Write the 7 in units' place, under the column added. 2 tens and 6 tens are 8 tens. The amount is 87 cents. Write the 8 in tens' place. Note. The learner will perceive that the operation upon the slate is essentially the same as the mental solution of the same question; (Art. 16. Ex. 41;) and that both give the same result. 2. A butcher purchased two droves of sheep, the first containing 436, and the second 243: how many sheep did both droves contain? Write the numbers, &c., and proceed as before. Thus 3 units and 6 units are 9 units; 4 tens and 3 tens are 7 tens; 2 hundreds and 4 hundreds are 6 hundreds. The amount is 679. Operation. 436 First drove. 679 Whole number. 22. It will be perceived, from these examples, that units are added to units, tens to tens, and hundreds to hundreds; that is, figures of the same order are added to each other. This is the only way numbers can be added; for figures standing in different orders or columns, express different values. (Art. 8.) Thus 3 units and 3 tens will neither make six units nor six tens, any more than 3 oranges and 3 apples will make six apples or 6 oranges. In like manner it is plain that 4 tens and 4 hundreds will neither make 8 tens nor 8 hundreds. OBS. In writing numbers to be added, great care should be taken to place units under units, tens under tens, &c., in order to prevent mistakes which would otherwise be liable to occur from adding different orders to each other. 3. A man found two purses of money, one containing 425 dollars, the other 361 dollars: how many dollars did both purses contain ? QUEST. What is its use? How are all arithmetical calculations performed? What direction is given to those who wish to become proficients in arithmetic? How do you write numbers for addition? Which column do you add first? Which next? Note.-Does the operation upon the slate differ from the mental solution of the same question? 22. Can figures standing in different columns be added to each other? Why not? Illustrate by an example. Obs. What is the object in writing units under units, &c.? 4. What is the sum of 3261 and 5428? 5. What is the sum of 45436 and 12321 ? 6. What is the sum of 420261 and 231204? 8. What is the sum of 730043000 and 268900483 ? Write the following examples upon the slate, and find the sum of each: 23. When the sum of a column does not exceed 9, it must be written, as we have seen, under the column added. But when the sum of a column exceeds 9, it requires two or more figures to express it; (Art. 7;) consequently, it cannot all be written under the column added. What then must be done? We will now illustrate this case. 13. A man paid 98 dollars for a horse, and 65 dollars for a wagon how much did he pay for both? Operation. 98 price of horse. 163 Amount. Hence Directions. Write the numbers, and begin at the right hand, as before. Thus 5 units and 8 units are 13 units. Now 13 is 1 ten and 3 units, and requires two figures to express it; (Art. 7;) consequently it cannot be written under the column of units. we write the 3 units in the units' place, and reserving the 1 ten or left hand figure in the mind, add it with the tens in the next column. Thus 1 ten (which was reserved) and 6 tens are 7 tens, and 9 are 16 tens, which are equal to 1 hundred and 6 tens. Write the 6 tens under the column added, and the 1 hundred in the place of hundreds. The amount is 163 dollars. QUEST.-23. When the sum of a column does not exceed 9, where is it written? Can the whole sum be written under the column when it exceeds 9? Why not? In the 13th example, what is the sum of the units" column? How do you dispose of it? What do you do with the sum of the next column ? OBS. It will be perceived that the operation upon the slate is substantially the same as the mental solution of the same question. (Art. 16. Ex. 40) In each case, we add the orders separately; in each, finding the sum of the units' column to be 13, or 1 ten and 3 units, we add the 1 ten to the number of tens which are contained in the example; and in each we obtain the same result. 14. A gentleman bought a span of horses for 645 dollars, a carriage for 467 dollars, and a set of harness for 158 dollars: how much did he give for the whole establishment? Operation. 645 price of horses. carriage. 467 158 66 harness. Proceed as before. Thus, 8 units and 7 units are 15 units, or we simply say, 8 and 7 are 15, and 5 are 20. Set the 0 under the column added, and, reserving the 2, add it with the next column. (which was reserved) and 5 are 7, are 17. Set the 7 under the column added, and add the 1 with the next column. 1 (which was reserved) and 1 2 1270 dollars. Ans. and 6 are 13, and 4 are 2, and 4 are 6, and 6 are 12. Set the 2 under the column added, and since there is no other column to be added, write the 1 in the next place on the left. The amount is 1270 dollars. 24. The process of reserving the tens or left hand figure, when the sum of a column exceeds 9, and adding it mentally to the next column, is called carrying tens. 25. When the sum of a column exceeds 9, set the units or right hand figure under the column added, and carry the tens or left hand figure to the next column. In adding the last column on the left, it will be noticed we set down the whole sum. This is done for the obvious reason that there are no figures in the next column to which the left hand figure can be added, and is in fact carrying it to the next order. QUEST.-Obs. Does the operation upon the slate differ essentially from the mental solution of the same example? In what respects do they coincide? 24. What is the process of reserving the tens and adding them to the next column, called? 25. When the sum of any column exceeds 9, what is to be done with it? When the sum is 20, what do you set down, and what do you carry? ILLUSTRATION OF THE PRINCIPLE OF CARRYING. Operation. 26. To illustrate the principle of carrying, let us take the thirteenth example, and as we add the columns, write down the whole sum of each in a separate line. Thus the sum of the units' column is 13, or 1 ten and 3 units; the sum of the tens' column is 15 tens, or 1 hundred and 5 tens. Now adding these results together as they stand, i. e. 98 price of horse. 65 66 66 wagon. 13 sum of units. 15* 66 "tens. adding units to units, tens to tens, 163 Amount. &c., the amount is 163, the same 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 numbers 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. 10;) 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 adopted 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? |