Examples. 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? Ans. 2, 2, 17. Ans. 2, 5, 7. Ans. 2, 3, 5, 7. 8. What are the prime factors of 735? Ans. 3, 5, 7, 7. MULTIPLICATION BY FACTORS. 91. 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. 7. Multiply 301 by 50, or 10 × 5. Ans. 15050. 8. What cost 42 horses, at $165 each? Ans. 36000 dollars. 10. Required the weight of 56 bags of grain, each weighing 112 pounds. Ans. 6272 pounds. 11. Required the cost of 249 bales of goods, at $28 per bale. 12. A farm has been divided into 49 lots, each containing 570 square rods; how large was the farm? Ans. 27930 square rods. 92. 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? To prove this principle, let the dividend be 16 and the divisor 4. Then, 1644. Now, if we divide the dividend and divisor by some number, as 2, the dividend becomes 8, and the divisor, 2; and 824, the same result as before. 93. 1. Divide 60 by 15. OPERATION. 60 X 20 4, Ans. 15 $ X 5 For convenience, we indicate the division, by writing the dividend over the divisor. divisor. Factoring, the dividend becomes 3 X 20, and the divisor 3 X 5. Canceling, by crossing the factor 3, common to both dividend and divisor, we have 20, or 20÷ 5, equal to 4, the quotient required. 9, common to both dividend and divisor, we have left in the dividend 5 X 2, or 10, which is the quotient. the dividend and divisor, we have left in the dividend 3, which is the quotient. Rule.-Cancel all the factors common to both dividend and divisor, and then divide the product of the remaining factors of the dividend by the product of the remaining factors in the divisor. When a factor is canceled, 1 is understood to remain, but does not require to be written, except in the quotient, when there are no other factors. Examples. 4. Divide the product of 16 X 14 by 8 X 7. Ans. 4. 5. Divide the product of 21 × 15 × 5 by 18 X 10 X 3. Ans. 211. 6. Divide the product of 33 X 35 X 6 by 11 × 7 X 3. 7. Divide the product of 72 × 8 by 12 × 6. Ans. 8. 8. How many bushels of potatoes, at 60 cents a bushel, must be given in exchange for 4 calico dresses, each containing 12 yards, at 30 cents a yard? 9. How many bushels of peaches, at $1.60 a bushel, must be given for 5 bushels of apples, at $1.28 a bushel? Ans. 4 bushels. 10. Exchanged 3 firkins of butter, each containing 56 pounds, at 24 cents a pound, for 6 bags of coffee, each containing 32 pounds; how much was the coffee a pound? Ans. 21 cents. What is the Rule? When a factor is canceled, what remains? When is the factor 1 to written? 94. ANALYSIS. Analysis, in Arithmetic, is the process of solving a question or example by considering the relation of its numbers. No rules, therefore, are required for the solution of questions by analysis. MENTAL EXERCISES. 1. If 6 oranges cost 18 cents, what will 8 oranges cost? SOLUTION. — If 6 oranges cost 18 cents, 1 orange will cost 1 sixth of 18 cents, which is 3 cents, and 8 oranges will cost 8 times 3 cents, which are 24 cents. Therefore, if 6 oranges cost 18 cents, 8 oranges will cost 24 cents. 2. If 5 peaches cost 10 cents, how much will 10 peaches cost? 3. If 7 yards of velvet cost $28, how much will 9 yards cost? 4. If 8 pounds of sugar cost 72 cents, how much will 10 pounds cost? 5. When 36 cents are paid for 4 quarts of berries, how much must be paid for 7 quarts? 6. What will 10 quarts of milk cost, if 5 quarts cost 40 cents? 7. When $6 will buy 24 yards of cloth, how many yards will $8 buy? 8. If 9 pounds of veal cost 54 cents, how many pounds can be bought for 42 cents? 9. If 7 plows can be bought for $77, how many can be bought for $44? What is Analysis? |