of Ex. 1. Suppose a man gave 475 dollars for a span horses, and 352 dollars for a carriage: how much more did he pay for his horses than for his carriage? Operation. hund. tens units Horses, 4 7 5 Dolls. Carriage, 3 5 2 Dolls. 1 2 3 Dolls. Direction. Write the less number under the greater, so that units may stand under units, tens under tens, &c. Now, beginning with the units, proceed thus: 2 units from 5 units leave 3 units; write the 3 in units' place, under the figure subtracted. 5 tens from 7 tens leave 2 tens; set the 2 in tens' place, &c. 3 hundreds from 4 hundreds leave 1 hundred; write the 1 in hundreds' place, &c. The remainder is 123 dollars. Rem. OBS. It is important for the learner to observe, that we subtract units from units, tens from tens, &c.; that is, we subtract figures of the same order from each other. This is done for the same reason that we add figures of the same order to each other. (Art. 22.) Hence, in writing numbers for subtraction, great care should be taken to set units under units, &c., in order to prevent the mistake of subtracting different orders from each other. 2. A merchant bought 268 barrels of flour. On examination, he found that only 123 barrels were fit for use: how many were damaged? Ans. 145. Suggestion. Write the less number under the greater, &c., and proceed as before. 3. A traveler, having 576 dollars, was robbed of 344 dollars: how many dollars had he left? 4. What is the difference between 648 and 235 ? QUEST.-How do you write numbers for subtraction? Where begin to subtract? Obs. What orders do you subtract from each other? Why not subtract different orders from each other? Why place units under units, &c., in subtraction? 35. When the figures in the lower number are each of them smaller than those directly over them, each lower figure, as we have seen in the preceding examples, must be subtracted from that above it, and the remainder must be placed under the figure subtracted. But it often happens that a figure in the lower number is larger than that above it, and consequently cannot be taken from it. 13. It is required to find the difference between 75 and 48. Operation. 75 48 27 Rem. It is plain that we cannot take 8 units from 5 units, for 8 is larger than 5. What then shall we do? Since 75 is composed of 7 tens and 5 units, we can take 1 ten from the 7 tens, and adding it mentally to the 5 units, it will make 15 units. Then subtracting the 8 units from 15 units, will leave 7 units; write the 7 under the units' column. Now, as we took 1 ten from the 7 tens, we have but 6 tens left; and 4 tens from 6 tens leaves 2 tens: write the 2 under the tens' column. The whole remainder, therefore, is 2 tens and 7 units, or 27. 36. The process of taking one from a higher order in the upper number, and adding it to the figure from which the subtraction is to be made, is called borrowing ten, and is the reverse of carrying ten. (Art. 24.) The 1 taken from a higher order is always equal to 10 in the next lower order to which it is added. (Art. 8.) 37. The principle of borrowing may be illustrated by the following analytic solution of the last example. Thus, QUEST.-35. When the figures in the lower number are each smaller than those over them, how proceed? Where do you place the remainder? Is a figure in the lower number ever larger than that above it? 36. What is meant by borrowing 10? 75=60+15 Rem.=20+ 7, or 27. Taking 1 ten from 7 tens, and uniting it with the 5 units, we number. And we simply separate have 60 plus 15 for the upper the lower number into the tens and units of which it is composed. Now subtracting, as in the last article, 8 from 15 leaves 7: 40 from 60 leaves 20. Thus the remainder is 20+7, or 27, the same as before. OBS. It is manifest that this process of borrowing ten does not change the value of the upper number; for it consists simply in transposing a part of one order to another order in the same number, which can no more diminish or increase the number, than it will diminish or increase the amount of money a man has, if he takes a part from one pocket and puts it into another. It is advisable for the pupil to analyze several examples as above, until the process of borrowing becomes familiar. 