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be placed in the quotient, so that the quotient figures, to the left of it, may stand in their proper places.

10. A man paid 802 dollars for land, at 2 dollars an acre. How many acres did he buy?

Ans. 401. 11. I buy flour to the amount of 884 dollars, giving 4 dollars a barrel. How many barrels do I buy ?

Ans. 221. 12. Divido 777 by 7

Divide 8,642' by 2 606 3

6,006 6 936 3

9,999 482 2

3,906 3 [Thus far each order has been divisible by the divisor, without any remain

The pupil should recite the process of division, in these and the followiny cxaraples, aloud, giving the names of the several orders, in succession.)

In the preceding examples, it is unimportant on which side we begin to divide, if the quotient figures are correctly arranged. Here. aiter, we shall see, that it will be necessary to begin on the left. It may happen that the figures of some of the crders will leave a remainder, when divided. For example,

13. At $4.00, a box, how many boxes of oranges can I buy for $72.00 ?

4) 72 (18 Here the figure in the highest order is 7. 4 is in 7, 4 only once. Therefore, 1 is the greatest quotient figure I can obtain for that order. There is a remainder of 3. 32 I join this 3 (tens,) that is, 30 units, to the 2 units, mak. 32 ing 32 units, and divide both together. 4 is in 32, 8 cimes. Therefore 18 is the answer. 14. In 3,134 pints, how many quarts ?

Ans. 1,567. 15. In 9,836 gills, how many pints ? (4 gills make a pint.)

Ans. 2,459. 16. In 9,872 lbs. of cheese, how many cheeses of 8 pounds each ?

Ans. 1,234. 17. I put 816 lbs. of butter in boxes of 6 lbs. each. How many bores are thero ?

Ans. 136. 18. In 783,954 barley-corns, how many inches ? (3 barley-corns make 1 inch.)

Ans. 261,318. 19. At 7 dollars a barrel, how many barrels of flour can I buy for 525 dolls. ?

20. A merchant sold 7 cwt. of rice for $22.75. What was that 2 cwt. ?

21. Divide 125 by 5. 3,375 by 9. 5,274 by I. 83,136 by 6.

Hence, when there is a remainder left, on dividing any order, consider it as so '

many tens of the next lower order, to which prefix it, and divide the whole together. This is the opposite of carrying, and may be called carrying downwards.

If it should happen, that any order of the dividend is too small to contain the divisor, a cypher must be placed in the quotient, for that order; and the order itself must

4 | 48

be treated exactly like a remainder, in the preceding cases : that is, it must be considered as so many tens of the next lower order, and prefixed accordingly.

In case it should be the highest order, we must divide the first two figures of the dividend, in order to obtain the first quotient figure.

22. In 136 quarts, how many gallons ? Ans. 34. 23. In 216 pints, how many quarts ?

Ans. 108. 24. In 824 quarts, how many pecks ? Ans. 103. 25. In 8,412 quarters, how many cwt. ? Ans. 2,103.

The process will be shorter, if, instead of writing down. each product in order to subtract it, we perform the subtraction in the mind : and if, when there is a remainder after dividing any order, we prefix it, as directed above, to the next lower order, without writing it down.

Thus, 26. In 48 pecks, how many bushels ?

12 Quotient. We must divide by 4. 4 is contaited in 4, 1 time, and in 8, 2 times. In this case we may set the quotient immediately under the dividend. 27. In 56 pecks; how many bushels ? Ans. 14.

28. Divide 7,984 by 4. 32. Divide 66,672 by 9.
29.
8,76311 3.

33.

33,222 * 6. 30. 56 91,888

.8.

34. 882,924 " 6.
31.
" 47,332

" 4. 35. " 9,999,333 “ 9. This kind of mental Division is called Short Division.

When each product is actually written down and subtracted, the process is called Long Division. Short Division is commonly employed, when the Divisor is less than 12.

Thus far every divisor has been a single figure, and the following is the process of Division, Short Division being employed.

I. PLACE THE DIVISOR AT THE LEFT OF THE DIVIDEND.

II. BEGIN AT THE LEFT, DIVIDE EACH ORDER OF THE DIVIDEND BY THỂ Divisor, AND WRITE THE RESULT IN ITS PROPER PLACE IN THE QUOTIENT.

III. IF FOR ANY ORDER OF THE DIVIDEND, A SIGNIFICANT QUOTIENT FIGURE CANNOT BE OBTAINED, WRITE A CYPHER IN THE QUOTIENT ; AND IF ANY THING REMAINS AFTER DIVIDING ANY ORDER, PREFIX IT TO THE

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NEXT LOWER ORDER.

EXAMPLES FOR PRACTICE. 36. Divide 18,945 by 5. Ans. 3,789. 37. Divide 14,756 by 7. Ans. 2,108. 38. Divide 27,954 by 9. Ans. 3,106. 39. Divide 108,423 by 9. Ans, 12,047.

40. Divide 259,362 by 6. Ans. 43,227.
41. Divide 271,835 by 5. Ans. 54,367.
42. Divide 409,876,960 by 5. Ans. 81,975,392.
43. Divide 290,874,224 by 8. Ans. 36,359,278.
44. Divide 321,942,201 by 3. Ans. 107,314,067.
45. Divide 225,631,832 by 4. Ans. 56,407, 958.

46. Divide 200,000,000,007 by 9. 800,000,202 by 6. 400,001,000 by 8. 372,984 by 4. 361,060, 707 by 7. 18,325,429,002 by 2. 987,654,321,005 by 5.

MENTAL EXERCISES. Ş XXVI. 1. George had a top, but by accident le split it into 2 equal parts. What part of the whole top. would one of those pieces be called ? Ans. ONE HALF.

