25. 6 from 18 leave how many? 7 from 18? 8 from 18? 10 from 18? 9 from 18? 5 from 18? 4 from 18? 3 from 18? 2 from 18? 11 from 18? 12 from 18? 13 from 18? 14 from 18? 15 from 18? 16 from 18? 26. 7 from 19 leave how many? 8 from 19? 6 from 19? 5 from 19? 4 from 19? 3 from 19? 2 from 19? 9 from 19? 10 from 19? 11 from 19? 12 from 19? 13 from 19? 14 from 19? 15 from 19? 27. Repeat the 1 from 1 leaves 04 from 4 leaves 0 7 from 7 leaves 0 2 5 8 3 4 7 28. 2 from 9 are how many? 2 from 19? 2 from 291 2 from 39? 2 from 49? 2 from 59? 2 from 69? 2 from 79? 2 from 89? 2 from 99? 2 from 109? [Let the teacher here take 3 from 9, 3 from 19, and so on, until he has gone through all the combinations, that can be made in this way, in the same manner as directed in Addition, $VII.] 29. 4 from 7 leave how many? 4 from 17? 4 from 27? 4 from 37? 4 from 57? 4 from 87? 4 from 47? 4 from 67? 4 from 77? 4 from 97? 4 from 107? 30. 8 from 16 leave how many? 8 from 26? 8 from 36 8 from 46? 8 from 56? 8 from 66? 8 from 76? 8 from 86? 8 from 96? 8 from 106? 31. 7 from 15 leave how many? 7 from 25? 7 from 35? 7 from 45? 7 from 55? 7 from 65? 7 from 85? 7 from 957 from 105? 32. 9 from 9 leave how many? 9 from 19? 9 from 29? 9 from 39? 9 from 49? 9 from 59? 9 from 69? 9 from 79? 9 from 89? 9 from 109 ? 33. 6 from 15 leave how many? 6 from 25? 6 from 45 ? 6 from 35? 6 from 55? 6 from 65? 6 from 75? 6 from 85 6 from 95 ? 6 from 105? 34. 5 from leave how many? 5 from 6? 5 from 9? 5 from 8? 5 from 11? 5 from 10? 5 from 12? § XXI. 1. Write the following numbers on your slate, take away, or subtract the lower one from the upper, and set down the difference in figures. 7 8 6 5 8 9 4 5 7 9 8 5. 3 5 2 3 4 7 3 2 5 6 6 4 500 1000 6000 70000 200000 6000000 40000000 3. Take an example where several significant figures are used. 20800 38009 570308 60805 47023 580321469 10400 17006 Hence, it is evident, that, in finding the differences of numbers, units of the same orders must be taken from each other, and their differences put in the places of the orders employed. [The pupil should be taught to explain operations in Subtraction by repeating the process aloud as in Addition.] 16. Here are some further examples. 93570224 76583424 82350113 29763022 56249879 45987688 5795786314 14372312 33126663 32744352 2133332202 67896789 23232323 78978978 592800394267 12341011 12341234 12111211 23562467 220000013002 The pupil will observe, that it matters not on which side he be gins to perform the above examples. He will find it best, however, to accustom himself to commence on the right, as this will be necessary in most of the examples, which follow. Neither is it of any consequence how the given numbers are set down, nor which is written uppermost, if units are always taken from units, tens from tens, &c. But the mode whieh we have used above, is altogether the most convenient; viz., to place the larger number above the other, in such a manner that the same orders may stand under each other; that is, units under units, tens under tens, &c. We must caution the pupil against subtracting units of different orders from one another, since 6 tens from 9 hundreds are neither 3 tens nor 3 hundreds. But 6 tens from 9 tens are 3 tens, and 6 hundreds from 9 hundreds are 3 hundreds. 19. A man, having 26 acres of land, sold 9. How many had he left? 9 units cannot be taken from 6 units. 26 9 But 9 units can be taken from 26 units, and the remainder is 17. 20. A merchant had 53 yards of cloth, and sold 16 yards; how many had he left? 53 6 units cannot be taken from 3 units. But 6 16 units may be taken from 53 units, leaving 47 units, or 4 tens and 7 units. From this, take the 37 Ans. 1 ten, and there remain 3 tens, and 7 units, or 37. 21. Let the following examples be performed in this way." 72 91 34 76 321 465 317 645 239 34 62 15 68 113 219 222 173 164 38 29 1542 2187 58632 32984 53890 3124 1213 1234 13472 15763 41763· 1216 We might perform all examples in Subtraction in this manner: but an easier method is the following. 22. In a cistern there were 73 gallons of water, and 25 were used out. How many were left? Instead of proceeding as before, I first separate the 73, which is 7 tens and 3 units, into 6 tens, and 1 ten and 3 units; or 6 tens and 13 units, thus, I then separate the 25 into 2 tens, and 5 units, thus, Then 5 units from 13 units, leave 8 units; 60 13 73 20 5 25 40 +8 48 and 2 tens from 6 tens, leave 4 tens. The answer therefore is 4 tens and 8 units, or 48. This process is nothing more than taking one of the tens, and writing it with the units, for the sake of convenience. This is called BORROWING, and is the reverse of carrying in Addition. 