18. Divide 675 yd. 3 qr. by 17. Ans. 39 yd. 3 qr. 19. If 15 loads of hay weigh 16 T. 10 cwt., what is the weight of 1 load? Ans. 1 T. 2 cwt. 20. If the moon moves in 29 days 382° 6′ 55′′, how much is the daily motion? Ans. 13° 10′ 35′′. MISCELLANEOUS. 1. Reduce 15 cwt. 3 qr. 11 lb. to pounds. 2. Reduce 13120 rods to miles. Ans. 41 miles. 3. Required the cost of 5 tons of iron, at 6 cents a pound. Ans. $600.4. How many hundred-weight of flour can be bought, at 4 cents a pound, for $7.48? Ans. 1 cwt. 3 qr. 12 lb. 5. If a man travels one day 30 m. 2 fur. 23 rd., and another day 27 m. 3 fur. 17 rd., how far does he travel in all? Ans. 57 m. 6 fur. 6. If a merchant cut 11 yd. 3 qr. from a piece of calico containing 30 yd. 1 qr. 2 na., how much will remain? 7. If, from a cask containing 64 gal. 2 qt. 1 pt. 2 gi. of molasses, there should leak out 11 gal. 3 qt. 1 pt. 1 gi., how much would remain in the cask? Ans. 52 gal. 3 qt. 0 pt. 1 gi. 8. What is the weight of 3 dozen silver forks, each fork weighing 4 oz. 0 pwt. 6 gr.? Ans. 12 lb. 2 oz. 5 pwt. 9. How much must be paid for 2 ch. 28 bu. of coal, at 25 cents a bushel? Ans. $25. 10. Bought 3 pieces of cloth, each containing 42 yd., of which were sold one piece, and 27 yd. 1 qr. 2 na. of another; what quantity remained? REVIEW. — What is Reduction? (112.) Reduction Descending? (113.) Reduction Ascending? (116.) 11. How much older is John than James, John's birth having been on the 20th December, 1858, and James' on the 10th of August, 1863? Ans. 4 yr. 7 mo. 21 d. 12. If a man can lift 201 lb. 12 oz., and that is 8 times what a boy can lift, how much can a boy lift? Ans. 1 qr. 3 oz. 8 dr. 13. How many square yards of carpeting will it take Ans. 92 sq. yd. 4 sq. ft. for a room 26 ft. by 32 ft.? 14. If a steamer can cross the Atlantic, the distance being 3174 miles, in 15 days, what is her daily speed? Ans. 211 m. 4 fur. 32 rd. 15. A milkman left 2 gal. 3 qt. 1 pt. of milk at a boarding house, every morning for 6 days; what quantity did he leave in that time? Ans. 17 gal. 1 qt. 16. Into how many house-lots, of 76 square rods each, can a field of 29 acres 40 square rods be divided? Ans. 61, and 44 square rods remaining. FACTORING. 127. An Integer, or Integral Number, is a whole number. 128. A Prime Number is an integer which cannot be divided without a remainder by any integer other than itself, or 1; as 1, 2, 3, 5, 7. 129. A Composite Number is an integer which can be divided without a remainder by some integer other than itself, or 1; as 4, 6, 8, 8. What is an Integer? A Prime Number? A Composite Number? 130. A Factor of a number is any integer contained in it without a remainder; thus, 2 is a factor of 4. A Prime Factor is a prime number. 131. Factoring is the process of finding the factors of a composite number. A PRINCIPLE OF FACTORING. 132. A number is equal to the product of its prime factors. WRITTEN EXERCISES. 133. 1. Find the prime factors of 84. OPERATION. 2)84 2)42 3)21 7 By trial, we find that 84 is composed of two factors, 2 and 42; of which 2 is prime, and 42 is composite. The 42 we find composed of the factors 2 and 21; of which 2 is prime, and 21 is composite. The 21 is composed of two factors, 3 and 7, both prime. Therefore, the prime factors required are 2, 2, 3, and 7. Rule.-Divide the given number by any prime number greater than 1, that will divide it without a remainder, and the quotient, if composite, in the same manner; and thus continue until the quotient is prime. The divisors and the last quotient will be the prime factors required. PROOF. The product of the prime factors will equal the given number, if the work is right. What is a Factor? A Prime Factor? What is Factoring? How do you divide the given number? If the quotient is composite, how do you proceed? How long do you continue? What will be the prime factors? Recite the Rule. What is the Proof? Examples. Ans. 2, 2, 17. 2. What are the prime factors of 44? Ans. 2, 2, 11. 3. What are the prime factors of 57? 4. What are the prime factors of 68? 5. What are the prime factors of 70? 6. What are the prime factors of 95? 7. What are the prime factors of 210? 8. What are the prime factors of 735? Ans. 2, 5, 7. Ans. 2, 3, 5, 7. Ans. 3, 5, 7, 7.. MULTIPLICATION BY FACTORS. 134. 1. Let it be required to multiply 452 by 63. OPERATION. 452 3164 9 Ans. 28476 63 is equal to 7 times 9; hence 63 times 452 is equal to 9 times 7 times 452. 7 times 452 gives 3164, and 9 times 7 times 452, or 9 times 3164, gives 28476. Therefore, 452 multiplied by 63 is equal to 28476. Rule. - Multiply the multiplicand by one factor, and this product by another, and so on, until all the factors have been used; the last product will be the one required. Examples. 2. Multiply 365 by 35, using its factors. When the multiplier is a composite number, how can you multiply? Ans. By its factors. What is the Rule? 3. Multiply 205 by 27, or 3 X 9. Ans. 8480. Ans. 8336. Ans. 67440. Here, we annex a cipher to the multiplicand for the product of it by 10 (Art. 54), and then multiply by the 6; that is, we annex the cipher to the multiplicand, and multiply by the number expressed by the significant figure. Ans. 15050. 7. Multiply 301 by 50, or 10 × 5. 8. What cost 42 horses, at $165 each? 9. Multiply 24 d. 8 h. 42 m. 40 s. by 15, using its factors. Ans. 365 d. 10 h. 40 m. 10. Required the weight of 56 bags of grain, each weighing 1 cwt. 0 qr. 12 lb. Ans. 62 cwt. 2 qr. 22 lb. 11. Required the cost of 249 bales of goods, at $28 per bale. 12. A farm has been divided into 49 lots, each containing 3 A. 2 R. 10 P.; how large was the farm? Ans. 174 A. 2 R. 10 P. 135. CANCELLATION. Cancellation is the process of shortening operations in division, by rejecting equal factors from the dividend and divisor. Cancellation depends upon the principle that Dividing both the dividend and divisor by the same number will not change the quotient. What is Cancellation? Upon what principle does Cancellation depend? |