Casting out 9's Addition, subtraction, multiplication, and division may be proved by casting out 9's. In addition, the sum of the excesses of I's in the several addends, should equal the excess of 9's in the sum. 5678 8 2 4934 We add the digits of the first addend, 5, 11, 18, 26; then the digits of the sum, 26, and write 8. This is the excess of the 9's, for 5678 ÷ 9 gives 8 for a remainder. We proceed in the same way with the other addends: 4, 13, 16, 20; 2: 3, 9, 11, 18; 9; 0 (a sum of 9, or any multiple of 9, we count as 0). We proceed in the same way with the excesses, 8, 10; 1. 14,239 1 We proceed in the same way with the sum, 1, 5, 7, 10, 19; 10; 1. The sum of the excesses of 9's in the several addends is 1; the excess of 9's in the sum is 1; the answer is probably correct. In subtraction, the excess of 9's in the minuend, should equal the excess of 9's in the subtrahend and the remainder. 7683 6 3997 1 3686 5 Minuend, 7, 13, 21, 24; 6. Subtrahend, 3, 12, 21, 28; 10; 1: remainder, 3, 9, 17, 23; 5: their sum, 6. In multiplication, the excess of 9's in the product of the excesses of the factors, should equal the excess in the answer. 968 5 453,024 0 Multiplicand, 9, 15, 23; 5: multiplier, 4, 10, 18; 9; 0: their product, 0. Product, 4, 9, 12, 14, 18; 9; 0. In division, the excess of 9's in the product of the excesses of the ́quotient and divisor, plus the excess in the remainder, should equal the excess in the dividend. RELATIONS Give 166. When 13,825, or 13,820 +5, is divided by 2, why is the remainder the same as when its last digit is divided by 2? the rule for the divisibility of a number by 2. The first part, 13,820, or 1382 tens, is divisible by 2 because ten is divisible by 2. Since the first part is divisible by 2, the divisibility of the number depends upon the last digit. 167. When 13,825, or 13,800+25, is divided by 4, why is the remainder the same as when the number denoted by its last two digits is divided by 4? Give the rule. 168. When 13,825, or 13,000 + 825, is divided by 8, why is the remainder the same as when the number denoted by its last three digits is divided by 8? Give the rule. 169. What is the remainder when 1 with any number of ciphers is divided by 9? 170. What is the remainder when 2, 3, 4, 5, 6, 7, or 8, times 1 with any number of ciphers, is divided by 9? 171. When 13,825, or 10,000 + 3000 + 800 + 20 + 5, is divided by 9, why is the remainder the same as when the sum of its digits is divided by 9? When 10,000 is divided by 9 the remainder is 1; when 3000 is divided by 9 the remainder is 3; etc. See Ex. 170. 172. If a number divided by 9 gives the same remainder as the sum of its digits divided by 9, what is the rule for the divisibility of a number by 9? 173. Show that 1 with any odd number of ciphers lacks 1 of being a multiple of 11. 174. Show that 1 with any even number of ciphers exceeds by 1 a multiple of 11. 175. How much does 2, 3, 4, 5, 6, 7, 8, 9, times 1 with any odd number of ciphers, lack of being a multiple of 11? 176. How much does 2, 3, 4, 5, 6, 7, 8, 9, times 1 with any even number of ciphers, exceed a multiple of 11? 177. When 75,316, or (70,000+300+6)+(5000+10), is divided by 11, why is the remainder the same as when the sum of the digits in the odd places minus the sum of the digits in the even places, is divided by 11? 70,000+300+6 exceeds a multiple of 11 by 7+3+6 (Ex. 176); 5000+10 lacks 5 + 1 of being a multiple of 11 (Ex. 175); 75,316 exceeds a multiple of 11 by (7+3+6) −(5 + 1). 178. If a number divided by 11, gives the same remainder as the difference between the sum of its digits in the odd places and the sum of its digits in the even places, divided by 11, what is the rule for the divisibility of a number by 11? 179. A contractor is to build houses 24, 36, 48, and 60 feet long, and 16, 32, 32, and 48 feet wide. What length of clapboard can be used most conveniently for the sides? for the ends? State the relations. Relation number of feet in length of clapboard for the sides the largest number that is exactly contained in 24, 36, 48, and 60, or their G. C. D.... 180. A lady wishes to buy a piece of cloth which she can cut without waste into an exact number of pieces either 3, 4, or 5 yards long, as she may decide later. What is the smallest number of yards the piece can contain? 181. A real estate agent wishes to divide 3 pieces of land 325, 675, and 950 feet wide, into town lots of equal width. What is the largest possible width for each lot? 182. Ropes 48, 52, and 56 feet long are to be cut into the longest possible equal lengths. How long must each piece be? 183. A, B, and C start together around a circular track. A goes once around in 6 minutes; B, in 8 minutes; C, in 9 minutes. What is the least number of minutes before they will be together again at the starting point? State terms and relations. Given terms: number of minutes passed when A is at the starting point is 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ; B, 8, 16, 24, 32, 40, 48, 56, 64, 72, . .; C, 9, 18, 27, 36, 45, 54, 63, 72. Relation number of minutes before they are again together - least number that will exactly contain 6, 8, and 9, or their L. C. M. AMER. ARITH.- 6 184. After how many minutes will A and B first be together at the starting point? A and C? Band C? 185. D, E, and F start together around a circular track 5280 feet in length. D rides 1760 feet per minute; E, 1320; and F, 1056. How many times must D ride around the track before they are all together again at the starting point? State the relations. = = Relations: number of minutes D goes once around 5280 1760; number minutes B 5280 ÷ 1320; number minutes C 5280 ÷ 1056. = The minutes when they are first together the L. C. M. of the minutes each makes the circuit; number times A goes around = L. C. M. number minutes A makes the circuit. 186. By counting eggs, 4, 6, or 10 at a time, a farmer had none left over in each case. What is the least number he could have had? State the relations. 187. By counting eggs 4, 6, or 10 at a time, a farmer had 3 left over in each case. What is the least number he could have had ? State the relation. 188. By counting eggs 4, 6, or 10 at a time, a farmer had 3 eggs left over in each case; counting 11 at a time, he had none left. What is the least number he could have had? State the relations. Relations: possible numbers 3+ common multiples of 4, 6, and 10; the the least of these results divisible by 11. least number Solution the L. C. M. of 4, 6, and 10 is 60; 63, 123, 183, 243, 303, are numbers representing in order 3+ common multiples of 4, 6, and 10; 363 is the least of these which contains 11. 363, COMMON FRACTIONS FIRST CONCEPTION AN EXPRESSION OF DIVISION Division may be expressed by writing the dividend above, and the divisor below, a horizontal line. Such an expression is a common fraction; the dividend is the numerator; the divisor, the denomi nator. The numerator, or the denominator, or both, may contain fractions; such an expression is a complex fraction. We sometimes speak of a fraction of a fraction; a compound fraction. An integer plus a fraction is a mixed number. The plus sign is usually omitted. ILLUSTRATIONS common fraction. 4, numerator. State the terms, and the meaning denominator; it means 3 ÷ 4. 3 is the numerator; 4, the of the fractions: NOTE. ; is the numerator;, the denominator; it means - ਤੂੰ . The pupil should read the first part of p. 45, the whole of p. 49, and the explanation of Ex. 55 on p. 51. |
