30. A testator provided in his will, that his estate, valued at 25479 dollars, should be divided as follows, viz: his only son to have 13500 dollars, his only daughter 5375 dollars, and his widow the remainder. What sum did the widow receive? Ans. 6604 dollars. 31. A gentleman who owned a tract of land containing 15735 acres, sold from it, at different times, 3050 acres, 521 acres, 200 acres, and 2370 acres. How many acres of the tract had he then remaining? Ans. 9594 acres. 32. Borrowed of my neighbor at one time 175 dollars, at another 340 dollars, and at another 520 dollars. Having paid him 685 dollars, what balance have I yet to pay? From the sum 175+340+520, subtract 685. Ans. 350 dollars. 33. Put in store at one time 500 pounds of hemp; at another time 3800 pounds; and at another 2005 pounds. Having withdrawn 3473 pounds, what quantity remains in store? Ans. 2832 pounds. 34. A merchant bought flour at one time for 325 dollars; and at another time for 460 dollars. Having become damaged, the whole was sold at a loss of 184 dollars; for what sum was it sold? Ans. 601 dollars. 35. A grocer purchased brandy for 195 dollars, and wine for 370 dollars. He sold the former at a loss of 35 dollars, and the latter at a loss of 80 dollars. What did he get for the whole ? Ans. 450 dollars. 36. A manufacturer sold two bales of cotton, which together contained 2000 yards. The first bale containing 985 yards, how many yards were in the second bale? Ans. 1015 yards. 37. A bought of B 875 acres of land, for 23400 dollars. For 500 acres of the tract, he paid 11379 dollars; how many acres were in the remainder of the tract, and for what sum was it purchased? Ans. 375 acres; and 12021 dollars. 38. One farmer had two fields of corn, each producing 230 barrels; and another had three fields, each producing 195 barrels. Which of the two raised the larger crop, and by how many barrels ? Ans. The second, by 125 barrels. 39. Three persons, A, B, and C, propose to purchase a manufactory, valued at 25850 dollars. A agrees to pay 5000 dollars, B twice as much as A, and C the remainder; what sum will C have to pay ? Ans. 10850 dollars. 40. A merchant exchanges a stock of goods worth 6725 dollars, and a house worth 3120 dollars, with a farmer, for a tract of land valued at 5900 dollars, the deficiency on the part of the land to be made up in money. What sum will the merchant receive? Ans. 3945 dollars. 41. A has 2070 dollars more than B, and 375 dollars less than C, who has 4000 dollars; what sum then has B ? Ans. 1555 dollars. 42. Two travelers start together from the same place, and journey in the same direction, for three successive weeks,— the first at the rate of 240, the second 275 miles, per week; how far are the travelers apart at the end of the time named ? Ans. 105 miles. 43. Having 4800 dollars on hand, I wish to add to this a sum which will enable me to purchase a farm at 5390 dollars, and leave 300 dollars for other purposes. Required the sum to be added to the first one. Ans. 890 dollars. 44. A planter sold cotton amounting to 3460 dollars, and out of these proceeds purchased groceries for 150 dollars, and other provisions for 375 dollars. How much of the first sum had he remaining? Ans. 2935 dollars. 45. A cabinet maker sold furniture to the amount of 4000 dollars, and received, in part payment, at different times, 100 dollars, 200 dollars, 475 dollars and 904 dollars. How much of the debt remains to be paid? Ans. 2321 dollars. 46. A says to B, "in my fruitful pastures, I have 300 fat B replies, "then I have 120 more than you." Says C, "I lack but 35 of having as many as you both." How many had C? Ans. 685 oxen. oxen." 47. In 1820, the population of the United States was 9 millions, 638 thousand, one hundred and sixty-three; in 1830, it was 12 millions, 856 thousand, one hundred and sixty-five; and in 1840, 17 millions, 63 thousand, three hundred and fifty-three. What was the increase from 1820 to 1830, and what from 1830 to 1840? 3218002, from 1820 to 1830; 4207188, from 1830 to 1840. Ans. 48. The population of Europe being estimated at 238 millions, 473 thousands, 957; of Asia at 390 millions; of Africa at 65 millions; of Oceanica at 20 millions; of N. America at 35 millions; and of S. America at 15 millions, 240 thousand;— how much does the population of Asia exceed that of all the other grand divisions of the Earth? Ans. 16286043 inhabitants. MULTIPLICATION. § 37. MULTIPLICATION consists in finding the sum or product of a number taken any given number of times. The number multiplied is called the multiplicand; and the multiplying number, the multiplier; the two together are also called the factors of the product. Thus, 3 times 5 is 15; then 3 is the multiplier, 5 the multiplicand, and 15 the product. Also 3 and 5 are the factors of 15. Which is the multiplicand? The multifactors? How many is 3 times 2? plier? The product? The How many is 3 times 3? 