The process of the last example is the shortest foi finding the greatest common divisor of small numbers; but when the numbers are large and the common factors not so readily found, the following method is generally adopted. 5. Find the G. C. D. of 84 and 147. 84 ) 147 (1 Divide the smaller number 84 into the larger, and the re63 ) 84 ( 1 mainder into the last divisor; 63 and again the remainder into the last divisor, until there is 21 ) 63 ( 3 63 no remainder; the last divisor is the G. C. D. of the two numbers. ANALYSIS.—As each number is a multiple of the G. C. D., so the difference of the numbers is also a multiple of the G. C. D.; hence in every case the dividend and divisor are multiples of the G. C. D., and whenever the divisor is contained in the dividend, that divisor is the G. C. D. 6. Find the G. C. D. of 323 and 425. G. C. D = 17. 17. Find the G. C. D. of 2310 and 4626. G. C. D. = 6. CANCELLATION. THEOREM. The dividend contains all and exactly the same factors as the divisor and quotient. Any composite number is the product of all its prime factors, and may be resolved into them. The prodnct of any two integral numbers is a composite number and must contain all the factors of both numbers; and as a dividend is the product of its divisor and quotient, it must contain the same factors as its divisor and quotient. COR. 1.—The same is true if one or both divisor and quotient be fractional ; for when reduced to a common denominator, their numerators may be regarded as integral. COR. 2.-Every factor of the divisor will cancel the same factor in the dividend. COR. 3.— The factors which are not canceled by those of the divisor will be the factors of the quotient. COR. 4.—Canceling a factor in the dividend divides the quotient by the same factor. COR. 5.-Canceling a factor in the divisor multiplies the quotient by the same factor. PROBLEMS. 1. Divide 648 by 36. = 18, Ans. 2, 2, 3, 3. 2. Divide 625 by 125. 625 中 = 5, Ans. 599 5, Ans. 100 4. A man bought 30 yards of cloth at $5 a yard; he then exchanged it for other cloth at $3 a yard. How many yards of the latter did he get ? PRACTICAL EXAMPLES. 1. A man bought 30 cows at 25 dollars each; he then exchanged the cows for horses at 50 dollars each ; how many horses did he get ? Ans. 15 horses. 2. A farmer sold 1500 bushels of wheat at 125 cents per bushel, and received in return barley at 75 cents per bushel. How many bushels of barley did he get ? Ans. 2500 bu. barley. 3. How many acres of land, at 25 dollars per acre, can be obtained for 5 houses and lots at 750 dollars each ? Ans. 150 acres. 4. How many yards of flannel, three-quarters of a yard wide, will line a coat made of 3 yards of cloth six-quarters wide ? Ans. 6 yards. 5. Sold 320 acres of land at 60 dollars per acre, and invested the proceeds in other land at 40 dollars per acre; how many acres did I get ? Ans. 480 acres. 6. Exchanged 432 pieces of cloth at 18 dollars each, for linen at 6 dollars a piece. How many pieces of linen did I get ? Ans. 1296 pieces. 7. Sold a farm of 477 acres at 48 dollars per acre, and invested the returns of sale in another farm at $36 per How many acres did I buy? Ans. 636 acres. 8. Sold 15 horses at 732 dollars each, and bought sheep for the proceeds at 4 dollars each. How many sheep did I buy? Ans. 2745. 9. Multiply the following numbers: 12, 15, 27, 28, 32, and divide the product by 2, 3, 4, 5, 6, 7, 8, and 9. Ans. 12. acre. FRACTIONS. DEF. 1.- If a unit or any other number is divided into equal parts, one or more of these parts is a fraction of the whole, and all the parts constitute the whole. If a unit is divided into two equal parts, each part is called one-half, and is written t; and the two halves constitute the whole; thus, t=1. If a unit is divided into three equal parts, each part is one-third (}); two of the parts, f; and the three thirds constitute the whole; thus, j=1. If 5 is divided into two equal parts, each part is five-halves (6); two of the parts, 10 = 5, etc.; if 6 is divided into three equal parts, each part is f= 2; and ; two-thirds of 6 is 4, etc. Fourths, fifths, sixths, etc., are similarly constructed. 2. When a unit is divided into equal parts, any number of the parts less than the whole, expressed fractionally, is called a Proper Fraction; as, 1, $, 1, 1%, etc. The quotient is the same in division when the dividend is less than the divisor; as, Ps, 1, etc.; but when the divisor is less than the dividend, the quotient is called an Improper Fraction; as, , , y, etc. 3. When the division indicated by an Improper Fraction is performed, and the divisor is not contained an exact number of times in the dividend, the quotient is partly integral and partly fractional, and is termed a = Mixed Number; thus, =11; 5 0 = 8; and 50 25 = 10%. COR.—The denominator expresses the number of parts into which a unit or any other number is divided, and the numerator expresses the number of parts of a unit taken, or the number divided. EXEMPLIFICATION.-If each half is divided into two equal parts, the whole number of parts is four, and the one-half has made two of those parts; hence, t= t; if each half is divided into three equal parts, the whole number of parts is six, and I =*;:: =i=== == 총 Po, etc., and f = = = = *, etc.; hence, if both terms of a fraction are multiplied by the same number, the value of the fraction is not changed, and by this principle fractions are reduced to a Common Denominator. PROBLEMS. 1 x2=2 2x3 26=Y 1X4=_4 35 5=25 7x5=3 1. Reduce j and to a common denominator. IX33, and XIX. 2. Reduce 3, 5, and 1 to a common denominator. 本 x4, and 19. 3. Reduce 4 and ķ to a common denominator. 4x12%), and . COR. 1.- The least common multiple of all the denominators is the least common denominator. COR. 2.-Fractions are reduced to a common denominator thus: Multiply both terms of each fraction by the quotient obtained by dividing its denominator into the least common denominator ; or, when all the denom |