10. A gentleman gave his son 3692 dollars, and his daughter 1212 dollars less than his son; how much did his daughter receive? Ans. 2480 dollars.

ART. 30. Method of operation when any figure in the subtrahend is greater than the figure above it in the minuend.

Ex. 1. If I have 624 dollars, and lose 342 of them, how many remain? Ans. 282.

OPERATION.

Minuend 624
Subtrahend 3 4 2
Remainder 28 2

We first take the 2 units from the 4 units, and find the difference to be 2 units, which we write under the figure subtracted. We then

proceed to take the 4 tens from the 2 tens above it; but we here find a difficulty, since the 4 is greater than 2, and cannot be subtracted from it. We therefore add 10 to the 2 tens, which makes 12 tens, and then subtract the 4 from 12, and 8 tens remain, which we write below. Then, to compensate for the 10 thus added to the 2 in the minuend, we add one to the 3 hundreds in the next higher place in the subtrahend, which makes 4 hundreds, and subtract the 4 from 6 hundreds, and 2 hundreds remain. The remainder, therefore, is 282.

The reason of this operation depends upon the self-evident truth, That, if any two numbers are equally increased, their difference remains the same. In this example 10 tens, equal to 1 hundred, were added to the 2 tens in the upper number, and 1 was added to the 2 hundreds in the lower number. Now, since the 3 stands in the hundreds' place, the 1 added was in fact 1 hundred. Hence, the two numbers being equally increased, the difference is the same.

NOTE. - This addition of 10 to the minuend is sometimes called borrowing 10, and the addition of 1 to the subtrahend is called carrying 1.

ART. 31. From the preceding examples and illustrations in subtraction, we deduce the following general

RULE. - Place the less number under the greater, so that units of the same order shall stand in the same column.

Commencing at the right hand, subtract each figure of the subtrahend from the figure above it.

If any figure of the subtrahend is larger than the figure above it in the minuend, add 10 to that figure of the minuend before subtracting, and then add 1 to the next figure of the subtrahend.

QUESTIONS. Art. 30. How do you proceed when a figure of the subtrahend is larger than the one above it in the minuend? How do you compensate for the 10 which is added to the minuend? What is the reason for this addition to the minuend and subtrahend? How does it appear that the 1 added to the subtrahend equals the 10 added to the minuend? What is the addition of 10 to the minuend sometimes called? The addition of 1 to the subtrahend?- Art. 31. What is the general rule for subtraction?

ART. 32. Second Method of Proof. Subtract the remainder or difference from the minuend, and the result will be like the subtrahend if the work is right.

This method of proof depends on the principle, That the smaller of any two numbers is equal to the remainder obtained by subtracting their difference from the greater.

14. From 671111 take 199999.
15. From 1789100 take 808088.
16. From 1000000 take 999999.
17. From 9999999 take 1607.
18. From 6101507601061 take 3806790989.

Ans. 471112.
Ans. 981012.
Ans. 1.

Ans. 9998392.

Ans. 6097700810072.

QUESTIONS. Art. 32. What is the second method of proving subtraction? What is the reason for this method of proof, or on what principle does it depend?

19. From 8054010657811 take 76909748598,

Ans. 7977100909213.

20. From 7100071641115 take 10071178.

Ans. 7100061569937.

21. From 501505010678 take 794090589.

Ans. 500710920089.

Ans. 1.

22. Take 99999999 from 100000000. 23. Take 44444444 from 500000000. Ans. 455555556. 24. Take 1234567890 from 9987654321.

Ans. 8753086431. 25. From 800700567 take 1010101. Ans. 799690466. 26. Take twenty-five thousand twenty-five from twenty-five millions. Ans. 24974975.

27. Take nine thousand ninety-nine from ninety-nine thousand. Ans. 89901. 28. From one hundred one millions ten thousand one hundred one take ten millions one hundred one thousand and ten.

Ans. 90909091.
Ans. 9999991.

