75-60+15 48=40+ 8

Taking 1 ten from 7 tens, and uniting it with the 5 units, we have 60 plus 15 for the upper Rem=20+7, or 27. number. And we simply separate 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 Take 2367

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. Then, borrowing 1 of the 10 hundreds and adding it to the 3 tens, it will make 13 tens, and 6 from 13 leaves 7. 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. u. There is another method of borrowing, or rather of paying, which the learner will often find more con

QUEST.-How illustrate the principle of borrowing upon the blackboard? Obs. Is the value of the upper number increased by borrow37. a. When we ing? Is it diminished? How does this appear? borrow 10, what other way is there to compensate for it?

venient in practice than the preceding, and less liable to lead him into mistakes, especially, when the figure in the next higher order is a cipher.

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 mani festly 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 the second 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.

Proof.

Lost 48 dolls.

127 Smaller No.
48 Remainder.
175 Larger 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 the operation.

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?

1

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. 37, 38.)

EXAMPLES FOR PRACTICE.

1. A man bought a piece of cloth containing 37 yards, and sold 24 yards of it. How much had he left?

2. A merchant had on hand a quantity of flour, for which he asked 245 dollars; but for ready money he made a deduction of 24 dollars. How much did he receive for his flour?

3. In a certain Academy there were 357 scholars, 168 of whom were young ladies. How many young gentlemen were there?

4. A farmer raised 4879 bushels of wheat, and sold 3876 bushels. How much had he left?

5. A man purchased a farm for 4687 dollars, but the times becoming hard he 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 of them. How much did he give for them?

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 gentleman's income is 3268 dollars a year, and his expenses are 2789 dollars. How much does 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 in the same time. 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, and from New York to Albany it is 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 A. D. 1790, and was 84 years old when he died in what year was he born?

18. General Washington was born A. D. 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 sailed having on board a cargo valued at 100000 dollars, but being overtaken by a storm, 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.

did the population increase during that period?

How much

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?