3. The reason we begin to subtract at the right hand, is because when we have occasion to borrow, it is necessary to pay before the next figure is subtracted.

SECOND METHOD OF BORROWING.

37. Required to find the difference between 75 and 48 ?

Operation.

75-6 tens (60) +15 units. 48=4 tens (40) + 8 units. Rem. 2 tens (20) + 7, or 27.

Analytic Solution.-Since 8 units cannot be taken from 5 units, we must borrow 10. Now, taking 1 ten from 7 tens, and uniting it with the 5 units, the upper number becomes 6 tens or 60 plus 15 units. Again, separating the lower number into the units and tens of which it is composed, it becomes 4 tens or 40 plus 8 units. Now 8 units from 15 units leave 7 units. Next, since the 7 tens have lent 1 ten there are but 6 tens left, and 4 tens from 6 tens leave 2 tens. The remainder is 2 tens (20)+7 units, or 27. That is,

38. When a figure in the lower number is larger than that above it; take a unit from the next higher order in the upper number, and add it to the upper figure; then subtracting the lower figure from this sum, diminish the next upper figure by 1, and proceed as before.

OBS. This method of borrowing does not affect the difference between the two numbers. This is manifest from the fact that it does not change the value of the upper number, but simply transposes a part of one of its orders to another, which can no more increase or diminish it, than it can increase or diminish the amount of money a man has, if he takes a part from one pocket and puts it into another.

19. A man having 6042 acres of land, sold 2364 acres: how many acres did he have left? Ans. 3678.

20. From 8796 subtract 2675.

21. From 6210896 subtract 3456809.

22. From 1000000 subtract 67583.

23. From 7834501 subtract 1000000. 24. From 68436907 subtract 59476012.

25. From 8006754231 subtract 7975663417

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 under the figure subtracted. (Art. 34.)

III. When a figure in the lower number is larger than that above it, add 10 to the upper figure; then subtract as before, and add 1 to the next figure in the lower number. (Arts. 35, 36.)

PROOF.-Add the remainder to the smaller number; and if the sum is equal to the larger number, the work is right.

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.

Note.-As soon as the learner becomes familiar with the process of subtraction, he should drop the intervening words as in addition, and simply pronounce the results. Thus, in the ninth example, instead of saying 4 from 6 leaves 2: and 2 from 7 leaves 5, &c., he should say two, five, three, &c., setting down each result while pronouncing it.

EXAMPLES FOR PRACTICE.

1. A man bought a piece of cloth containing 237 yards, and How much had he left?

sold 124 yards of it.
2. A merchant had
he asked 245 dollars;
tion of 24 dollars.

on hand a quantity of flour, for which but for ready money he made a deducHow much did he receive for his flour? 3. In a certain academy there were 357 scholars, 168 of whom were young ladies. How many gentlemen were there? 4. A farmer raised 4879 bushels of wheat, and sold 3876 bushels. How much had he left?

QUEST.-40. How do you write numbers for subtraction? How proceed when each figure in the lower number is smaller than that above it? How, when a figure in the lower line is larger than that above it? 36. What is meant by borrowing ten? When you borrow 10, why do you add 1 to the next figure in the lower number? Obs. Upon what principle does this method of borrowing depend? How does it appear that the two numbers are equally increased? Why do you borrow 10 instead of 8, 12, or any other number? 38. What is the second method of borrowing? Obs. How does it appear that this method does not affect the difference between the two given numbers? 34. Obs. Why write the less number under the greater? Why place units under units, tens under tons, &c.? 36. Obs. Why begin to subtract at the right hand?

5. A man purchased a farm for 4687 dollars, but was obliged to sell it for 896 dollars less than he gave for it. How much did he sell it for?

6. A merchant bought 2268 dollars worth of goods, which, in consequence of getting damaged, he sold for 848 dollars less than cost. How much did he sell them for?

7. A merchant sold a lot of silks for 561 dollars, which was 179 dollars more than the cost. How much did they cost?

8. A man bought an estate for 8796 dollars, and sold it again for 9875 dollars. How much did he gain by his bargain?

9. A farmer raised 1389 bushels of wheat one year, and 1763 the next. How much more did he raise the second year than the first?

10. A man bought a house and lot for 5687 dollars. The house was worth 3698 dollars: how much was the lot worth?

11. Suppose a man's income is 3268 dollars a year, and his expenses are 2789 dollars. How much can he save in a year? 12. The United States declared their independence in 1776: how many years is it since?