14. From 6042 Rem. 3675 Since 7 units cannot be taken from 2 units, we borrow 10, which, added to the 2, will make 12: then 7 (units) from 12 (units) leave 5. Now, having borrowed 1 of the 4 tens, it becomes 3 tens; and 6 from 3 is impossible: hence we must borrow again. But the next figure in the upper number, i. e. the figure in the hundreds' place, is a 0, and consequently has nothing to lend. We must therefore borrow 1 from the next order still, i. e. from thousands, and adding it to the 0, it will make 10 hundreds. Now, borrowing 1 of the 10 hundreds and adding it to the 3 tens, it will make 13 tens, and 6 from 13 leaves 7. Now, diminishing the 10 hundreds by 1, (which we borrowed,) it becomes 9, and 3 from 9 leaves 6. Again, diminishing the 6 thousands by 1, (which we borrowed,) it becomes 5, and 2 from 5 leaves 3. The answer is 3675. 37. a. There is another method of borrowing, or rather of paying, which the learner will often find more conven QUEST.-What is the 1 taken from the higher order equal to? How illustrate the principle of borrowing upon the black-board? Is the value of the upper number increased by borrowing? Is it diminished? How does this appear? 37. a. When we borrow 10, what other way is there to compensate for it? ient in practice than the preceding, and less liable to lead him into mistakes. When we borrow 10, that is, when we add 10 to the upper figure, instead of considering the next figure in the upper number to be diminished by 1, the result will manifestly be the same, if we simply add 1 to the next figure in the lower number. Thus in the last example, instead of diminishing the 4 tens in the upper number by 1, we may add 1 to the 6 tens in the lower number, which will make 7; and 7 from 14 leaves 7, the same as 6 from 13. Again, adding 1 to the 3 hundreds (to compensate for the 10 we borrowed) makes 4 hundreds; and 4 from 10 leaves 6, the same as 3 from 9. Finally, adding 1 to the 2 (because we borrowed) makes 3; and 3 from 6 leaves 3. The remainder is 3675, the same as before. 15. From 574 Take 326 Rem. 248 6 from 4 is impossible: add 10 to the 4, and it will make 14; then 6 from 14 leaves 8. Adding 1 to the 2 makes 3, and 3 from 7 leaves 4. 3 from 5 leaves 2. Ans. 248. OBS. This method of borrowing depends on the self-evident principle, that if any two numbers are equally increased, their difference will not be altered. That the two given numbers are equally increased by this process, is evident from the fact that the 1 added to the lower number is of the next superior order to the 10 added to the upper number, and will compensate for it; for 1 in a superior order is equal to 10 in an inferior order. (Art. 8.) Hence, 38. When a figure in the lower number is larger than that above it, borrow 10, i. e. add 10 to the upper figure, and from the number thus produced, subtract the lower figure to compensate this, add 1 to the next figure in the lower number; or, diminish the next figure in the upper number by 1, and proceed as before. QUEST.-Obs. Upon what does this method of borrowing depend? How does it appear that you increase the given numbers equally? 39. PROOF.-Add the remainder to the smaller number; and if the sum is equal to the larger number, the work is right. 19. A man bought a horse for 175 dollars, and sold it for 127 dollars: how much did he lose by his bargain? Operation. Paid 175 dolls. Rec'd 127 dolls. Lost 48 dolls. Proof. 127 Smaller No. Since the sum of the smaller number and remainder is equal to the larger number, the operation is correct. OBS. This method of proof depends upon the obvious principle, that if the difference between two numbers be added to the less, the sum must be equal to the greater. 20. From 8796 subtract 2675, and prove the operation. 21. From 6210896 subtract 3456809, and prove the operation. 22. From 1000000 subtract 67583, and prove ration. the ope 23. From 7834501 subtract 1000000, and prove the operation. 24. From 68436907 subtract 59476012, and prove the operation. 25. From 8006754231 subtract 7975663417, and prove the operation. 40. From the preceding illustrations and principles we derive the following GENERAL RULE FOR SUBTRACTION. I. Write the less number under the greater, so that units may stand under units, tens under tens, &c. II. Beginning at the right hand, subtract each figure in the lower number from the figure above it, and set the remainder directly under the figure subtracted. (Art. 35.) QUEST.-38. How then do you proceed, when a figure in the lower number is larger than the one over it? Why do you add 1 to the next figure in the lower line? 39. How is subtraction proved? Obs. Upon what principle does the proof of subtraction depend? 40. What is the general rule for subtraction? |