23. The United States declared themselves independent on the 4th of July, 1776. How many years from that time, to the 4th of July, 1829? Here, the tens in the less number are more than those of the greater. We must, therefore, borrow one of the hundreds. Separating then as before, we have And, 1000+700+120+9=1829 Therefore, the answer is 0+0+ 50+3= 53 years. Now, this process may be performed in the mind, without any actual separation of the parts. For, we see, that the figure from which we borrow, becomes one less, and, the number to which we unite what we borrowed, becomes 10 greater, because it is of a lower order. 24. A man raised 763 bushels of wheat, one year, and 851, the next. How much more did he raise the second year than the first? Here, we increase the 1 unit by 10, mak. ing it 11 units, from which we subtract the 3 units, leaving 8. We then diminish the 5 tens by 1 ten, leaving 4 tons. Therefore, we borrow again in like manner. 851 763 88 Ans. But we cannot take 6 tens from this. In the same manner, perform the following. 25. From a garrison containing 9,563 men, there were sent away 2,764. How many were left? Ans. 6,769 26. In 1,800 there were 903 post-offices in the United States, and in 1,828 there were 7,530. How much had their number increas. ed. Ans. 6,627 27. During the whole American revolution, the continental troops employed were 231,971, and the militia 56,163. How many more continental troops, than militia? Ans. 175,808 You have now been learning to perform SUBTRACTION. Then, SUBTRACTION IS FINDING THE DIFFERENCE BETWEEN TWO NUMBERS. Of course, it is exactly the reverse of Addition. THE DIFFERENCE FOUND IS CALLED THE REMAINDER, THE NUMBER, FROM WHICH THE SUBTRACTION IS MADE, IS CALLED THE MINUEND. -between two numbers, signifies that the latter is to be subtracted from the former; thus, 5-2-3 signifies that 5 less 2 equals 3. This sign is called minus, which is a Latin word meaning less. From the preceding examples, we derive the following rule.. PLACE THE SUBTRAHEND UNDER THE MINUEND, SO THAT THE SAME ORDERS MAY STAND UNDER EACH OTHER. BEGIN AT THE RIGHT, AND TAKE EACH FIGURE IN THE SUBTRAHEND FROM THE ONE ABOVE IT. IF THE FIGURE IN THE MINUEND BE TOO SMALL, INCREASE IT BY 10, AND DIMINISH THE NEXT HIGHER ORDER OF THE MINUEND BY 1. § XXII. 1. A man bought a farm for 7,608 dollars, and being in want of money, sold it for 6,139 dollars. 7000+500+90+18 1000+400+60+ 9 What did he lose? 6139 1469 Ans. Here, we are obliged to borrow in the units place. But there are no tens in the upper number. Now, as we must borrow a ten, and there are no tens to take it from, we must take it from the hundreds. We have separated the orders, to show how this is done. We take 1 hundred from the 6 hundreds, leaving 5 hundreds. This 1 hundred contains ten tens, of which we borrow one ten, and set down the remaining nine tens in the tens' place. We then subtract each order of the lower separately. It is plain that, without separating the orders, we should have obtained the same answer, if after borrowing, we had called the 6 hundreds, 5 hundreds, and the 0 tens, 9 tens. 2. A man owns stock in a bank, to the amount of 5,003 dollars, but the bank failing, he sold it for 1,328 dollars. What did he lose? 4000+900+90-+13 3000+600+70+ 5 5003 3675 Ans. The result is here, evidently the same as it would have been, if, after borrowing, we had called the 5 thousands, 4 thousands, and the O hundreds and the 0 tens, 9 hundreds and 9 tens. 3. A man, having 60,000 dollars, lent 13,846. he left? 4. In a storm, the crew of a vessel were obliged goods to the amount of $69,893. The whole $200,000. How much was left ? How many had Ans. $46,154. to throw overboard cargo was worth Ans. $130,107. 5. A man had $30,000 of which he invested $7,600 in bank stock. How much was left? Ans. $22,400. 6. A person owed $9,000.00 and paid $6,872.32. How much did he then owe? 7. Take 936,859 from 3,000,331 "230,470,000 Ans. $2,127.68. 9. Take 87,964,321 from 230,449,000 " 230,112,780 "1,275,003,006 10. |