33. · A man, failing in trade could only pay £666; 4; 14. on a debt of £706. How much did his creditor lose ? Ans. 39 ; 15; 101. 34. Bought a hhd. of wine for £67; 4. What was that a gal. ? Ans. £1; 1; 4. 35. Bought 48 yds. of cloth for £11; 2. What was that a yd. ? Ans. 4s. 7d. 36. If a dozen silver spoons weigh 3 lb. 2 oz. 13 dwt. 12 gr. What is the weight of 1 spoon? Ans. 3 oz. 4dwt. 11 gr. 37. If 15 yds. of cloth cost $45, what will 47 yds. come to at that rate ? Ans. $141. Nota. If 15 yds. cost $45, 1 yd. will cost one fifteenth as much. 45+15=$3. If lyd. cost $3 47 yds. will cost 47 times as much. 3x47=$141, Ans. Questions like the last belong to PROPORTION or the RULE OF THREE. (see ở xci.) The above mode of solving them is called the SOLUTION BY ANALYSIS, or the ANALYTIC SOLUTION. 38. If 2 gal. of brandy cost $1, what will 50 gal. cost ? Ans. $25. 39. If 100 men can do a piece of work in 12 days, how many will it take to do it in 3 days ? Ans. 400. Note. It will take 12 times as many to do it in 1 day; and f as many to do it in 3 days, as in 1. How many yds. of cloth, 3 qrs. wide will line 30 yds. that is 5 qrs. wide ? Ans. 50. 41. How many yde. of matting, 2 ft. 6 in. broad, will cover a floor, 27 ft. long, and 20 ft. broad? Ans. 70. 42. If 30 men can perform a piece of work in 11 days, how many men will accomplish a piece, 4 times as large, in of the time? 43. If 13 gal. of molasses cost $6.50, what cost 25 ? Ans. $12.50. 44. 1f 15 lb, of sugar cost $1.875, what cost 29 ? Ans. $3.625. 45. If 21 lbs. of coffee cost $5.25, what cost 57 ? Ans. $14.25. 46. If 5 yards of cloth cost $38.40, what cost 639 yards ? 47. If 18 lbs. of sugar cost $1.875, what will 7 barrels come to, cach weighing 3 cwt. 3 qrs. 15 lbs. 9 oz. ? FRACTIONS. S XLI. From what has already been said of Fractions, the learner will understand that they are, I, ExpressIONS FOR PARTS OF AN INTEGER OR WHOLE UNIT, or II. EXPRESSIONS INDICATING THE DIVISION OF ONE NUMBER BY ANOTIIER. Bef re proceeding farther, we will lay down a few principles, resulting from the nature of Division, which will assist us in pur. suing the subject of Fractions. If a divisor is contained in a dividend, a certain number of times, the same divisor will be eontained i: twice that dividend, twice as often; in 3 times that dividend, three times as often; in 4 times that dividend, 4 times as often, and so on. If, then, the divison remains the same, MULTIPLYING THE DIVIDEND BY ANY NUMBER, MULTIPLIES THE QUOTIENT BY THE SAME NUMBER. If a divisor is contained in a dividend a certain number of times, the same divisor is contained in half that dividend one half as often ; in one third of that dividend, one third as often, and so on. If, then, the DIVISOR remains the same, DIVIDING THE DIVIDEND BY ANY NUMBER, DIVIDES THE QUOTIENT BY THE SAME NUMBER. If a divisor is contained in a dividend, a certain number of times, twice that divisor is contained in the same dividend, one half as often ; 3 times that divisor is contained in the same dividend, one third as often, and so on. If, then, the DIVIDEND remains the same, MULTIPLYING THE DIVISOR BY ANY NUMBER, DIVIDES THE QUOTIENT BY THE SAME NUMBER. If a divisor is contained in a dividend, a certain number of times, half that divisor is contained in the same dividend, twice as often; one third of that divisor, 3 times as often, and so on. If, then, the DIVIDEND remains the same, DIVIDING THE DIVISOR BY ANY NUMBER, MULTIPLIES THE QUOTIENT BY THE SAME NUM BER. It seems then, that multiplying the dividend has the same effect on the quotient as dividing the divisor ; and that dividing the dividend has the same effect as multiplying the divisor. Of course, multiplying both divisor and dividend produces opposite effects upon the quotient, and dividing both divisor and dividend produces opposite effects upon the quotient. Then, IF THE DIVISOR AND DIVIDEND BE BOTH MULTIPLIED, OR BOTH DIVIDED BY THE SAME NUMBER THE QUOTIENT WILL NOT BE ALTERED. Now as every Fraction is an instance of division, in which the numerator is the dividend, the denominator the divisor, and