EXERCISES ON CHAPTER II. Sign of Aggregation. § 2. A parenthesis () is used to denote the Aggregate of the inclosed expression; and this aggregate connects with whatever sign immediately precedes or follows the parenthesis. Thus (13+7)×5 denotes the sum of 13 and 7 multiplied by 5; that is, 20X5. 1. Find the value of the expression, (246+34+9+16+1)×4. 2. Find the value of the expression, (370+65+100+3—90)×5. 3. Find the value of the expression, 4. Find the value of the expression, 5. Find the value of the expression, Ans. 1224. Ans. 2240. Ans. 3175. Ans. 2355584. Ans. 280 71 6. Find the value of the expression, (4093757-307609)X(5083—3). Ans. 19233631840. 7. Find the value of the expression, (73017600-189976)÷(763+7). 8. Find the value of the expression, Ans. 94581354. (83073769+23764)X(7307—4). Ans. 606861283499. 9. Find the value of the expression, (7360793-283746+3848-500)÷(75013—113). 10. The sum of two numbers being 35745, and one of the numbers 1740, what is the other number? Ans. 34005. 11. The difference of two numbers being 13000, and the less number 635, what is the greater number? Ans. 13635. 12. The difference of two numbers being 37073, and the greater number 739860, what is the less number? Ans. 702787. 13. The product of two numbers being 96350, and one of the numbers 470, what is the other number? Ans. 205. Two Numbers Found from their Sum and Difference. § 63. The Sum+the difference of two numbers is TWICE the greater number; and the Sum--the difference of two numbers, is TWICE the less number. For example, take the numbers 10 and 6. Their sum 16+their difference 4 is 20, which is twice 10. and their sum 16-their difference 4 is 12, which is twice 6. 14. The sum of two numbers is 80, and their difference is 20. What is the greater, and what the less number? 80+20 is 100, twice the greater; and 80-20 is 60, twice the less; hence the greater number is of 100, and the less is of 60. Ans. 50 and 30. 15. The sum of two numbers is 1000, and their difference is 200. What are the two numbers ? Ans. 600 and 400. 16. The sum of two numbers being 1840, and their difference 500,-what are the two numbers? Ans. 1170 and 670. 17. The product of two numbers being 11600 and the multiplicand 80, what is the multiplier ? Ans. 145. 18. The product of two numbers being 46400, and the multiplier 240, what is the multiplicand? Ans. 193 19. The dividend being 23200, and the quotient 160, what is the divisor? The dividend is a product, and the quo. a factor. Ans. 145. 20. The sum of 1728 dollars having been divided equally among a number of men, each man received 24 dollars. What was the number of men? Ans. 72 men. 21. A and B together have 2300 dollars, and A has 500 dollars more than B. What sum has each ? Ans. A has 1400, and B 900 dollars. 22. An ironmonger bought 15 tons of ton, and 23 tons at 37 dollars per ton. by selling the whole at 42 dollars per ton? 23. C and D together have 4348 dollars, and C has 375 dollars less than D. What sum has each ? Ans. C has 19861⁄2, and D 23611⁄2 dollars. 24. A merchant collected from A 230 dollars, from B 303 dollars, and from C 95 dollars. If he lay out the whole sum collected, for cloth at 7 dollars a yard, how much can he buy? Ans. 89 yards. 25. A farmer wishes to fill three kinds of sacks, containing 3 bushels, 4 bushels, and 5 bushels, and the same number of each kind, with 1728 bushels of corn. How many sacks can he fill? Ans. 144 of each kind. 26. A farmer sold wheat for 900 dollars, corn for 274 dollars, and other produce for 329 dollars. Out of these proceeds he bought three pair of oxen at 55 dollars a pair, and paid the remainder for 65 acres of land. What did the land cost him per acre? Ans. 20 dollars. 27. A bought a building lot in town for 75 dollars, which was at the rate of 200 dollars per acre, and B purchased 3 pasture lots, each containing 13 acres, for 950 dollars. What quantity of ground was in A's lot, and what did B pay per acre? Ans. of an acre; and B 24 200 dollars. 