ILLUSTRATION OF THE PRINCIPLE OF CARRYING. 26. To illustrate the principle of carrying, let us take the thirteenth example, and as we add the columns, write Operation. 98 price of horse. 65 66 66 wagon. down the whole sum of each in a 13 sum of units. 15* 66 66 tens. as before. Thus, it will be seen that the 1 ten or left hand figure in the sum of the first column, is added to the sum of the next column or the 15 tens, in the same manner as it was in the solution above. Again, the principle of carrying may be illustrated by separating the numbers to be added into the parts or orders of which they are composed. Thus, 98 is composed of 9 tens or 90, and 8 units. 6 tens or 60, and 5 units. QUEST.-If the sum is 36, what? If 70, what? What do you do with the sum of the left hand column? Why? Does this differ from carrying? Here it will also be noticed, that when the sum of any column exceeds 9, the tens or left hand figure is added, in every instance, to the same column or order to which it is carried in the solution. 27. From these illustrations it will be seen, that the process of carrying tens is, in effect, simply adding the tens to tens, the hundreds to hundreds, &c., which are contained in the given example; or adding figures of the same order together, which is the only way they can be added. (Art. 22.) For, if the sum of any column exceeds 9, and thus requires two or more figures to express it, (Art. 7,) the right hand figure denotes units of the same order as the column added, and the left hand figure denotes units of the next higher order; (Art 8;) consequently, it is of the same order as the next column to which it is carried. The result will obviously be the same, whether we add the tens in their proper place, as we proceed in the operation, or reserve them till we have added the respective columns, and then add them to the same orders. The former method is the more convenient and expeditious, and is therefore adoped in practice. 15. What is the sum of 473 and 987? Ans. 1460. QUEST.-27. What, in effect, is the process of carrying the tens to the next column? How does this appear? Does it make any difference with the result, when the tens are added to the next column? When are they commonly added? Why? 28. PROOF.-Beginning at the top, add each column downwards, and if the second result is the same as the first, the work is supposed to be right. OBS. The object of beginning at the top and adding downwards, is that the figures may be taken in a different order from that in which they were added before; otherwise, if a mistake has been made the first time adding, we should be liable to fall into the same again. But the order being reversed, the presumption is, that any mistake which may have been made will thus be detected; for it can hardly be sup posed that two mistakes exactly equal will occur. 20. Find the sum of 256, 763, and 894, and prove the operation. 21. Find the sum of 8054, 5730, and 3056, and prove the operation. 22. Find the sum of 74502, 83000, and 62581, and prove the operation. 23. Find the sum of 68056, 31067, 680, and 200, and prove the operation. 24. Find the sum of 50563, 8276, 75009, 31, and 856, and prove the operation. 25. Find the sum of 65031, 2000, 35221, and 870, and prove the operation. 29. From the preceding illustrations and principles we derive the following GENERAL RULE FOR ADDITION, I. Write the numbers to be added under each other; so that units may stand under units, tens under tens, &c. (Art. 21, Ex. 1.) II. Begin at the right hand, and ald each column sepa rately. When the sum of a column does not exceed 9, write it under the column; but if the sum of a column exceeds 9, write the units' figure under the column alded, and carry the tens to the next column. (Arts. 23 25.) QUEST.-28. How is addition proved Obs. Why add the colunms downwards, instead of upwards? 29 What is the general rule for addition? III. Proceed in this manner through all the orders, and finally set down the whole sum of the last or the left hand column. (Art. 25.) EXAMPLES FOR PRACTICE. 1. A man bought a quantity of flour for 38 dollars, a ton of hay for 14 dollars, and a firkin of butter for 12 dollars. How much did he give for the whole? 2. A grocer bought three boxes of honey; the first contained 22 pounds, the second 15, and the third 9 pounds. How many pounds were there in all? 3. A man being asked his age, answered that it was equal to the united ages of his three children, the oldest of whom was 18, the second 16, and the third 14 years old. What was his age? 4. A man bought 5 hogsheads of molasses for 238 dollars, and sold it so as to gain 75 dollars. How much did he sell it for? 5. A lady purchased materials for 3 dresses; for the first she paid 15 dollars, for the second, 9 dollars, and for the third, 7 dollars. How much did she pay for them all? 6. A boy bought a cap for 12 shillings, a pair of gloves for 6 shillings, a pair of boots for 16 shillings, and a book for 6 shillings. How much did he give for the whole ? 7. A gentleman owns 3 houses; for the first he receives a rent of 150 dollars, for the second 175, and for the third 225 dollars. What is the sum of all his rents? 8. A shopkeeper commenced business with 1530 dollars; after trading some time, he found he had gained 950 dollars. How much had he then ? 9. A man bought a horse for 87 dollars, a carriage for 75 dollars, and a harness for 28 dollars. How much did he give for the whole? 10. What number of dollars are there in four purses; the first containing 25 dollars, the second 73, the third 84, and the fourth 96 dollars? H. A poor man having lost his house by fire, to help him repair his loss, one man gave him 25 dollars, another 15, another 10, another 5, and another 3. How much did he receive from all? 12. In a certain school there were three classes in arithmetic; the first class contained 8 scholars, the second 11, and the third 14. How many scholars were study. ing arithmetic? 13. A merchant, on closing his business for the day, found he had received 23 dollars from one customer, 57 from another, 31 from another, and 25 from various oth How much did he receive that day? ers. 14. A laborer, in pursuit of employment, walked 7 miles the first day, 10 the second, 12 the third, 15 the fourth, and 20 the fifth day. How far had he then walked? 15. A man, owning a large farm, gave to one of his sons 112 acres, to another 123, to the third 147, and had 200 acres left. How large was his farm at first? 16. A man bought a barrel of oil for 30 dollars, and sold it so as to gain 15 dollars. How much did he sell it for? 17. A lad bought a geography for 50 cents, a grammar for 25 cents, an arithmetic for 13 cents, and a slate for 10 How much did he give for them all? cents. 18. A gentleman purchased a carpet for 38 dollars, a dozen chairs for 36 dollars, a bureau for 15 dollars, and a table for 12 dollars. What did his bill amount to? 19. A merchant had 4 notes; one for 157 dollars, an other for 368, another for 576, and another for 1687 dol lars. What was the whole amount of his notes? 20. A gentleman bought a cloak for 56 dollars, a coal for 25 dollars, a vest for 9 dollars, a hat for 7 dollars, and a pair of boots for 5 dollars. What did he give for the whole ? 21. A fashionable lady purchased a cashmere shawl for 469 dollars, a watch for 237 dollars, a pocket hand kerchief for 87 dollars, and a bonnet for 53 dollars. What was the amount of her bill? 22. A farmer had 375 sheep and 168 lambs in one pasture, in another 379 sheep and 197 lambs. How many sheep had he? How many lambs? How many sheep and lambs together? 23. Four men entered into partnership; one furnished 2878 dollars, another 1784 dollars, a third 1265 dollars, and the fourth 894 dollars. What was the amount of their stock? |