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and ğ, and from their sum subtract of

8. A. owns of of a vessel; B. owns of . greater is A.'s share than B.'s? 9. Subtract 133 from 153.

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9 8

1512 =12, 12.

1312
111 Ans.

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Having reduced the fractions to a common denominator, and found new numerators, as in Addition, we have to be taken from. We therefore borrow a unit, and say from 13, and add 3, the remainder, to 8, the numerator of the subtrahend. 3+8=11, which we write under the fractions, and carry 1 to 13, the whole number, which makes 14; and 14 from 15, and 1 remains. The answer, then, is 111.

10. A man bought a horse for of of $150, and sold him for of of of $60. Did he gain or lose, and how much? Ans. $40 gain.

To find the least Common Multiple.

Art. 86.-The common denominator found by the preceding rule, is a common multiple of the denominators of the given fractions; for every product must be divisible by all its factors; but it was not the least common multiple.

1. What is the least common multiple of 4, 6, 8, 10?

4 x 6 x 8 x 10=1920.

Operation. 2)4, 6, 8, 10

2)2, 3, 4,

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1, 3X2X5 X2 X2=120.

1920 is evidently a common multiple of 4, 6, 8, and 10, because they are its factors; but it is not the least common multiple. We find, also, that each of these numbers is a multiple of 2, or that 2 is a prime factor of each. Dividing by 2, we find the other factors, which are 2, 3, 4, 5. Again: As the quotient, 4, is a multiple of 2, we may substitute for it 2, one of its factors; and as we employ the other factor for a divisor, we erase the other quotient, 2. We now have 3, 2, 5, undivided numbers, which are prime factors of the dividends 6, 8, 10; the other prime factors are the divisors. If now we multiply to

gether these undivided numbers and the divisors, we shall have a combination of all the prime factors of each dividend, and consequently it must be divisible by them. Thus the divisors 2 and 2 are the factors of 4, the first dividend, 2 × 2=4. If 4 be divisible by 4, then 2, 3, and 5 times 4 must be divisible by 4; also 3, the first undivided number, is a factor of 6, the second dividend; and 2, the first divisor, is the other factor, 3×2=6. If 6 be divisible by 6, then 2 and 5 times 6 must be divisible by 6. The same may be said of 2 and 5, the other undivided numbers. Hence it appears, that the product of the continued multiplication of the remainders and divisors is divisible by the several dividends; and by examining the operation, it will be found to be the least number which can be divisible by them; for all repetition of prime factors beyond what is necessary to produce each dividend, is avoided. Therefore, to find the least common multiple of two or more numbers, we have the following

RULE.

Write the numbers in a horizontal line; divide them by the least prime number that will measure two or more of them; write the quotients and undivided numbers in a horizontal line under the given numbers; divide the numbers in this second line, in the same manner. Thus continue to divide until the quotients and undivided numbers are all prime to each other. The product of the continued multiplication of the divisors and undivided numbers will be the least common multiple required.

OBS. 1.-We divide by any number that will divide two or more of the numbers, to find first the least common measure of two or more.

EXERCISES.

2. What is the least common multiple of 3, 4, 9 and 12?

Ans. 36.

3. What is the least number which can be divided by 7, 8, 10, and 12, without a remainder?

Ans. 840.

4. What is the least common multiple of 7, 14, 28, 35 ? Ans. 140.

5. What is the least number which can be divided by the nine digits without a remainder? Ans. 2520.

QUESTIONS.-1. How is the least common multiple of two or more numbers found? 2. Why do you divide by any number that will divide two or more without a remainder?

6. Reduce,,, to equivalent fractions having the least common denominator.

214 5 6

2 × 5 × 3 × 2=60

60 being the least common multiple of 4, 5, and 6, it is, therefore, the least common denomi

nator of the fractions,, . The remaining part of the process is performed by the Rule under Art. 84. Still other illustrations may be given. The value of the fraction, is three-fourths of a unit. That is, the unit is divided into 4 parts, and the fraction expresses of them. If we divide the unit into 60 parts, and wish to express the same part of a unit, we must take of 60. If we divide 60 by 4, we have onefourth; if we multiply one-fourth by 3, we have three-fourths, 60-4=15, and 15x3=45. Now 3 is three-fourths of 4, and 45 is three-fourths of 60; therefore 45=2, an equivalent fraction.

