« ΠροηγούμενηΣυνέχεια »
Exercises. 1. Multiply £12 11s. 6d. by 12%. 2. What is the value of acre at £21 per acre ? 3. Multiply 178. 5 d. by 7543. 4. A person walks 77% miles in 104 hours, at what rate is
that per hour ? 5. Seven pieces of cloth, each containing 11% yards, cost
£549, what was the price per yard ? 6. Three persons, A, B, C, purchase property worth £625 ;
A purchases ļ, B , and C the rest : what is the worth
of each person's share ? 7. The total amount of gold exported from San Francisco,
in California, during the month of October, 1851, was 6884875 American dollars; what was its worth in English money, an American dollar being of an English one ? [American dollars differ in value in
different States. ] 8. A person has ģ of a cargo, worth £900, and wishes to
sell z of his share ; what is it worth? 9. A person holding of property worth £864 108. wishes
to dispose of as much of it as will produce £160; what
share will he possess after the sale ? 10. The true length of the year is very nearly 365 days 5
hours 49 minutes; what is the length of the part, or 7 weeks, consideriug a week to be the 52nd part of a year?
(62.) To find the Greatest Common Measure of two or more
Whole Numbers. By the greatest common measure of a set of numbers is meant the greatest number wbich will divide them all without leaving any remainder; a number being said to measure another only when it is contained in that other a certain number of times exactly: 3, for instance, measures 9, 12, 15, &c., but it does not measure 10, 13, 16, &c. Of the numbers 8 and 12, 4 is the greatest common measure; that is, the greatest divisor common to them both: 6 is the greatest common measure of 12 and 18, as also of 12, 18, and 36; and so on. The words greatest common measure are, for shortness, replaced by the letters G. C. M.
The chief use of a rule for finding the G. C. M. of two numbers is to enable us to discover with certainty the lowest terms in which any proposed fraction can be written. In the fractions hitherto considered, this object has been attained by a simple inspection of the numerator and denominator, as the factors common to both are often sufficiently obvious to become readily detected by a little examination; in which examination the table of factors at the end will frequently be of assistance. I have postponed the general rule about to be given till now, because you should always endeavour to discover the greatest common measure in this way,
without resorting to any such rule ; for the operation by it is sometimes long, though not difficult, and frequently ends by showing that no common measure exists. Moreover, it is not indispensably necessary that a fraction be reduced to its lowest terms : factors which obviously enter both numerator and denominator should always be removed; but when the detection of them requires the aid of the rule below, they might, in most cases, be suffered to remain, without bringing any discredit upon the computer; for the simplified result is seldom worth the trouble of obtaining it. It is right, however, that you
should know the rule, more especially as it may be applied to other purposes.
(63.) RULE. To find the G. C. M. of two numbers, divide the greater by the less; make the remainder a new divisor, and the former divisor a new dividend ; then make the second remainder a divisor, and the preceding divisor a dividend, and so on, always dividing the last divisor by the last remainder till the remainder disappears, or becomes 0. The divisor which thus leaves no remainder is the G. C. M. of the two numbers. If there be a third number, apply the rule to the G. C. M. of two and the third : and so on.
Note.—You may very often stop the process before it terminates in this way of itself. The divisors continually diminish in magnitude, and on this account, though the original numbers may be too large for you to detect their G. C. M. without a rule, yet the smaller numbers, which form the several divisors, may be small enough, after a step or two, to enable you to see by simple inspection what the G. C. M. of a pair of them is, and it so happens, that the G. C. M. of any two of the divisors is always likewise the G. c. M. of the original numbers. And it is strange that this way of curtailing the work is not expressly pointed out in books of arithmetic. In the first example on next page, the process is carried on till it terminates of itself
, for the purpose of showing you all the steps in full; but after the first of these steps, the remainder of the work is unnecessary. Your first remainder 92 will obviously divide by 2; the quotient 46 will also divide by 2, giving 23; the only different factors of 92 are therefore 2 and 23. Now 161 will not divide by 2 ; but you find, upon trial, that it will divide by 23. Hence 23 is the G. C. M.
Ex. 1. Find the G. C. M. of 161 and 253; and thence reduce the fraction of to its lowest terms.
Proceeding by the rule, as in the margin, the G. C. M. is found to be 161) 253(1 23; we infer, therefore, that both 161 numerator and denominator of the fraction have 23 for their highest 92) 161(1 common divisor. Dividing them,
92 therefore, by this number, we find the fraction in its lowest terms to
69) 92(1 be 11, which is certainly a much
69 simpler form than 16s. And I may here observe, that we can often
23) 69 (3 form a much more correct estimate
69 of the value of a fraction after it is reduced to its lowest terms: you can form a better notion of 7 elevenths of a thing than of 161 parts of it out of 253.
