In the preceding examples, the numbers, when added, have been less than 10, and, of course, have required but one figure to express them. In the last example it will be seen that the numbers in the unit column, when added, amount to more than 10, and in the column of tens, the amount is more than tenthat is, ten tens. Let the student write the numbers to be added on the blackboard, and illustrate in the following manner: Seventy-eight equals seven tens plus eight units, and ninetysix equals nine tens plus six units. Operation 1st, Tens. Units. 7+ 8 = 96 16 +14= 2d, 3d. Tens. Units. 7+8 78 9 + 6 96 Writing the numbers, units under units, and tens under tens, and adding, we have sixteen 17 + 4 = 174 Ans. but fourteen units equal tens plus fourteen units, one ten plus four units; the left-hand figure of the units therefore belongs in the column of tens. It will be seen by this operation, that what is called carrying for ten, is simply adding numbers to the column where they belong. In practice, numbers are written as in operation 3d, and a part of the operation is carried on in the mind. The same may be illustrated, thus : 78 96 14 16 Placing the numbers as before directed, and adding the right-hand column, we find it amounts to 14 units, or 1 ten and 4 units. The next column amounts to 16 tens, or 100 and 6 tens, which, when added, make 4 units, 7 tens, and 174 Ans. 100, or 174, the answer. From the foregoing it is evident, that one in the column of tens is equal to ten in the column of units, and one in the column of hundreds is equal to ten in the column of tens. This is the reason why we carry for 10 rather than any other number. RULE. Add each column, beginning at the right hand, and set down the amount directly under the column, if it be less than 10; but if it be 10 or more, set down the right-hand figure, and add the left to the next column. Under the last left-hand column set down the whole amount. This is the same as carrying one for every ten. QUESTIONS.-11. What is the rule for Simple Addition? 12. How is Addition proved? Proof.-Perform the addition downwards, and if this last amount corresponds with the sum total, the work is supposed to be right. The following method may be adopted when the scholar has become acquainted with the rule of Division : Add the digits in the top line, and from their sum reject the nines, and write down the excess at the right hand, directly even with the figures in the line. Proceed in the same manner with each proposed line of figures. Then add the several remainders, or excess of nines, and from their sum reject the nines and write down the excess. Add the digits in the sum total, and reject the nines, and the excess, if the work be right, will be the same as the excess last obtained. Thus, The sum of the digits in the sum total 3497 8295 18304 Excess of 9's. 6 7Ex. 9's. This method of proof depends upon a property of the number nine, which belongs to no other digit but 3, which is a factor of 9; -namely, that any number divided by 9 will leave the same remainder as the sum of its digits divided by 9. This peculiar property of the number 9 grows out of the decimal relation of place. Were the ratio of increase any other than ten fold, it would belong to the number next before the ratio. E. g. Were the ratio five fold, it would belong to 4; were it six, it would belong to 5, &c. It will be observed, that if we remove a digit from the units' place to the tens, we increase its value by as many 9's as there are units in the digit. Thus: 2 units, 20 units, or two 9's plus 2 units.— If we remove the digit still further to the left, we increase the number of 9's ten fold at each remove. Thus: 200-22 nines plus 2 units, and 2000-222_nines plus 2 units. Hence it appears that whatever be the number of 9's expressed by the digit, the digit itself is always a remainder; and since the proof depends not upon the number of 9's, but upon the remainders, it is plain that if we add any number of columns of units upwards, and then add the several lines from right to left, we shall have the sum of the same digits. If therefore we cast out the nines from each sum, the remainders, if the work is right, must be the same. OBS. This method of proof, although not a demonstration, is, nevertheless, very satisfactory, for it presents an entirely different combination of digits; nor should we be likely to drop a 9, or a multiple of 9. EXERCISES. Art. 16.