2. For the method of proof by casting out the 9's, (too difficult for a work of this grade,) see "Ray's Higher Arithmetic.”

3. Add the numbers 853, 516, 29, 6, 34, 580.

SOLUTION.-Writing the numbers and adding them, their sum is 2018.

PROOF. If the numbers be separated by a line between 29 and 6, the sum of the numbers in the first division will be 1398, and in the second, 620. Adding these numbers together, their sum is 2018, as before.

853

516

29

-1398

6

34

620

580 2018

2018

4. Find the sum of 3745, 2831, 5983, 7665.

In adding long columns of figures, it is necessary to retain the numbers carried. This may be done by placing them in small figures under their proper columns, as 8, 2, 1, in the margin.

3745

2831

5983

7665

20224

16. 23+41+74+83+16= how many? 17. 45+19+32+74+55= how many? 18. 51+48+76+85+4 = how many? 19. 263+104+321+155= how many? 20. 94753+2847+93688+9386+258+3456 are how many? Ans. 204388.

Ans. 237.

Ans. 225.

Ans. 264.

Ans. 843.

21. January has 31 days, February 28, March 31, April 30, and May 31: how many days are there in these five months?

Ans. 151.

22. June has 30 days, July 31, August 31, September 30, October 31: how many days in all?

Ans. 153.

23. The first 5 months have 151 days, the next 5 have 153 days, November has 30, and December 31: how many days in the whole year?

Ans. 365.

24. I bought 4 pieces of muslin: the first contained 50 yards, the second 65, the third 42, the fourth 89: how many yards in all? Ans. 246 yds.

25. I owe one man $245, another $325, a third $187, a fourth $96: how much do I owe?

26. General Washington was born A. lived 67 years in what year did he die?

Ans. $853.

D. 1732, and

Ans. 1799. B

the flood was

27. From the creation of the world to 1656 years; thence to the siege of Troy, 1164 years; thence to the building of Solomon's Temple, 180 years; thence to the birth of Christ, 1004 years: in what year of the world did the Christian era begin? Ans. 4004.

28. A has 4 flocks of sheep; in the 1st are 65 sheep and 43 lambs; in the 2d, 187 sheep and 105 lambs; in the 3d, 370 sheep and 243 lambs; in the 4th, 416 sheep and 95 lambs: how many sheep and lambs has he?

Ans. 1038 sheep, and 486 lambs.

29. A man bought 30 barrels of pork for $285, 18 barrels for $144, 23 barrels for $235, and 34 barrels for $408 how many barrels did he buy, and how many dollars did he pay? Ans. 105 barl., and $1072.

:

3

2

30. The first of four numbers is 287; the second, 596; the third, 841; and the fourth, as much as the first three: what is their sum? Ans. 3448.

31. The Pyramids of Egypt were built 337 years before the founding of Carthage; Carthage was founded 49 years before the destruction of Troy; Troy was destroyed 431 years before Rome was founded; Carthage was destroyed 607 years after the founding of Rome, and 146 before the Christian era. How many years before Christ

were the Pyramids built?

Ans. 1570. 32. Add three thousand and five; forty-two thousand, six hundred and twenty-seven; 105; three hundred and seven thousand and four; 800,791; three hundred and twenty thousand, six hundred. Ans. 1474132.

33. Add 275,432; four hundred and two thousand and thirty; three hundred thousand and five; 872,026; four million, two thousand, three hundred and fortyAns. 5851840.

seven.

34. Add eight hundred and eighty millions, eight hundred and eighty-nine; 2,002,002; seventy-seven million, four hundred and thirty-six thousand; two hundred and six million, five thousand, two hundred and seven; 49,003; nine hundred and ninety million, nineteen thousand, nine hundred and nineteen.

Ans. 2155513020.

III. SUBTRACTION.

ART. 23. 1. If you have 9 apples, and give 4 away, how many are left? Ans. 5. Why? Because 4 and 5 are 9.

2. Frank had 15 cents; after spending 7, how many were left? Ans. 8. Why?

3. If you take 8 from 13, how many are left? Ans. 5.

The operation in the preceding examples is termed Subtraction. Hence, Subtraction is the process of finding the difference between two numbers.

The larger number is called the Minuend; the less, the Subtrahend; and the number left after subtraction, the Difference or Remainder.

REMARKS.-1. The word Minuend means, to be diminished; Subtrahend, to be subtracted.

2. In Addition (See Art. 20), numbers of the same kind are added; in Subtraction they are taken from each other; therefore, 'Subtraction is the reverse of Addition.

3. When the given numbers are of the same denomination, the operation is called Simple Subtraction.

ART. 24. The Sign, is called minus, meaning less. Placed between two numbers, it denotes that the one on the right is to be taken from that on the left.

Thus, 8—5=3, shows that 5 is to be taken from 8; it is read, 8 minus 5 equals 3. Here, 8 is the minuend, 5 the subtrahend, and 3 the remainder.

ART. 25. When numbers are small, the difference between them may be ascertained in the mind; when large, the operation is most easily performed by writing them.

EXAMPLES.

1. A man having $135, spent $112: what sum had he left?

Since only things of the same unit value can be added (Art 20), the difference between things of the same unit value only can be found; hence,

Place units under units, tens under tens, &c., that the figures between which the subtraction is to be made, may be in the most convenient position.

SOLUTION. After arranging the numbers,

say 2 (units) from 5 (units) leave 3 (units),

hund.

tens.

units.

112 subtrahend.

23 remainder.

which put in units' place; then 1 (ten) from 135 minuend. 3 (tens) leaves 2 (tens), which put in tens' place; then 1 (hundred) from 1 (hundred) leaves 0, and there being no figures on the left of this, the place is vacant; hence the number of dollars left is 23; that is, 135-112=23.

2. A farmer having 245 sheep, sold 123: how many had he left?

Ans. 122.

3. A man bought a farm for $751, and sold it for

$875 how much did he gain?

:

What is the difference

Ans. $124.

REVIEW.-23. What is Subtraction? Give an example. What is the minuend? The subtrahend? Remainder? REM. What does minuend mean? What subtrahend?

23. Why is subtraction the reverse of addition? 24. What does the sign minus mean? When placed between two numbers what does it denote?