2. Find the G. C. M. of 175 and 912.
Here we find that the second divisor is 37, 175)9125 which obviously has no factor, unity not being 875 considered as a factor; hence the numbers have no common measure, so that the fraction
37 ili is already in its lowest terms.
3. Reduce 3439 to its lowest terms.
Proceeding as in the margin, we arrive, after 12321) 54345(4 a few steps, at the re
49284 mainder 210, a number whose factors, it is very
5061) 12321(2 easy to see, are 10, 7,
10122 and 3, the 10 being composed of the factors
2199) 5061(2 2 and 5; so that the
4398 G. C. M., if any common measure exist, must be
663) 2199 (3 one or more of these
1989 factors. A glance at the last figure of 663
210 shows that neither 2 nor 5 can be a divisor of it. It is easy to try 7, which is found not to succeed; but 3 does succeed : hence the G. C. M. of the terms of the proposed fraction is 3. And thus several steps of work are saved. Dividing numerator and denominator by 3, we have 12321=41075, which, after all, is a degree of simplification too trilling to be worth the trouble by which it has been effected; but then this could not have been foreseen. The common measure 3 could have been easily found by inspection.
(64.) The truth of the rule for the g. C. M. depends upon two general principles; namely, l. If one number be divided by another, every factor common to dividend and divisor, must also be a factor of the remainder. For suppose two numbers, that have no factor in common, to be divided one by the other, and the remainder to be found : it is plain, that if the dividend and divisor were each to be multiplied by any number, 4, for instance, and the division then to be performed, we should get the same quotient as before ; but the remainder would be 4 times the former remainder, because if you subtract 4 times a number from 4 times another number, the remainder must be 4 times as great as it would have been if you had subtracted without multiplying the numbers by 4.
You see, therefore, that when a factor is introduced into dividend and divisor, it is also introduced into the remainder; so that whenever a common factor exists in dividend and divisor, you would be sure to find it out by trying all the factors of the remainder.
2. Whatever factor is common to remainder and divisor, also belongs to the dividend. For the dividend is equal to the product of quotient and divisor, with the remainder added : the portion of the dividend, furnished by the product of quotient and divisor, has, of course, whatever factor the divisor has; so that if the other portion, namely, the remainder, have the same factor, it follows, that the sum of these portions, that is, the dividend itself, must have that factor. These two principles suggest the rule; thus, referring to Example 1, the greatest common factor of the remainder 92 and divisor 161, is also the greatest common factor of 161 and 253: in like manner, the G. C. M. of 69 and 92, is the G. C. M. of 92 and 161, and, consequently, from what has just been inferred, the G. C. M. of 161 and 253: and, lastly, the G. C. M. of 23 and 69, which we see is 23 itself, is the G. C. M. of 69 and 92, and, therefore, of 92 and 161, and of 161 and 253.
Exercises. 1. Find the G. C. M. of 247 and 323. 2. Of 272 and 425.
3. Of 57 and 63.
4. Of 408 and 527.
5. Of 1164 and 1261. 6. Reduce 29? to its lowest terms. 7. Reduce 618 to its lowest termis. 8. Reduce 71.63 to its lowest terms. 9. Find the g. c. M. of 5283 and 176491. 10. A field is 169 rods long, and 156 rods wide: what is the
length of the longest chain that will exactly measure
both length and width? 11. Three persons, A, B, C, having, respectively, £323,
£456, and £551, lay it out in land, at the greatest price per acre that will allow each to spend the whole of his money: what was the price per acre, and how
many acres did each buy? 12. Find the least number of ounces of standard gold that
can be coined into an exact number of half-sovereigns : standard gold being £3 178. 10 d. an ounce.*
(65.) To find the Least Common Multiple of a Set of
Numbers. Any number that is exactly divisible by a set of other numbers is called a common multiple of those others, and the least number that is exactly divisible by a set of others, is called the least common multiple of those others : it is, for brevity, expressed by the initial letters, L. C. M.
From knowing how to find the L. C. M. of a row of numbers, we can always reduce a row of fractions to others equal to them in value, and, at the same time, having the lowest possible common denominator; for the lowest common denominator will, of course, be the least common multiple of the original denominators. Each changed fraction will have this L. C. m. for denominator, and therefore for numerator, it must have the product that arises from dividing the L. C. m. by the original denominator, and multiplying the original numerator
* As a hint towards the management of this question, the learner may be informed, that if the £3 178. 10 d. be reduced to half-pence, and the number of half-pence be divided by the number in 108., the quotient will give the number of half-sovereigns and the fraction of a half-sovereign that can be coined from one ounce; he will then have to find the lowest number the dividend must be multiplied by to prevent the entrance of a fraction in the quotient. It will not be difficult for him to see that £3 108. out of the £3 178. 10 d. may be neglected in the work. He will find the least number of ounces to be 80, and the corresponding number of half-sovereigns to be 623.