-—1. If a man pay 1496 dollars for a house, 734 dollars for a lot of land, 300 dollars for railroad stock, and 145 dollars for a share in a bridge, how much does he expend in the whole? Ans. 2675 dollars. 2. A man sold plank to the amount of 834 dollars; boards to the amount of 376; shingles to the amount of 400; timber 621; two masts, one for 30 and the other for 50 dollars; what was the amount of the whole? Ans. 2311 dollars. 3. A merchant, on settling his accounts, finds himself in debt to A. $100; to B. 60; to C. 78; to D. 80; to E. 447; how much does he owe in all? Ans. 765 dollars. 4. From the creation of the world to the Christian era was 4004 years; from that time to the Declaration of American Independence was 1776, and 64 years since that period. How many years since the Creation ? Ans. 5844 years. 5. A man by his will left his two sons 1450 dollars each; his four daughters 1200 each; to his wife 1500; to various charitable objects, 1834; what was the value of his estate? Ans. 11034 dollars. 6. If 1889 figures cover one side of a slate, how many will it take to cover both sides of 4 slates? Ans. 15112. 7. Bonaparte was born in the year 1769; lived 52 years. In what year did he die? Ans. 1821. 8. General Jackson took the Presidential chair in 1829 OCcupied it 8 years. In what year did his course terminate? Ans. 1837. 9. George Washington was born in the year 1732. lived 67 years. In what year did he die? Ans. 1799. He 10. The distance from New York to Rahway, N. J., is 20 miles, from Rahway to New Brunswick 12 miles, from New Brunswick to Princeton 18 miles, from Princeton to Trenton 12 miles, from Trenton to Bristol 10 miles, from Bristol to Philadelphia 20 miles. What is the distance to Philadelphia? 11. Lafayette was born in the year 1757. age of 78. In what year did he die ? from New York Ans. 92 miles. He died at the Ans. 1835. pounds; how Ans. 4320. strike in 24 12. A man sold five oxen, each weighing 864 much did they all weigh? 13. How many times does a common hours? 14. A gentleman left his two sons each clock Ans. 156. 1480 dollars; his only daughter 1500 dollars, and his wife 200 more than all his children; what was the wife's portion, and what was the value of the whole estate? { Wife's portion, 4660) } Ans. Whole estate, 9120 dolls. 15. There are two numbers the less is 1768; their difference is 961; what is the larger number? Ans, 2729. 16. From Boston to Providence it is 40 miles; from Providence to New York 198 miles; from New York to Philadelphia 92 miles; from Philadelphia to Wilmington 28 miles: from Wilmington to Baltimore 72 miles; from Baltimore to Richmond 110 miles; from Richmond to Raleigh 155 miles; from Raleigh to Charleston 256 miles; from Charleston to Savannah 113 miles; from Savannah to New Orleans 713 miles. How many miles from Boston to New Orleans, passing through the above places? Ans. 1777 miles. which 17. A man bought five firkins of butter; one firkin contained 150 pounds, another 60, another 75, another 98, another 125. How much did they all contain? Ans. 508. 18. There were five churches erected, one in cost 16,500 dollars, two in each, one in which cost which cost 12,640 dollars. the whole? When the columns to be added are long, the following method will be found convenient. Begin to add with the unit figure, as usual; and for every ten, place a dot against that figure which makes ten, or more than ten, and add the excess to the figure above it; and thus proceed to the top of the column. Write the excess of ten at the foot of the column added; then count the dots, and as many as they are, so many carry to the next left-hand column. —, which cost 18,350 dollars 19,386 dollars, and one in How much was the expense of Ans. 85,226 dollars. QUESTION.—When the columns to be added are long, how may you proceed ? SUBTRACTION. Art. 17.-1. John's father gave him 6 apples. He gave his brother 4 of them. How many had he left ? 2. Joseph bought sixpence worth of candies, and ninepence worth of hazel-nuts. How much more did he give for the hazel-nuts than for the candies?· 3. Henry was 10 years old when his mother died; his sister was 6. How much older was Henry than his sister? SUBTRACTION TABLE. Art. 18.-Signs. A short horizontal line traction. Thus: 7-4=3, is read: 7 minus 4 Latin word, which signifies less) equals 3. signifies sub(minus is a 5-50 6-5 1 7-5=2 8-5= 3 8-4-4 9-5= 4 8-3= 5 9-4=5 5 10-5= 5 QUESTIONS.-1. Two from 2-how many? 2. Two from 3-how many? 3. Two from 4-how many? 4. Two from 5-how inany? In this manner the scholar should be questioned, until he is familiar with the above Table. |