29. From one million take nine. 30. From three thousand take thirty-three. Ans. 2967. 31. From one hundred millions take five thousand.

Ans. 99995000. 32. From 1,728 dollars, I paid 961 dollars; how many remain? Ans. 767 dollars. 33. Our national independence was declared in 1776; how many years from that period to the close of the last war with Great Britain, in 1815? Ans. 39 years. and the next will

34. The last transit of Venus was in 1769, be in 1874; how many years will intervene?

Ans. 105 years. square miles, and

35. The State of New Jersey contains 6851 Delaware 2120. How many more square miles has the former State than the latter? Ans. 4731.

36. In 1840 the number of inhabitants in the United States was 17,069,453, and in 1850 it was 23,191,876; what was the increase? Ans. 6,122,423.

37. In 1850 there were raised in the State of Ohio 56,619,608 bushels of corn, and in 1853, 73,436,690 bushels; what was the increase? Ans. 16,817,082 bushels.

38. By the census of 1850, 13,121,498 bushels of wheat were raised in New York, and 15,367,691 bushels in Pennsylvania; how many bushels in the latter State more than in the former? Ans. 2,246,193 bushels.

39. The city of New York owes 13,960,856 dollars, and Boston owes 7,779,855 dollars; how much more does New York owe than Boston? Ans. 6,181,001 dollars. 40. From five hundred eighty-one thousand take three thousand and ninety-six. Ans. 577,904. 41. It was ascertained by a transit of Venus, June 3, 1769, that the mean distance of the earth from the sun was ninety-five millions one hundred seventy-three thousand one hundred twentyseven miles, and that the mean distance of Mars from the sun was one hundred forty-five millions fourteen thousand one hundred forty-eight miles. Required the difference of their distances from the sun. Ans. 49,841,021 miles.

ART. 33. Method of subtracting when there are two or more subtrahends.

Ex. 1. A man owing 767 dollars, paid at one time 190 dollars, at another time 131 dollars, at another time 155 dollars; how much did he then owe? Ans. 281 dollars.

hend, to be taken from the minuend. In the second, the subtrahends subtracted, as they

are

are added, at one operation, thus: beginning with units, 5 and 1 equal 6, which from 7 units leaves 1 unit; passing to tens, 5 and 9 and 3 equal 17 tens; reserving the left-hand figure to add in with the figures of the subtrahends in the next column, the right-hand figure, 7, being larger than 6 tens of the minuend, we add 10 to the 6, and, subtracting, have left 9 tens; and, passing to hundreds, we add in the left-hand figure 1 reserved from the 17 tens, and also add 1, equal 10 tens, to compensate for the 10 added to the minuend, and with the other figures, 1 and 1 and 1 equal 5 hundreds, which, taken from 7 hundreds, leave 2 hundreds; and 291 as the answer sought.

2. E. Webster owned 6,765 acres of land, and he gave to his oldest brother 2,196 acres, and his uncle Rollins 1,981 acres ; how much has he left? Ans. 2,588 acres.

3. The real estate of James Dow is valued at 3,769 dollars, and his personal estate at 2,648 dollars; he owes John Smith 1,728 dollars, and Job Tyler 1,161 dollars; how much is Dow Ans. 3,528 dollars.

worth?

ART. 34.

IV. MULTIPLICATION.

MENTAL EXERCISES.

WHEN any number is to be added to itself several times, the operation may be shortened by a process called Multi

cation.

Ex. 1. If a man can earn 8 dollars in 1 week, what will he earn in 4 weeks?

-

ILLUSTRATION. It is evident, since a man can earn 8 dollars in 1 week, in 4 weeks he will earn 4 times as much, and the result may be obtained by addition; thus, 8+8+8+8=32; or, by a more convenient process, by setting down the 8 but once, and multiplying it by 4, the number of times it is to be repeated; thus, 4 times 8 are 32. Hence in 4 weeks he will earn 32 dollars.

QUESTION.

Art. 34. How may the process of adding a number to itself several times be shortened?