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ARITHMETIC.

A. 1888-838-1138.

500

6. Multiply 83 by J. Note. If the denominator of any fraction be equal to the numerator of any other fraction, they may both be dropped on the principle explained is XXXVII.; thus of of may be shortened, by dropping the numeratos 3, and denominator 3; the remaining terms, being multiplied together, will pro duce the fraction required in lower terms, thus: 2 of 4 of 4 of 8 1., Ans.

The answers to the following examples express the fraction

7. How much isofofof?
in its lowest terms.

8. How much is of

of? times 5?

A. 8.

A. .

A. 301.

times 161?

A. 272.

11. How much is 20

times of ?

A. 223-388.

9. How much is 5 10. How much is 16

↑ XLII.

To find the Least Common Multiple of
Q. 12 is a number produced by multiplying 2 (a factor) by
two or more numbers.
some other factor; thus 2x6-12; what, then, may the 12 be
called? A. The multiple of 2.

Q. 12 is also produced by multiplying not only 2, but 3 and
6, likewise, each by some other number; thus, 2x6-12; 3×4
=12; 6x2=12; when, then, a number is a multiple of seve
ral factors or numbers, what is it called? . The commo
multiple of these factors.

Q. As the common multiple is a product consisting of tw or more factors, it follows that it may be divided by each of these factors without a remainder; how, then, may it be de termined, whether one number is a common multiple of two o more numbers, or not? A. It is a common multiple of thes numbers, when it can be divided by each without a remainder Q. What is the common multiple of 2, 3, and 4, then? A. 24 Q. Why? A. Because 24 can be divided by 2, 3, and 4, with

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out a remainder.

Q. We can divide 12, also, by 2, 3, and 4, without a remair der; what, then, or more numbers, called? A. The least common multiple o these numbers. the least number, that can be divided by

ral other numbers, without a remainder; as, for instance, 3 wi Q. It sometimes happens, that one number will divide sev! divide 12, 18, and 24, without a remainder; when, then, sev

numbers can be thus divided by one number, what is th number called? A. The common divisor of these num

Q. 12, 18, and 24 may be divided also, each, by 6, even, what, then, is the greatest number called, which will divide 2 or more numbers without a remainder? A. The greatest common divisor.*

* In T XXXVII., in reducing fractions to their lowest terms, we were sometimes obliged, in order to do it, to perform several operations in dividing; but, had we only known the greatest common divisor of both terms of the fraction, we might have reduced them by simply dividing once; hence it may sometimes be convenient to have a rule

To find the greatest common divisor of two or more numbers. 1. What is the greatest common divisor of 72 and 84 ? OPERATION.

72

12 84 (1

72

12) 72 (6

72

A. 12, common divisor.

In this example, 72 is contained in 84, 1 time, and 12 remaining; 72, then, is not a factor of 84. Again, if 12 be a factor of 72, it must also be a factor of 84; for, 72+12-84. By dividing 72 by 12, we do find it to be a factor of 72, (for 72÷12

6 with no remainder); therefore 12 is a common factor or divisor of 72 and 84; and, as the greatest common divisor of two or more numbers never exceeds their difference; so 12, the difference between 84 and 72, must be the greatest common divisor.

Hence, the following RULE. Divide the greater number by the less, and, if there be no remainder, the less number itself is the common divisor; but, if there be a remainder, divide the divisor by the remainder, always dividing the last divisor by the last remainder, till nothing remain: the last divisor is the divisor sought.

Note. If there be more numbers than two, of which the greatest common divisor is to be found, find the common divisor of two of them first, and then of that common divisor, and one of the other numbers, and so on.

A. 12.

A. 84.

A. 24. lowest terms.

2. Find the greatest common divisor of 144 and 132. 3. Find the greatest common divisor of 168 and 84. 4. Find the greatest common divisor of 24, 48, and 96. Let us apply this rule to reducing fractions to their See T XXXVII. In this example, by using the common_div sor, 12, found in the answer to sum No. 2, we have a number that will reduce the fraction to its lowest terms, by simply dividing both terms but once.

5. Reduce to 'ts lowest terms.

12), Ans.

After the same manner perform the following examples.

50

000

6 Find the common divisor of 750 and 1000; also reduce to its lowest, terms. A. 250, and 4.

7. Reduce 38 to its lowest terms.

660

8. Reduce to its lowest terms.

A. Z.
A. 88

Should it be preferred to reduce fractions to their lowest terms by XXXVII., the following rules may be found serviceable.

Any number ending with an even number or cipher is divisible by 2.

Any number ending with 5 or 0 is divisible by 5; also if it end in 0, it is Avisible by 10

1. What is the least common multiple of 6 and 8?

OPERATION.