28. If a person's income be 5000 dollars a year, and his expenses be at the rate of 5 dollars per day, at what rate would he save money per day,—there being 365 days in a year? Ans. 8 dollars per day. 29. An army of 5000 men have provisions for 3 months. If 1625 men be discharged, how long will the same provisions suffice for the remainder? Ans. 4 months. 1500 3375 30. A carpenter can earn 45 dollars a month, but his necessary expenditures are at the rate of 24 dollars a month. He wishes to purchase a certain lot of ground, which contains 19 acres, and is held at 35 dollars per acre. In what time can he save enough to make the purchase? Ans. 3114 months. 31. A sold to B 15 cords of wood at 3 dollars per cord, 53 barrels of corn at 2 dollars per barrel, and 2 beeves at 30 dollars each. In payment, A takes 160 dollars in cash, 3 sacks of coffee at 14 dollars a sack, and 20 gallons of molasses. What did A's sales amount to, and what did the molasses cost him per gallon? Ans. 211 dollars; and of a dollar per gallon. CHAPTER III. COMPOSITE NUMBERS.-PRIME FACTORS.-COMMON MEASURE. COMMON MULTIPLE. COMPOSITE NUMBERS. $ 64. A composite number is one which is the product of two factors, each greater than a unit. Thus 4 is a composite number, being 2×2. Is 6 a composite number? Is 7? Is 12? Is 19? Is 36? Is 45? Decomposition of Numbers. $65. Decomposing a number consists in resolving the number into its factors. Thus 6 is decomposed when resolved into the factors 3 and 2. Into what two factors may 15 he resolved? 21? 33 84? 99? Into what three factors may 24 be resolved? 30 70? 36? 100? § 66. In Division, the dividend is resolved into two factors, one of which is the divisor, and the other the quotient. Taking 4 as a factor of 20, what is the other factor? 7 being one factor of 56, what is the other factor? 9 being one factor of 108, what is the other factor? 12 being one factor of 144, what is the other factor? Any number whatever may be resolved into itself multiplied by a unit. Thus 5 is 5X1; 7 is 7X1, &c. Sign of Equality. § 67. The sign =, equal to, placed between two numbers, or numerical expressions, signifies that they are equal to each other. Thus 12+8=4X5 signifies that the sum of 12 and 8 is equal to the product of 4 and 5; and is read 12 plus 8 is equal to 4 into 5. Constant Product of Several Factors. 68. The Product of several factors remains the same in whatever order the factors are multiplied together. Take, for example, the product 2×3×5. Since 2X3=3×2, we have 2×3×5=3×2×5; and since 2X5=5×2, we have 3×2×5=3X5X2; and so on, there being six different ways in which the factors may be multiplied together. Division by the Canceling of Factors. § 69. A Product is divided by either of its factors by canceling that factor; or by the product of any two or more of its factors, by canceling those factors. For example, take 30=2×3×5. If we divide it by 2, the quotient will be 3×5, or 15; and if we divide it by 2X3, or 6, the quotient will be 5 (§ 66). If the product is to be divided by itself, all its factors must be canceled, and their place supplied by a unit; for a number is contained in itself once. The cancellation of a number is denoted by a line drawn across it. Thus 2×3×5 denotes that the 2 in this product is canceled, which is equivalent to dividing said product by 2. COMPOSITE MULTIPLIERS AND DIVISORS. When a multiplier or divisor can be resolved into factors, each of which shall be a number not exceeding 12, or such number with Os annexed, it will sometimes shorten the operation to multiply or divide by means of such factors. RULE IX. $ 70. To multiply by means of FACTORS. Resolve the multiplier into two or more factors; multiply by one of the factors, and the product thence arising by another factor; and so on, until all the factors are employed. The last product will be the one required. To multiply 345 by 18. EXAMPLE. Resolving 18 into the factors 3 and 6, we have 345X3=1035; and 1035×6=6210. Then 345X18=345×3×6=6210 (§ 68). |