INTRODUCTION TO CANCELLATION. MENTAL EXERCISES. 1. If we omit the factor 2 from 12 and 6, what factors will remain 1 2. Divide 24 by 6. Divide 24 by of 6. Divide of 24 by 6. 3. Divide of 24 by of 6. Divide 36 by 18, first taking out the common factor 6. 4. Is there any difference in the quotient of 48 divided by 12, and of 48 divided by of 12? 5. Divide 72 by 48, first omitting common factors. Divide 90 by 60 in the same way; 144 by 96. 6. Divide 2×2×2 by 2×2; 3×3×4 by 2×3; 3×4×5 by 3×5. 7. Divide 2×3×7 by 2×7; 2×3×4 by 2×3; 3×5×8 by 3×8; 6× 5×3 by 3×6; 2×79×10 by 9×2. CANCELLATION. 131. Cancellation is a process of abbreviating arithmetical operations by rejecting common factors in both dividend and divisor. PRINCIPLES. 1. The cancelling of a factor from any number divides the number by that factor. DEM. Thus if we take the factor 3 out of 24 we shall divide 24 by 3. 2. The cancelling of a factor in both dividend and divisor will not change the quotient. DEM.-Cancelling a factor in both dividend and divisor is the same as dividing them both by the same number, which, by the principles of division, does not change the quotient. 1. Divide 84 × 60 by 24 × 63. 2 OPERATION. 4 5 24×83 3 3 SOLUTION. We cancel the common factor 12 from 60 and 24, writing 5, the other factor of 60, above 60, and 2, the other factor of 24, below 24; we then cancel the common factor 21 from 84 and 63, writing 4, the other factor of 84, above 84, and 3, the other factor of 63, below 63; we then cancel 2 from 2 and 4, writing 2 above the 4; the product of the remaining factors of the dividend is 10, the product of the remaining factors of the divisor is 3; hence the quotient is 10 divided by 3, or 3. Rule.-I. Cancel the common factors from the dividend and divisor. II. Then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor. NOTES-1. The unit 1 takes the place of a cancelled factor, but need not be written, except in the dividend of the quotient, when there are no other factors of the dividend remaining. 2. A factor in one term will cancel two or more factors in the other term, when their product is equal to the former. 2. Some prefer to place the dividend upon the right and the divisor upon the left, of a vertical line. 2. Divide 12× 14 × 16 by 6x7x8. 3. Divide 20 × 32 × 35 by 4× 5 × 16. PRACTICAL PROBLEMS. Ans. 8. Ans. 70. Ans. §. Ans. 10. Ans. 42. 1. How many yards of muslin, worth 12 cents a yard, may be bought for 16 pounds of butter, worth 15 cents a pound? SOLUTION.-If one pound of butter is worth 15 tents, 16 pounds are worth 15x15 cents; for 16×15 cents, at 12 cents a yard, we can get as many yards of meslin as 12 is contained times in 15x16, which we find, by cancellation, to be 20. OPERATION. 5 4 13×18 =20 12 4 2. How many bushels of corn, worth 45 cents a bushel, must be exchanged for 125 pounds of butter, at 18 cents a pound? Ans. 50. 3. A exchanged rye, worth 84 cents per bushel, for 78 bushels of wheat, worth 98 cents per bushel; required the number of bushels of rye. Ans. 91. 4. How many bushels of corn, at 42 cents a bushel, must be given in exchange for 7 pieces of cloth, each containing 40 yards, at 36 cents a yard? Ans. 240. 5. How many boxes of tea, each containing 24 pounds, at 90 cents a pound, must be given for 27 firkins of butter, of 56 pounds each, at 20 cents a pound? Ans. 14. 6. A farmer sold a grocer 9 loads of apples, each load containing 18 bags, and each bag 2 bushels, at 35 cents a bushel, and received in payment 12 boxes of sugar, each con taining 135 pounds; what was the sugar worth a pound? Ans. 7 cents. PRIME NUMBERS. 132. No general method of determining prime numbers, beyond a certain limit, has yet been discovered, althoug much time has been spent in the investigation. 133. We give the following practical method, which consists in writing a series of numbers, and sifting out those which are composite. METHOD.