Then if any thing, or number, be divided into 2 equal parts, each of those parts is called ONE HALF of that thing

We have no single character to express a balf. We therefore employ a different mode of Notation, from that used for whole num. bers. As there is one thing divided, and this one thing is divided into two parts; we take a 1, to signify the one thing, and write a 2 under it, to signify that it divided into two parts ; thus, s.

2. How many halves make a whole one ? Ans. Two halves, written

3. A man divided an acre of ground into 3 equal parts. What part of the whole acre was one of the portions? Ans. ONE THIRD. What part were 2 of the portions ? Ans. Two THIRDS.

Then if a thing or number be divided into three equal parts, one of those parts is called ONE THIRD, and two of those parts TWO THIRDS.

To write one third we take a 1, because there is one thing divided, and write a 3 under it, because the one thing is divided into 3 equal parts. Thus one third is written f. Two thirds are twice as much as one third: we therefore take a 2, instead of a 1, and write a 3 under it ; thus, j.

Or, it is plain, that if 2 things be divided into 3 equal parts, each part will be twice as great as if only one thing were divided. Now, if 1 thing be divided into 3 equal parts, each part will contain one third of that thing. Of course, if 2 things be divided, each part will contain two thirds. In order, therefore, to write two thirds, we take a 2, because 2 things are supposed to be divided, and write un. der it a 3, because the 2 things are supposed to be divided into 3 equal parts. Thus, two thirds are written 3.

4. How many thirds make a whole one? Ans. Three thirds, written

5. A man cut a stick of timber into 4 equal parts. What part of the whole stick was one of the pieces? Ans. ONE FOURTH, or ONE QUARTER. What part were two of them? What part were 3? Then, if a thing or number, be divided, fc.

To write one fourth, we take 1, &c. (Let the reasons be given.] Then, one fourth is written ļ, two fourths, , and three fourths, 1.

6. How many fourths make a whole one? How written ?

7. If a yard of cloth be divided into five equal parts, what part of the whole, will one of the portions be? Ans. ONE FIFTH. What part will 2 ? 3? 4 ?

Then, if a thing or number be divided, &c.
One fifth is written }; two fifths, ; three fifths, ; and four

fifths," .

8. How many fifths make a whole one? How written?

9. A box, capable of holding a bushel, is divided by partitions into 6 equal parts. What part of a bushel will one of these divisions hold ? What part will 2? 3? 4? 5?

Then, if a thing or number be divided, fc.

One sixth is written á; two sixths, & ; three sixths, ; ; four sixths, * ; five sixths, ã.

10. How many sixths make a whole one? How written ?

11. There are seven days in a week. What part of a week is 1 day? What part are 2 days ? 3 days? 4? 5? 6?

Then, if a thing or number be divided, c.

12. How is one seventh written ? Ans. 1. Why? How are two sevenths written ? Why? Three sevenths ? Why? Four sevenths? Why? Five sevenths ? Why? Six sevenths? Why?

13. Hòw many sevenths make a whole one? How written ?

14. In a mile are 8 furlongs. What part of a mile is one furlong? 2 furlongs ? 3? 4? 5? 61 7?

Then, if a thing or number be divided, fc.

15. How is one eighth written ? Ans. 1. Why? How are two eighths written ? Why? Three eighths ? Four cighths? Five eighths? Sixeighths? Seven cighths? Why?

16. How many eighths make a whole one ? How writ

ten ?

feet? 3 square

17. In one square yard are 9 square feet. What part of a square yard is a square foot.? 2 square feet? 4? 5? 6? 7? 8?

Then, if a thing or number be divided, &c.

18. How is one ninth written ? Ans. Why? Two minths ? Three ninths ? Four ninths? Five ninths? Six ninths ? Seven ninths ? Eight ninths ? Why?

19. How many ninths make a whole one? How written ?

20. In one cent, are 10 mills. What part of a cent is a mill? 2 mills ? 3 mills ? 4? 5? 6? 7? 8? 9?.

21. How is one tenth written ? Ans. . Why? Two tenths ? Three tenths ? Four tenths? Five tenths ? Six tenths ? Seven tenths ? Eight tenths ? Nine tenths? Why?

22. How many tenths make a whole one? How written ?

We have now been learning the Notation, (that is the manner of writing) for part of numbers, as far as tenths. It is unnecessary to go farther, for, by observing the above, the pupil will be able to write, for himself, ay part, or number of parts of a unit, or any single part of a higher number. For he will observe, that two num. bers ero always employed, with a line between them; and that the number, from which the name of the part is derived, is always writ. cen below this line, If, then, he wishes to take one such part of a unit, he writes a 1 above the line; if he wishes to take 2 such parts, he writes a 2 above the line ; if 3, he writes a 3; if 4, a 4; if 5, a 5; if 723, he writes 723 above the line, Or, if he wishes to take one such part of two units; he writes a 2 above the line ; if of 3, a 3; if of 4, a 4; if of 947, he writes 347 above the line.

EXAMPLES FOR PRACTICE. 1. Write six twentieths. Ans. z. Write one twentieth of 6. Ans. 27

Note. We see that the two answers are the same, that is, that one twentieth of sig, is equal to six twentieths of 1.

2. Write 15 forty sevenths. Ans. 1. One forty seventh of 15. Ans. 14.

3. Write 12 twenty sixths, 13 four hundred and fifty eighths. 4. Write 25 eighty fourths. 3 fifty ninths.' 2 fortieths.

PROMISCUOUS EXAMPLES FOR MENTAL EXERCISE. 1. What do you understand by one half of any thing ? Ans. If any thing, or number bę divided, &c.

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