13. Two brothers commenced business at the same time; one gained 3678 dollars in five years, the other gained 2387 dollars. How much more did one gain than the other?

14. The distance from Boston to Springfield is 98 miles, and from Boston to Pittsfield it is 151 miles. How far is it from Springfield to Pittsfield?

15. From New York to Utica it is 243 miles, from New York to Albany 150 miles. How far is it from Albany to Utica? 16. America was discovered by Columbus in 1492: how many years is it since?

17. Dr. Franklin died in 1790, and was 84 years old when he died in what year was he born?

18. General Washington was born in 1732, and died in 1799: how old was he when he died?

19. The first settlement in New England was made at Plymouth, in the year 1620: how many years is it since?

20. A ship having a cargo valued at 100000 dollars, was overtaken by a storm, and 27680 dollars worth of goods were thrown overboard. How much of the cargo was saved?

21. The population of Massachusetts in 1840, was 737699, and that of Connecticut was 309978. How many more inhabitants were there in Massachusetts than in Connecticut?

22. In 1840, the population of Massachusetts was 737699, and in 1820 it was 523287. How much did the population increase during this period?

23. In 1840, the population of the state of New York was 2428921, and in 1820 it was 1372812. How much did the population increase during that period?

24. In 1840, the population of the New England States was 2234822, and that of the State of New York was 2428921. How many more inhabitants were there in the State of New York than in New England?

25. In 1800, the population of the United States was 5305925, and in 1840 it was 17069453. How much did it increase in forty years?

26. A farmer having 389 acres of land, sold to one man 126 acres, and to another 163. How many acres had he left?

27. A gentleman having 1768 dollars deposited in the bank, gave a check for 175 dollars to one man, to another for 238 dollars, and to another for 369 dollars. How much remained on deposit?

28. A man bought a horse for 87 dollars, a carriage for 75 dollars, and a harness for 16 dollars, and sold them all together for 200 dollars. How much did he gain by the bargain?

29. A man bought a quantity of sugar for 25 dollars, a quantity of molasses for 27 dollars, and a quantity of raisins for 29 dollars, for which he paid a hundred dollar bill. How much change ought he to receive back?

30. An orchard contained 120 apple-trees, 47 peach-trees, and 28 pear-trees. Of the apple-trees 26 were cut down for a railroad to pass through, 18 of the peach-trees died, and 5 of the pear-trees were blown down. How many trees were left in the orchard?

31. A gentleman had 2700 dollars which he wished to distribute among his three sons. To the eldest he gave 825 dollars, to the second 785 dollars, and the remainder to the young. est. How much did the youngest son receive?

32. A man owing 5648 dollars, paid at one time 536 dollars, at another 378 dollars, and at another 896 dollars. How much did he then owe?

33. A man having 7689 dollars, invested 689 dollars in railroad stock, 500 dollars in a woolen factory, and 1250 dollars in bank stock. How much had he left?

34. A man bought a quantity of oil for 1763 dollars, and a lot of candles for 598 dollars. He afterwards sold them both for 2684 dollars. How much did he gain by the bargain?

35. A man owning 3789 acres of land, gave to one son 869 acres, and to another 987 acres. How much had he left? 36. A ship of war sailing with 650 men, lost in one battle 29 men, in another 37, and by sickness 19 more. How many were still living?

37. A merchant owes one man 2684 dollars, another 1786 dollars, another 987 dollars. The whole amount of his property is 4684 dollars. How much more does he owe than he is worth?

38. A man bought three farms: for the first he gave 4673 dollars, for the second 5674 dollars, and for the third 9287 dollars. He sold them all for 37687 dollars. How much did he

gain by the bargain?

39. A man bought 86 dollars worth of wheat, 48 dollars worth of butter, and a fine horse worth 148 dollars. He gave his note for 128 dollars, and paid the rest in cash. How much money did he pay?

40. A gentleman left a fortune of 18864 dollars, to be divided between his two sons and one daughter; to one son he gave 6389 dollars, to the other 6984 dollars. How much did the daughter receive?

41. A man owing 8648 dollars, paid at one time 486, at another 684, at another 729 dollars. How much did he still owe?

42. Suppose a man gains by one speculation 867 dollars, by another 687; another time he gains 563 dollars, and then loses 479; still another time he gains 435 dollars, and loses 378. How much more has he gained than lost?

43. A man borrowed of a friend 684 dollars at one time, 786 at another, 874 at another, and 976 at another. He has paid 568 dollars. How much does he still owe?