17069453 in 1840; and 23263488 in 1850. How many

more did they contain in 1800 than in 1790?

16. How many more in 1810 than in 1800? 17. How many more in 1820 than in 1810? 18. How many more in 1830 than in 1820? 19. How many more in 1840 than in 1830? 20. How many more in 1850 than in 1840? 21. How many more in 1850 than in 1790? 22. How many more in 1850 than in 1800? 23. How many more in 1820 than in 1790? What is the value of

24. 27 bu. 2 pk. 2 qt.

18 bu. 3 pk. 3 qt.?

25. 83 yd. 2

qr.

1 na.

47 yd. 3

na.

qr. 2 ? 26. 8 cwt. 2 qr. 15 lb. 7 oz. 3 dr.

3 oz. 9 dr.?

2 cwt. 3 qr. 24 lb.

12 gal. 3 qt. 1 pt. 3 gi.

4 lb. 7 oz 5 dwt. 13 gr.?

5 lb 113 43 29 14 gr.?

2 qr. 3 na.?

£28 14 s. 7 d. 2 qr. ?

40. 307 T. 8 cwt. 2 qr. 23 lb. 8 oz. 12 dr. 213 T. 15 cwt.

23 lb. 11 oz. 6 dr.?

41. 527 yd. 1 qr. 2 na. 1 in.

---

431 yd. 2 qr. 3 na. 2 in.?

27 m. 7 fur. 39 rd. 5 yd. 2 ft. 3 in. ?

43. Bought 7 T. 14 cwt. 1 qr. 19 lb. of hay, from which I sold 3 T. 7 cwt. 3 qr. 26 lb. How much had I left?

44. A goldsmith bought 7 lb. 7 oz. of gold. How much will he have left after manufacturing and selling 3 lb. 5 gr. of it?

45. A trader sold 9 yd. 3 qr. 2 na. of cloth from a piece containing 27 yd. 1qr. 1na. How much was left in the piece?

46. A man set on foot to travel from Boston to Springfield, the distance being 98 miles. The first day he travelled 28 m. 7 fur. 19 rd., the second 24 m. 6 fur. 28 rd., the third 29 m. 4 fur. 36 rd. How far was he from Springfield at the end of the third day?

47. A man undertook to dig a ditch for a certain price per rod. On completing it he demanded payment for a ditch 37 rd. 0 ft. 3 in. long. His employer, doubting his honesty, measured it, and found it to be but 36 rd. 5 yd. 1 ft. 9 in. long. A dispute arising between them, they called in Mr. Jenks to settle it, agreeing to abide by his decision. He measured the ditch, and found its length to be 36 rd. 16 ft. 9 in. What was the difference in their measurements?

72. Subtraction from Left to Right.

We can begin at the left to subtract as well as at the right, if we are only careful to reserve one for reduction from each denomination in the minuend when it is required by the lower denominations. This reduction will be necessary when the figures in the subtrahend at the right of the denomination considered are greater than the corresponding ones of the minuend.

Example. How many are 508.935

Explanation.

508.935 = Minuend.

249.748 = Subtrahend.

259.187 Remainder.

Reserving 1 hundred from the 5 hundreds, we have, 2 hundreds from 4 hundreds = 2 hundreds. Reducing the 1 hundred reserved to tens, and reserving 1 ten for further reduction, we have, 4

tens from 9 tens 5 tens. Reducing the 1 ten to units, and adding 8 inits, we have, 9 units from 18 units 9 units. Reserving 1 tenth, we have 7 tenths from 8 tenths = = 1 tenth. Reducing the 1 tenth reserved to hundredths, and adding 2 hundredths, (1 hundredth being reserved,) we have, 4 hundredths from 12 hundredths 8 hundredths. Reducing 1 hundredth to thousandths, and adding the 5 thousandths, we have, 8 thousandths from 15 thousandths = 7 thousandths. The answer is,

therefore, 259.187.

=

As soon as the process is sufficiently well understood to justify it, the explanations should be omitted, and the results only named.

