONS.

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204. A FRACTION is an expression denoting one or more equal parts of a unit.

205. A fractional unit is one of the equal parts into which the whole thing or integral unit has been divided. Thus halves, thirds, &c., being equal parts of integral units or whole things, are fractional units.

206. The unit of a fraction is the unit or whole thing from which its fractional parts have been derived.

207. A COMMON FRACTION is expressed by two numbers one above the other, with a line between them.

208. The number below the line is called the denominator. It shows into how many parts the whole number has been divided. It gives name to the fraction and value to the fractional unit. Thus, in the expression 4, the denominator is 7, indicating that the unit of the fraction has been divided into 7 equal parts, and that the value of the fractional unit is one seventh.

The number above the line is called the numerator. It shows how many parts have been taken, or numbers the fractional units expressed by the fraction. Thus, in the expression 4, the numerator is 2, indicating that the fractional unit, which is one seventh, has been taken 2 times.

209. The terms of a fraction are its numerator and denominator. Thus, the terms of the fraction are the numerator 2 and the denominator 3.

210. A proper fraction is one whose numerator is less than the denominator ; as , , ž.

211. An improper fraction is one whose numerator is equal to, or greater than, the denominator; as 7, it, 4.

212. A mixed number is a whole number with a fraction; as 31, 163, 90.

213. A simple or single fraction has but one numerator and one denominator. It may be either proper or improper; as 1, 11, 11

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214. A compound fraction is a fraction of a fraction, or two or more fractions connected by the word of; as of } of yo,

of } of

215. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as 17 3 24

47 13 41

216. A fraction is an expression of division ; the numerator answering to the dividend, and the denominator to the divisor (Art. 67); and the value of a fraction is the quotient arising from the division of the numerator by the denominator (Art. 80). Thus, in the fraction 42, the numerator 15 is the dividend, the denominator 7 is the divisor, and the value expressed 21, or the quotient arising from the division of the 15 by the 7.

217. Since a fraction is an expression of division, it follows,

1. That, if the numerator be multiplied, or the denominator be divided, by any number, the fraction is multiplied by the same number (Art. 81).

2. That, if the numerator be divided, or the denominator multiplied, by any number, the fraction is divided by the same number (Art. 82).

3. That, if the numerator and denominator be both multiplied, or both divided, by the same number, the fraction will not be changed in value (Art. 83).

REDUCTION OF COMMON FRACTIONS. 218. Reduction of fractions is the process of changing their form of expression without altering their value.

219. A fraction is in its lowest terms, when its numerator and denominator are prime to each other (Art. 166).

220, To reduce a fraction to its lowest terms. Ex. 1. Reduce is to its lowest terms.

By dividing both terms of the fraction by 4)

4, a factor common to them both, it is re4) * = 4 Ans. } Ans. duced to iš. Dividing both terms of is by

4, a factor common to them both, it is re.

Ans. .

FIRST OPERATION.

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SECOND OPERATION.

duced to ]. Now, as 1 and 3 are prime to each other, the fraction } is in its lowest terms.

The same result is often more readily ob

tained by dividing the terms of the fraction 16 ) 18 = } Ans. by their greatest common divisor, as by the

second operation.
Since dividing the nun ator and denominator of a

ction by the same number, or cancelling equal factors in both, changes only the form of the fraction, while the value expressed remains unchanged (Art. 217).

RULE. - Divide the numerator and denominator by any number greater than 1 that will divide them both without a remainder, and thus proceed until they are prime to each other. Or,

Divide both the numerator and denominator by their greatest common divisor.

EXAMPLES. 2. Reduce je to its lowest terms.

Ans. 1 3. Reduce to its lowest terms.

Ans. 1
4. Reduce to its lowest terms.

Ans. .
5. Reduce 1ung to its lowest terms.
6. Reduce 15 to its lowest terms.
7. Reduce jen to its lowest terms.
8. Reduce to to its lowest terms.
9. Reduce $15 to its lowest terms.

Ans. 479.
10. Reduce 116 to its lowest terms.

Ans. 15.

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11. Reduce sis to its lowest terms.
12. Reduce 8 16 to its lowest terms.

Ans. os.
13. Reduce to its lowest terms.

221. To reduce an improper fraction to an equivalent whole or mixed number. Ex. 1. How many yards in 147 of a yard ?

Ans. 63 Since 19 nineteenths make one yard,

it is evident there will be as many 19 )117 ( 62 Ans. yards in 117 nineteenths as 19 is con 114

tained times in 117, which is 6 times.

Therefore, 61 yards is the answer re. 3

quired. RULE. - Divide the numerator by the denominator. NOTE. Should there a remainder occur, write it over the denominator, and make this fraction a part of the answer.

Ans. . ਝੰs
Ans. 2.

Ans. 152

OPERATION.

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