The reason of this proof depends on the principle, that the sum of all the parts into which any number is separated is equal to the whole. (Art. 24, Ax. 13.) EXAMPLES. 2. 2. 3. OPERATION OPERATION AND PROOF. OPERATION. OPERATION AND PROOF. Dollars. Tons. 7 6 5 7 6 5 1 2 6 381 3 81 384 87 2 87 2 8 7 6 315 315 2 43 Ans. 2 3 3 3 First am't 2 3 3 3 Ans. 1 6 2 9 First am't 1 6 2 9 1 5 6 8 150 3 Ans. 2 3 3 3 Ans. 1 6 2 9 32. What is the sum of 15 +26+ 18+91 ? Ans. 150. 33. What is the sum of 6789 + 5832 + 4671 + 8907 ? Ans. 26199. 34. Required the sum of 76 +48 +59 +81. Ans. 264. 35. Required the sum of 123456 + 789012 + 345678 + 901234 + 567890 + 987654 + 321032 + 765437. 36. Required the sum of 876543 + 789112 + 345678 + 965887 + 445566 + 788743 + 399378 + 456789. 37. Required the sum of 789012 + 345678 + 901234 + 789037 + 891133 + 477666 + 557788 + 888878. 38. Required the sum of 987654 + 456112 + 222333 + 456789 + 987654 + 321178 + 123456 + 789561. 39. Required the sum of 678953 + 467631 + 117777 + 888888 + 444444 + 667679 + 998889 + 671236. 40. Required the sum of 783256 + 7128 + 39 + 432815 + 99 + 67851 + 125 + 641236 + 801 + 4328. 41. Required the sum of 12004 + 32 +1+7836 + 100 +46 + 3 + 6176 + 32 +91876. 42. Add together 763, 4663, 37, 49763, 6178, and 671. Ans. 62075. 43. Add together 15, 7896, 1, 13, 106, 113, 156, 100, 2201. 44. A butcher sold to A 369 lbs. of beef, to B 169 lbs., to C 861 lbs., to D 901 lbs., to E 71 lbs., and to F 8716 lbs. ; what did they all receive? Ans. 11087 lbs. 45. A owes to one creditor 596 dollars, to another 3961, to another 581, to another 6116, to another 469, to another 506, to another 69281, and to another 1261. What does he owe them all ? Ans. $ 82871. 46. If a boy earn 17 cents a day, how much will he earn in 7 days? Ans. 119 cts. 47. If a man's wages be 19 dollars per month, what are they per year? Ans. $ 228. 48. If a boy receive a present every New Year's day of 1783 dollars, how much money will he possess when he is 21 years old ? Ans. $ 37443. 48. Method of adding two or more columns at one operation. Ex. 1. A merchant paid for cloth 219 dollars, for flour 416 dollars, for hardware 711 dollars, and for rent 93 dollars. How much did he pay in all ? Ans. $ 1439. Beginning with the number last written down, we add the units and tens ; thus, 93 and 1 94, 2 19 and 10 - 104, and 6 110, and 10 - 120, and 416 9=129, and 10 = 139. Of this amount we write 711 the 9 units and 3 tens under the columns added ; 93 and add in the 1 hundred with the column of hundreds; thus, 1 (carried) and 7 = 8, and 4 = Ans. 1439 12, and 2 14 hundred, of which we write the 4 hundred under the column of hundreds and the 1 thousand in the thousands' place; and find the whole amount to be 1439. In like manner may be added more than two columns at one operation. NOTE. — The examples that follow can be performed as the above, or by the common method, or by both. 2. How many were the members of Congress in 1856, there being 2 Senators from each State, and Maine sending 6 Representatives, New Hampshire 3, Massachusetts 11, Rhode Island 2, Connecticut 4, Vermont 3, New York 33, New Jersey 5, Pennsylvania 25, Delaware 1, Maryland 6, Virginia 13, North Carolina 8, South Carolina 6, Georgia 8, Alabama 7, Mississippi 5, Louisiana 4, Tennessee 10, Kentucky 10, Ohio 21, Indiana 11, Illinois 9, Wisconsin 3, Iowa 2, Missouri 7, Arkansas 2, Michigan 4, Florida 1, Texas 2, California 2. Ans. 296. OPERATION. Dollars. 3. Water-mills were invented in the year 555 after Christ ; windmills 744 years after water-mills; pumps 126 years after windmills ; printing 15 years after watches 37 years after printing ; the spinning-wheel 53 years after watches ; the steam-engine 119 years after the spinning-wheel; the fireengine 14 years after the steam-engine; the spinning-frame 98 years after the fire-engine; and the electro-magnetic telegraph 71 years after the spinning-frame. What year was the electromagnetic telegraph invented ? Ans. 1832. 