the value of the Fraction, the quotient, we may apply the above princi. ples to them. Of course, MULTIPLYING THE NUMERATOR OF A FRACTION, MULTIPLIES THE VALUE, AND DIYIDING THE NUMERATOR DIVIDES THE VALUE. And, MULTIPLYING THE DENOMINATOR OF A FRACTION DIVIDES THE VALUE, AND DIVIDING THE DENOMINATOR MULTIPLIES THE VALUE. And, MULTIPLYING THE NUMERATOR PRODUCES THE SAME EFFECT UPON THE VALUE AS DIVIDING THE DENOMINATOR ; AND DIVIDING THE NUMERATOR PRODUCES THE SAME EFFECT AS MULTIPLYING THE DENOMINATOR. And, MULTIPLYING THE NUMERATOR PRODUCES AN OPPOSITE EFFECT UPON THE VALUE, FROM MULTIPLYING THE DENOMINATOR; AND DIVIDING THE NUMERATOR PRODUCES AN OPPOSITE EFFECT FROM DIVIDING THE DENOMINATOR. Therefore, IF THE NUMERATOR AND DENOMINATOR BE BOTH MULTII'LIED, OR DIVIDED BY THE SAME NUMBER, THE VALUE OF THE FRACTION WILL NOT BE ALTERED. org? MENTAL EXERCISES. $ XLII. 1. Which Fraction has the greatest value 1 ? Ans. Neither, because 6 is in 12, twice, and 3 is in 6 twice ; or 4=2, and =2. 2. Which is the greatest L or \ ? Why? ? Ans. Neither, for 6 is 3 times 2, or 3 twos, and 8 is 4 times 2, or 4 tuos. It is plain, that 3 twos is the same part of 4 twos that 3 ones is of 4 ones, or that 3 is of 4. 5. Which is the greatest it or j? Why? 8 or ? Why? y or ? Why? 48 ors? Why? jj or ? Why? We see then, that Fractions of the same value may be expressed by very different numbers. The numerator and denominator of a Fraction are called TERMS. WHEN A FRACTION IS Expressed BY THE SMALLEST NUMBERS POSSI. BLE, IT IS SAID TO BE IN ITS LEAST OR LOWEST TERME. When a Fraction is not in its lowest terins, it is plain that to inake it so, we must diminish the terms, in such a manner as not to alter the value of the Fraction. This can be done by dividing both, by some number which will divide them, without remainder. ($ XLI.) 1. Reduce the following Fractions to their lowest terms. ; 2. Reduce the following Fractions to their lowest terms. *; ; ; ; ; 38; ideias; i dini; 3. Reduce the following: Fraetions to their lowest terms. ; iš; ; ; ; ; 18; ; 19; 36; 13; 13; 1:. The following are to be written. 4. Reduce to its lowest terms. l'irst divide by 5, and then by 5 again. 5)73635)45, Ans. 5. Reduce 19 and 2; to their lowest terms. Ans. *; ; ; } and }. The rule then, seems to be DIVIDE BOTH TERMS OF THE PRACTION BY ANY NUMBER WHICII WILL DIVIDE THEM WITHOUT A REMAINDER, AND THE QUOTIENTS IN THE SAME MANNER, UNTIL NO NUMBER GREATER THAN 1 WILL DIVIDE BOTH, WITIIOUT A REMAINDER. The pupil will soon find, that there are somo numbers which Çamnot be divided without remainder, except by themselves or by 1. 24 20 40 35 18 4 5 24, 12 36 IR 27 14 18 36 Thus, 7 can be divided without remainder by no numbers except 7 and 1. 13, 17, &c. are such numbers. A NUMBER WHICH CAN ONLY BE DIVIDED BY ITSELF, OR BY 1, is CALLED A PRIME NUMBER. Numbers which admit of division, or of being resolved into fac. tors, we have already seen ($ x11.) are called composite numbers. AN EVEN DIVISOR OF A NUMBER, that is, one which will divide it without remainder, IS CALLED A MEASURE OF THAT NUMBER. Thus, 3 is a measure of 15, because 3 divides 15 without remainder. 7 is a measure of 35, &c. Every number is a measure or even didisor of itself. 10 may be divided by 2, 5, or 10. Of course, any number of 10s may be divided by 2, 5, or 10. Then, if the right hand figure of a number be a cypher, that number is divisible, that is, it may be divided by 2, 5, or 10. Of course, if the right hand figure of a number be divisible by 2, the whole number is divisi. ble by 2. And, if the right hand figure of a number be 5, the whole number is divisible by 5. 100 may be divided by 4. Of course, any number of 100s may be divided by 4. Then, if there be two cyphers at the right of a number, the number is divisi. ble by 4. Of course, if 4 will divide two figures on the right of a number, it will divide the whole number. 