-Since the even numbers after 2 are composite, we write the series of odd numbers; thus, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 3,5 7 3 3,7 5 3 3,5 7 3 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 51, 55, 57, 99. 5 3 7 3 5 83, 85, 87, 89, 91, 93, 95, 97, 99. Now, commencing at 3, since every third term is divisible by 3, every third number is composite, which we indicate by putting the tigure 3 over it. Commencing at 5, every fifth number is divisible by 5, and is therefore composite, hence we place a figure 5 over every fifth number. Proceed in the same manner with 7, and the numbers unmarked will be the prime numbers up to 100. This method was discovered by Eratosthenes, a Greek mathematician. He inscribed the series of odd numbers on parchment, and then cutting out the composite numbers, his parchment with its holes resembled ? sieve; hence the method has been called Eratosthenes Sieve. NOTE. This page is not to be studied and recited, but will be found o interest to teachers and advanced students. INTRODUCTION TO FRACTIONS. MENTAL EXERCISES. 1. If an apple is divided into two equal parts, what is one of these parts called? 2. What are two of these parts called? How many halves in anything? 3. What is of 6? of 4? of 12? of 16? of 10? of 18? of 20? of 24? of 28? of 36? 4. If I divide an apple into 3 equal parts, what is one of these parts called? 5. What are 2 and 3 of these parts called? How many thirds in anything? 6. The number of equal parts into which a unit may be divided is represented by a figure below the line; thus 2 represents halves; §, thirds; fourths, etc. 7. The number of fractional parts taken may be represented by a figure above the line; thus, represents 2 thirds; 3, 3 fourths; §, 5 sixths, etc. 8. What is of 6? of 9? of 12? of 18? of 15? of 21? of 27? Wnal are of 12? of 15? of 21? of 18? of 24? of 337 9. If I divide an apple into 4 equal parts, what is one of these parta called? If I divide in 5, or 6, etc., equal parts? 10. How many fourths make a whole? How many fifths? Sixthsi Sevenths? Eighths? Ninths? Tenths? 11. What is of 12? of 20? of 24? of 16? of 30? of 281 of 40?of 35? 12. What is of 20? of 15? § of 12? of 24? § of 271 of 30? of 22?of 64? 13. If a yard of muslin cost 24 cents, what will of a yard cost? What will of a yard cost? 14. Henry's age is 36 years, and his wife's agc is as much; what Is his wife's age? 15. If 5 melons cost 60 cents, what will 7 melons cost at the same rate? 16. What will 4 yards of satin cost at the rate of $6 for of a yard? 17. What must I ray for three-fourths of a ton of hay if five sixths of a ton cost $20? 18. What will 7 of a ton of coal cost at the rate of $4 50 for § of a ton? SECTION IV. COMMON FRACTIONS. 134. A Fraction is a number of the equal parts of a unit. 135. Fractions are divided into two classes; common fractions and decimal fractions. 136. A Common Fraction is one in which the unit is divided into any number of equal parts. 137. A Decimal Fraction is one in which the unit is divided into tenths, hundredths, etc. 138. A Common Fraction is expressed by two numbers, one written above the other, with a short line between them; thus, expresses 3 fourths. 139. The Denòminator of a fraction denotes the number of equal parts into which the unit is divided. 140. The Numerator of a fraction denotes the number of equal parts which are taken. 141. The Numerator and Denominator are called the Terms of the fraction. The numerator is written above the line, and the denominator below it. 142. Common Fractions consist of three principal classes; namely, Simple, Compound, and Complex. 143. A Simple Fraction is a fraction having a single integral numerator and denominator; as,, etc. 144. A Proper Fraction is a simple fraction whose value is less than a unit; as, 1. 145. An Improper Fraction is a simple fraction whose value is equal to or greater than a unit; as §, 7, 12, etc. 146. A Compound Fraction is a fraction of a fraction; as of, of § of 7, etc. 147. A Complex Fraction is one whose numerator, or denominator, or both, are fractional; as 5 of t ' of |