2)6.8

In this example, it will be perceived that the divisor 2 is a fuctor, both of 6 and 8, and that dividing 6 by 2 gives its other factor 3 (for 6÷2=3); likewise dividing the 8 by 2 gives its 3 4 other factor 4 (for 8÷2—4); consequently, if the divisors and quotients be multiplied together, their product must contain all the factors of the numbers 6 and 8; hence this product is the common multiple of 6 and 8, and, as there is no other number greater than 1, that will divide 6 and 8, 4×3×2 =24 will be the least common multiple of 6 and 8.

Note. When there are several numbers to be divided, should the divisor not be contained in any one number, without a remainder, it is evident, that the divisor is not a factor of that number; consequently, it may be omitted, and reserved to be divided by the next divisor.

2. What is the common multiple of 6, 3 and 4 ?

OPERATION.

3)6.3.4

2)2.1.4

1.1.2 Ans. 3X2X2=12|

In dividing 6, 3 and 4 by 3, I find that 3 is not contained in 4 even; therefore, I write the 4 down with the quotients, after which I divide by 2, as before. Then, the divisors and quotients, multiplied togeth er, thus, 2×2×3=12, Ans.

From these illustrations we derive the following

RULE.

I. How do you proceed first to find the least common multiple of two or more numbers? A. Divide by any number that will divide two or more of the given numbers without a remain der, and set the quotients, together with the undivided num bers, in a line beneath.

II. How do you proceed with this result? A. Continue divi ding, as before, till there is no number greater than 1 that will divide two or more numbers without a remainder; then multiplying the divisors and numbers in the last line together, will give the least common multiple required.

More Exercises for the Slate.

3. Find the least common multiple of 4 and 16.
4. Find the least common multiple of 10 and 15.
5. Find the least common multiple of 30, 35 and 6.
6. Find the least common multiple of 27 and 51.
7. Find the least common multiple of 3, 12 and 8.

A. 16

A. 30

A. 210,

A 469 A. H.

8. Find the least common multiple of 4, 12 and 20. 9. Find the least common multiple of 2,7, 14 and 49

1 XLIII.

A. 60.

A. 98.

To reduce Fractions of different Denominators to a Common Denominator.

Q. When fractions have their denominators alike, they may be added, subtracted, &c. as easily as whole numbers; for example, and are ; but in the course of calculations by numbers, we shall meet with fractions whose denominators are unlike; as, for instance, we cannot add, as above, and together; what, then, may be considered the object of reducing fractions of different denominators to a common denominator? A. To prepare fractions for the operations of addition, subtraction, &c., of fractions.

Q. What do you mean by a common denominator? A. When the denominators are alike.

1. Reduce and to a common denominator.

OPERATION.

Numer. 2×6=12, new numer.
Denom. 3×6=18, com. denom.
Numer. 5X3=15, new numer.
Denom. 6X3=18, com. denom.

In performing this example, we take 3, and multiply both its terms by the denom inator of; also, we multiply both the

terms of by 3, the denominator of; and, as both the terms of each fraction are multiplied by the same number, consequently the value of the fractions is not altered, ¶ XXXVII.

From these illustrations we derive the following

RULE.

I. What do you multiply each denominator by for a new de nominator? A. By all the other denominators.

II. What do you multiply each numerator by for a new numerator? A. By the same numbers (denominators) that I multiply its denominator by.

Note. As, by multiplying in this manner, the same denomi nators are continually multiplied into each other, the process may be shortened; for, having found one denominator, it may be written under each new numerator. This, however, the in' elligent pupil will soon discover of himself; and, perhaps, it is best he should.

More Exercises for the Slate.

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6. Reduce 3, and to a common denominator.

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Compound fractions must be reduced to simple fractions be fore finding the common denominator; also the fractional parts of mixed numbers may first be reduced to a common denomi nator, and then annexed to the whole numbers.

7. Reduce of and to a common denominator,

8. Reduce 14 and to a common denominator.

A. tit.

A. 14, 8.

9. Reduce 103 and of to a common denominator.

A. 1088, 18.

10. Reduce 8 and 144 to a common denominator.

A. 8777, 1474.

Notwithstanding the preceding rule finds a common denomi. nator, it does not always find the least common denominator But, since the common denominator is the product of all the given denominators into each other, it is plain, that this product ( XLII.) is a common multiple of all these several denomintors; consequently, the least common multiple found by T XLII. will be the least common denominator.

11. What is the least common denominator of, and †?

OPERATION.

3)3.6.2

2)1.2.2

1.1.1 Ans. 2X3=6

Now, as the denominator of each frac tion is 6ths., it is evident that the numer ator must be proportionably increased; that is, we must find how many 6ths each fraction is; and, to do this, we can take , &, and of the 6ths., thus:

of 64, the new numerator, written over the 6

of 65, the new numerator, written over the 6,

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of 6=3, the new numerator, written over the 6 = 8.

Ans. t. t. 1.

Hence, to find the least common denominator of several

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