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OBS. 2.-By this process the fractions are all reduced to the same denomination.

7. What is the least common denominator of, 1, 3,

20 40'

, and ?? Ans. 28, 18, 24, 13. 8. Reduce 8, 11, 12, to fractions having the least common denominator. Ans. 352 324 231

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396 396' 396*

9. Reduce,, 1, and 11, to fractions having the least 126 56 462 Ans. 4' 504' 504 common denominator.

10. A merchant buys 5 pieces of cloth. 403 yards; the second, 27; the third, 43; and the fifth, 394 yards. How many whole?

The first contains 347; the fourth, were there in the Ans. 1851

13

11. Which is the greater fraction, 11 or 18. Ans. 13 is greater by 144

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OBS. 3.-If the denominator of either of the given fractions be a multiple of each of the other denominators, it will be the least common denominator.

QUESTION.-3. How are fractions of different denominators reduced to equivalent fractions having the same denominator?

12. Reduce, and 32 to equivalent fractions having the least common denominator.

Multiplying the first by 4 and the second by 2, we have the answer required.

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Ans. 13, 12, 32.

or 20 ?

Dividing the terms of by 2, we have, and , the Answer.

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Fractions whose denominators are 10, 100, or 1000, etc. form a very important class of fractions, and will be treated under a separate head, called

DECIMAL FRACTION S.

Art. 87.-The term decimal signifies tenth. It is derived from the Latin word decem, which signifies ten. It is, therefore, applied to all fractions whose denominator is 10, or 1, with any number of ciphers. If a dollar be divided into ten parts, one of these parts, being worth ten cents, is one tenth of a dollar. If the dollar be divided into one hundred parts, one of these parts is the one hundredth part of a dollar. It is, nevertheless, a decimal fraction, because 100 is the product of 10's. The same may be said of a thousand, or ten thousand. A fraction is always known to be decimal, if its denominator be ten, a hundred, or a thousand. The denominator of a decimal fraction is not always expressed, but it can always be ascertained by the numerator. If it contains but one figure, the denominator is ten; if two, it is a hundred, etc. It is always one, with as many ciphers annexed as the numerator has places.

When the denominator is not expressed, the fraction is distinguished from a whole number by a period placed at the left of it. [The period is called the separatrix.] Example: .5, .50, is read five tenths, fifty hundredths, as though they were written 5 50 10, 100 If the numerator have not so many places as the denominator has ciphers, supply the defect by prefixing ciphers, thus: for 15, write .05; 1000, write .005. Ciphers placed at the right hand of a decimal do not affect its value,

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QUESTIONS.-4. What does the term decimal signify? 5. From what derived? 6. To what applied? 7. How is a fraction known to be decimal? 8. Is the denominator always expressed? 9. How, then, can it be known?

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5 50 are the same in value; for while the addition of the 100 cipher indicates a division into parts ten times smaller than the preceding, it makes the decimal express ten times as many parts. Thus, 5 tenths denotes 5 parts of a unit which is divided into 10 parts; and 50 hundredths denotes 50 parts of a unit which is divided into 100 parts. It is, therefore, plain, that the value is not altered, since 5 is half of 10, and 50 is half of 100.

The value of a decimal depends upon its distance from the unit's place. As whole numbers increase from the unit's place towards the left in a tenfold proportion, so decimals, in the same ratio, decrease from the unit's place towards the right hand; as will appear from the following

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From the above Table it is evident that each figure, whether a whole number or decimal, takes its value from the unit's place. If it be in the first place on the right of units, it is tenths; if in the second, it is hundredths, etc. Consequently, every decimal will have for its denominator 1, with as many ciphers as the decimal is places distant from the unit's place; thus, 2 in the Table is 2; 3 is 130; 4 is 1000, etc.

Art. 88. The manner in which decimal fractions are produced, and the relation they bear to whole numbers, may be seen by the following formula:

QUESTIONS.-10. On what does the value of a decimal depend? 11. In what proportion do decimals decrease from the unit's place towards the right? 12. From what does each decimal figure take its value? 13. What is the value of the first figure on the right of units ? 14. What effect have ciphers placed at the right hand of a decimal?

15. What effect at the left?

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