1,000 may be divided by 8. Of course, any number of 1,000s may be divided by 8. Then, if there be three cyphers at the right of a number, the number is divisible by 8. And, if 8 will divide three figures on the right of a number, it will divide the whole number. It has been already shown, ($ IX.) that if the sum of the figures, which com. pose a number, be divisible by 9 or by 3, the whole number is divisible by 9 or 20 is divisible by 4. Of course, any number of times 20 is divisible by 4. Then, if the tens be even, we need only try the right hand figure by 4, to dis. cover whether it will divide the whole number. 200 is divisible by 8. Of course, any number of times 200 is divisible by 8. Then, if the hundreds be even, we need only try the two right hand figures by 8, to discover whether it will divide the whole number. To discover whether a Prime number will divide, we must make actual trial. 1. Find the measures of the following numbers, omitting the number itself, as being of course a measure. 18, 27, 20, 21, 24, 48, 72. Ans. . Of 18, the measures are 2, 3, 6, 9. Of 27 ; 3, 9. Of 20; 2, 4, 5, 10. Of 21; 3, 7. Of 24; 2, 3, 4, 6, 8, 12. Of 48; 2, 3, 4, 6, 8 12, 16, 24. Of 72 ; 2, 3, 4, 6, 8, 9, 12, 18, 24, 36. 2. Find the measures of the following numbers, 108, 120, 432, 936, 846. 3. Reduce the following Fractions to their lowest terms. *720 ; 172'; ; 23 ; iii ; 1184 ; 384 33 i #fiji 111; 29. Ans. bis; i ; ; ; 29*; nou i någ; ; . 4. Reduce the following: 19; 21; 2011; 8 3 8 0 8 3429 541 756 36, 81 2323 ; T9888 iri 7328172 by 3. 4_1_4.6 117 94682 39972 998.8 11 103284 ģ XLIII. In almost every one of the above instances the pupil las found it necessary to perform several successive divisions. But if he could have found at once, the greatest number, which would divide both terms without remainder, only one division would have been necessary. A NUMBER WHICH IS A MEASURE OF TWO OR MORE NUMBERS IS CALLED THEIR COMMON MEASURE. THE GREATEST NUMBER WHICH WILL MEASURE TWO OR MORE NUM. BERS IS CALLED THEIR GREATEST COMMON MEASURE. MENTAL EXERCISES. 1. What is the greatest common measure of 4 and 6 ? of 8 and 12? of 14 and 16 ? of 6 and 9 ? 2. What is the greatest common measure of 4 and 10 ? of 4 and 14 ? of 9 and 15? of 10 and 15 ? of 7 and 21 ? 3. What is the greatest common measure of 3 and 27 ? of 30 and 9? of 15 and 20 ? of 20 and 30 ? of 30 and 40 ? 4. What is the greatest common measure of 2 and 20 ? of 8 and 26? of 6 and 16 ? of 18 and 24? 5. What is the greatest common measure of 12 and 16 ? of 16 and 20 ? of 12 and 18 ? of 16 and 18 ? 6. What is the greatest common measure of 35 and 28 ? of 28 and 21 ? of 21 and 14 ? of 14 and 2 ? In the case of small numbers like the above, there is no difficulty in finding the greatest common measure. The following are not so easy. 7. Find the greatest common measure of 125 and 375. We cannot tell what this is at first sight. But we see by the above examples, that the smaller number is sometimes, itself the measure sought. Let us try whether this is not the case in the present instance. We see by this illustration, that 125 is a measure 125)375(3 of 375, and as 125 cannot be measured by a number 375 greater than itself, it is, of course, the greatest common (measure sought. 8. Find the greatest common measure of 144 and 168. Try first, as before, whether 144 is the measure. 144)168(1 168---144 leaves 24 remainder. Now if 24 will 144 measure 144, it will also measure 144+24=168. 144:24=6. 24 then is a common measure of 144 24)144)6 and 168. It is also the greatest common measure, 144 for any number which will measure 144 and 168, will measure their difference. For it is contained an even number of times in 144, and also a greater even number of times in 168. But taking an even number of times any thing from an even number of times the same thing, will leave an even number of times that thing. But 24 cannot be measured by any number greater than itself. Therefore, there can be no greater common measure of 144 and 168, than 24. The explanation would be similar, if several successive divisions took place. Therefore, to find the greatest common measure of two numbers, DIVIDE THE GREATER NUMBER BY THE LESS, THAT DIVISOR BY THE REMAINDER, AND SO ON, ALWAYS DIVIDING THE LAST DIVISOR BY THE |