4. During the American Revolutionary war, the British lost at the battle of Lexington 273 men, at Bunker Hill 1054 men, at Long Island 400 men, at White Plains 300 men, at Fort Washington 1000 men, at Trenton 1020 men, at Princeton 400 men, at Hubbardton 200 men, at Bennington 800 men, at Brandywine 500 men, at Stillwater 600 men, at Germantown 500 men, at Saratoga 400 men, by Burgoyne's surrender 5791 men, at Fort Mercer 500 men, at Monmouth 400 men, at Rhode Island 260 men, at Brier Creek 23 men, at Stony Point 600 men, at Savannah 130 men, at Camden 325 men, at King's Mountain 1150 men, at Cowpens 800 men, at Guilford Court-House 600 men, at Hobkirk's Hill 250 men, at Eutaw Springs 700 men, at Yorktown 7000 men. What was the entire loss? Ans. 25976 men. 5. Required the number of square miles in the following States, there being in Maine 35,000, in New Hampshire 8,030, in Vermont 8,000, in Massachusetts 7,250, in Rhode Island 1,200, in Connecticut 4,750, in New York 46,000, in New Jersey 6,851, in Pennsylvania 47,000, in Ohio 39,964, in Michigan 56,243, in Indiana 33,809, in Illinois 55,405, in Wisconsin 53,924, in Iowa 50,914, and in California 188,982. Ans. 643,322. 6. Required the number of square miles in the following States, there being in Delaware 2,120, in Maryland 11,000, in Virginia 61,352, in North Carolina 45,500, in South Carolina 28,000, in Georgia 58,000, in Alabama 50,722, in Florida 59,268, in Mississippi 47,147, in Tennessee 44,000, in Kentucky 37,680, in Missouri 67,380, in Arkansas 52,198, in Louisiana 46,431, and in Texas 325,520. Ans. 936,318. 7. According to the census of 1850, Maine had 583,169 inhabitants, New Hampshire 317,976, Massachusetts 994,514, Rhode Island 147,545, Connecticut 370,792, Vermont 314,120, New York 3,097,394, New Jersey 489,555, Pennsylvania 2,311,786, Delaware 91,532, Maryland 583,034, District of Columbia 51,687, Virginia 1,421,661, North Carolina 869,039, South Carolina 668,507, Georgia 906,185, Florida 87,445, Alabama 771,623, Mississippi 606,526, Lousiana 517,762, Texas 212,592, Arkansas 209,897, Tennessee 1,002,717, Missouri 682,044, Kentucky 982,405, Ohio 1,980,329, Indiana 988,416, Illinois 851,470, Michigan 397,654, Wisconsin 305,391, Iowa 192,214, California 92,597, and the Territories 92,298. What was the whole number of inhabitants ? Ans. 23,191,876. SUBTRACTION. 49. SUBTRACTION is the process of taking one number from another to find the difference. When the two numbers are unequal, the larger is called the Minuend and the less number the Subtrahend ; and when the numbers are equal, either is the Minuend, and the other is the · Subtrahend. The result, or the number found by the subtrac. tion, is called the Difference, or Remainder. One number can be subtracted from another only when the units of both are of the same kind; and when the numbers are simple, the process is termed Subtraction of Simple Numbers. 50. To subtract simple numbers. Ex. 1. From 935 take 673. Ans. 262. OPERATION. We first take the 3 units from the 5 Minuend 9 3 5 units, and find the difference to be 2 units, Subtrahend 6 7 3 which we write under the figure subtracted. We then proceed to take the 7 tens from Remainder 2 6 2 the 3 tens above it; but we here find a difficulty, since the 7 is greater than the 3, and cannot be subtracted from it. We therefore add 10 tens to the 3 tens, which makes 13 tens, and then subtract the 7 from 13, and 6 tens remain, which we write below. Then, to compensate for the 10 tens, equal to 1 hundred, added to the 3 tens in the minuena, we add i hundred to the 6 hundred of the subtrahend, which makes |