4 times 3? 5 times 3? 6 times 3? Addition and Multiplication. § 38. The addition of the same number to itself, repeatedly, is a multiplication of that number. Thus 5+5+5 is 3 times 5; the sum or product being 15. How many is 6+6, or twice 6? How many is 8+8+8, or 3 times 8? or 3 times 6? 6+6+6, Recite the elementary products, once 1 is 1, once 2 is 2, &c., twice 1 is 2, twice 2 is 4, &c., 3 times 1 is 3, 3 times 2 is 6 &c., as given from left to right in the Multiplication Table. 8 Once 1 2 3 | 4 | 5 | 6 | 7 Twice is 2is 4is 6is is 10 is 12 is 14 is 16 is 3 times is 3 is 6 is 9 is 12 is 15 is 18 is 21 is 24 is 4 times is 4is Sis 12/1s 16 is 20 is 24 is 28 is 32 is 5 times is 5 is 10 is 15|is 20|is 25|is 30 is 35|is 40 is 6 times is 6is 12 is 18 is 24 is 30 is 36 is 42 is 48 is 7 times is 7 is 14 is 21|is 28 is 35 is 42 is 49 is 56 is 3 times is Sis 16 is 24|is 32|is 40|is 48¦is 56¦is 64¦rs 9 times is 9 is 18 is 27 is 36 is 45 is 54 is 63 is 72 is 10 times is 10 is 20 is 30 is 40 is 50|is 60 is 70 is 80 is 11 times is 11 is 22 is 33 is 41 is 55 is 66 is 77 is 8 is 12 times is 12 is 24 is 36 is 49 is 60 is 72 is 84 is 96 is 81 is 90 is 100 is 110 is 120 99 is 110 is 121 is 132 108 is 120 is 132 is 144| Constant Product. $39. The Product of two numbers remains the same, when the multiplicand and multiplier are taken the one for the other. Thus 3 times 15 gives the same product as 15 times 3. For, 3 times 15 must be 15 times as many as 3 times 1, which is 3; that is, 3 times 15 is just as many as 15 times 3. Prove that 4 times 23 is equal to 23 times 4. Prove that 5 times 17 is equal to 17 times 5. Prove that 6 times 39 is equal to 39 times 6. Sign of Multiplication. § 40. The sign X, called into, placed between two numbers, signifies that they are to be multiplied together. Thus 9×4, 9 into 4, signifies 9 times 4, or 4 times 9. § 41. The product of a concrete number taken any number of times, is a similar concrete number. Thus 3 times 5 cents is 15 cents. What is 3 times 9 pounds? 4 times 7 yards? 3 times 8 dollars? 6 times 9 miles? 7 times 10 acres? 8 times 11 furlongs? $ 42. A concrete number cannot be taken concretely as a multiplier; for, as a multiplier, a number can express nothing but repetitions of the multiplicand. For example, 3 hats at 5 dollars apiece would amount to 3 times 5 dollars, which is 15 dollars; the multiplier 3 expressing only repetitions of 5 dollars. We multiply by 3,-not by 3 hats. What would 4 slates amount to at 10 cents apiece? What would 5 books amount to at 8 cents apiece? What would 6 pounds of butter amount to at 12 cents a pound? What would 7 cords of wood amount to at 3 dollars a cord? What would 8 pair of boots amount to at 5 What would 9 tons of hay amount to at 8 dollars dollars a pair? a ton? If a man can walk 4 miles an hour, how far could he walk in 10 hours? If a yard of cloth sells for 6 dollars, what should be paid for 11 yards at the same rate? Allowing 12 men to do a piece of work in 5 days, in what time ought one man to do the same? RULE V. $ 43. To Multiply by a Number not exceeding 12; or by such Number with Os annexed. 1. Proceeding from right to left, multiply each figure of the multiplicand, and under each set its product, when less than 10. 2. When the product is 10 or more, set down its right hand figure, and add the left figure or figures to the next product. Set down the whole of the last product. 3. Ciphers in the right of either or both of the factors, are omitted in multiplying; but as many Os must be placed in the right of the product. 4. When the multiplier is 10, or 100, &c., the product will be formed by merely annexing to the multiplicand as many Os as there are Os in the right of the Multiplier. EXAMPLES. 1. To Multiply 5070 by 3; that is, to find 3 times 5070. 5070 15210 We place a 0 in the right of the product, for the 0 in the right of the multiplicand; and then say, 3 times 7 is 21; setting down the 1, we say 3 times 0 is 0, and the 2, from 21, added, makes 2; 3 times 5 is 15. The required product is 15210. 2. To Multiply 243 by 30; or to find 30 times 243. 243 Having placed a 0 in the right of the product, for the 0 in the right of the multiplier, we say 3 times 3 is 9; 3 times 4 is 12; 3 times 2 is 6, and 1 makes 7. 3. To multiply 359 by 100, we merely annex two Os to the multiplicand, and thus find the product 35900. |