Proof.-Perform the addition downwards, and if this last amount correspond 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 figures in the top line together, and find how many nines are contained in their sum. Reject those nines, and set down the remainder towards the right hand, directly even with the figures in the line. Do the same with each of the proposed lines of numbers, setting all these excesses of nines in a column on the right hand, as here,-5, 5, 6. Then, if the excess of 9's in this sum, found as before, be equal to the excess of 9's in the total sum, 18304, the work is right. Thus, the sum of the figures in the right-hand column, 5, 5, 6, is 16, the excess of which above 9 is 7. Also, the sum of the figures in the sum total, 18304, is 16-the excess of which above 9 is also 7, the same as the former. Thus: 34975 65125 82956 18304 This method of proof depends upon a property of the number 9, which belongs to no other digit but 3;—namely, that any number divided by 9 will leave the same remainder as the sum of its figures, or digits, divided by 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 from New York to Philadelphia? 11. Lafayette was born in the year 1757. age of 78. In what year did he die? Ans. 92 miles. 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 He died at the clock strike in 24 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? 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. 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. which 18. There were five churches erected, one in cost 16,500 dollars, two in, which cost 18,350 dollars each, one in — which cost 19,386 dollars, and one in which cost 12,640 dollars. How much was the expense of Ans. 85,226 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. 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 (minus is a Latin word, which signifies less) equals 3. QUESTIONS.-1. Two from 2-how many? 2. Two from 3-how many? 3. Two from 4-how many? 4. Two from 5-how many? In this manner the scholar should be questioned, until he is familiar with the above Table. Art. 19.—Susan had 6 frocks; 4 of them she burnt. How many had she left? Four from 6, and 2 remain; 6, the larger number, is called the Minuend, because it is the number to be diminished, or made less; 4 is called the Subtrahend, because it is the number to be subtracted; 2, the difference, is called the Remainder, because it is the number left after subtraction. The process of finding the difference between two numbers, is called Subtraction. Art. 20.-SIMPLE SUBTRACTION teaches to find the difference between two numbers of the same name or kind. (OBS. Art. 15.) The object in subtraction is to take the whole subtrahend from the whole minuend. Whenever the numbers are small, the operation may be performed in the mind; but when they are large, it is better to write them down, and subtract a part at a time. Thus, from 252 subtract 161. First Operation. Second Operation. 252 161 Ans. 91. Hunds. Tens. Units. Hunds. Tens. Units. = 16+1 9 +1 Ans. We first subtract 1 unit from 2, and write down the remainder. Then, because 6 tens, the next left-hand figure of the subtrahend, cannot be taken from 5, the figure above it, we take one from the next left-hand figure of the minuend, or place of hundreds, equal to 10 tens, which, added to 5 tens, makes 15 tens. Second Operation. Then, 6 from 15, and 9, or 9 tens remain. Subtracting tens from tens, the remainder is tens. Now, because we have taken 1 from 2, in the place of hundreds, and added it to the place of tens, we call the 2,as it really is,-1, and say, 1 from 1, and 0 remains; or, which is the same thing, we may add 1 to the lower figure, and say, 2 from 2 and 0 remains. Thus it appears that what is sometimes called borrowing ten is really making a new division of the minuend. This may be illustrated in the following man QUESTIONS.-1. What is Subtraction? 2. What does Simple Subtraction teach? 3. Why is it called simple? 4. How many numbers are required to perform the operation? 5. Which is the minuend? 6. Why called minuend? 7. Which is the subtrahend? 8. Why called subtrahend? 9. What is the remainder? 10. Why called remainder? |