Thus, in the example explained above, the pupil should say, 2 hundreds, 5 tens, 9 units, 1 tenth, 8 hundredths, 7 thousandths. The answer is, therefore, 259,187.

It is also a good mental exercise to read the results at once by inspecting minuend and subtrahend.

Having the numbers to be subtracted written thus,

perform the subtractions mentally, beginning at the left, and read at once 5916.78. Practice will make this very easy.

Perform the following subtractions by beginning at the left:

73. Subtraction of several Numbers at once.

A good method of proceeding when several numbers are to be subtracted is, to subtract the sum of each column of the subtrahend from the appropriate part of the minuend, redu. cing and changing denominations, as before explained.

How many are 862 28- 59 - 38 - 56?

-

Explanation. Adding the units of the subtrahends, we have, 6, 14, 23, 31 units, which cannot be taken from 2 units. To obtain as many as 31 units, we must take 3 tens from the 6 tens. 3 tens 30 units, which added to the 2 units equals 32 units; 31 from 32 leaves 1.

Now, it makes no difference whether we take 3 tens (on account of those we reduced to units) from the 6 tens, and afterwards take the tens in the tens column of the subtrahend, or whether we take all together. Adopting the latter method, and adding the tens column we have 3, 8, 11, 16, 18 tens, which cannot be taken from 6 tens, but reducing 2 hundreds to tens, and adding the 6 tens, we have 26 tens, from which 18 tens being taken, there will remain 8 tens.

2 hundreds from 8 hundreds =6 hundreds. The answer is, therefore, 681.

When the above is well understood, omit in practice a part of the explanation, as follows:

6, 14, 23, 31 from 32 leaves 1 unit; 3, 8, 11, 16, 18 tens from 26 tens = 8 tens; 2 hundreds from 8 hundreds = 6 hundreds.

The following form may also be taken :—

6, 14, 23, 31, and 1 are 32 units; write 1 in the units' place; 3, 8, 11, 16, 18, and 8 are 26 tens; write 8 in the tens' place;

2 and 6 are 8 hundreds; write 6 in the hundreds' place.

Perform the operations indicated in the following examples by the method explained above: —

1. 87642 273 4827 285 437 869

2. 98402

5876.

2701

2596

-

245.

6. Subtract .87 + 4.73 + 826 + 42.71 + 9.854 + 3.27 from 9012031.

7. Subtract 8837 + 1429 + 6372 + 8406 + 9785 + 4203 from 9120301.

8. Subtract, 8375.94 + 276.483 + 5427.98 +386.421 + 279.4381.679.4237 + 4598.7 from 846271.3.

9. Mr. Ewell has in his possession $9478.63, but he owes $143.27 to Mr. Webster, $549.71 to Mr. May, $581.375 to Mr. Kingsbury, and $378.875 to Mr. Bryant. How much will he have left after paying his debts?

10. A man travelled 8725.67 miles in the following conveyances, viz.: 1285.89 miles in railroad cars, 876.81 miles in a canal boat, 587.86 miles in a stage coach, 725.18 miles on horseback, 647.25 miles on foot, 3147.82 miles in a steamboat, and the rest in a ship. How many miles did he travel in a ship?

11. Messrs. Howes and Baker bought 27147 bushels of Indian corn. After selling, at private sale, 1438 bushels to one man, 2627 to another, 3781 to another, and 864 to another, they sold the rest at auction. How many bushels did they sell at auction? They received $719 for the first lot, $1313.50 for the second, $1890.50 for the third, $432 for the fourth, and enough to make up $13573.50 for what they sold at auction. How much did they receive for that which they sold at auction?

SECTION VI.

MULTIPLICATION.

74. Definitions and Illustrations.

(a.) MULTIPLICATION IS A PROCESS BY WHICH WE

IN

ASCERTAIN HOW MUCH ANY GIVEN NUMBER WILL AMOUNT TO, IF TAKEN AS MANY TIMES AS THERE ARE UNITS SOME OTHER GIVEN NUMBER.

(b.) The following are questions in multiplication: -How many are 7 times 6?

What is the value of 9 multiplied by 6 ?