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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 contained in 4, 1 time, and in 8, 2 times. In this cuse we may set the quotient immediately under the dividend. 27. In 56 pecks, how many busbels ? Ans. 14.

28. Divide 7,984 by 4. 32. Divide 66,672 by I.
29.
8,763 66 3.

33.

33,222 6 6.
30.
.66 91,888

8. 34. 882,924 66 6. 31. 66 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 THE DIVISOR, AND WRITE THE RESULT IN ITS PROPER PLACE IN 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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THE

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 hy 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. XXVI. 1. George had a top, but by accident he 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, cach of those parts is called ONE HALF of that thing.

We have no single character to express a half. 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, .

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 l, and write a 3 under it ; thus, ž.

Or, it is plain, that if 2 things be divided into 3 equal parts, each part will be twice as great as it only one thing were diyided. Now, if I thing be divided into 3 equal pirts, 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 ;.

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 }, two fourths, &, and three fourths, 1.

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

ten ?

7. If a yard of cloth be divided into five equal parts, what part of the whole, will one of the portions be? Ans.

What part will 2 ? 3? 4? Then, if a thing or number be dirided, fc. One fifth is written }; two fifths, 3 ;. three fifths, ; and four

ONE FIFTIH.

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 5 ; two sixths, i ; three sixths, i ; four sixths, 4 ; five sixths, 5.

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. . Why? How are two sevenths written? Why? Three sevenths ? Why? Four sevenths ? Why? Five sevenths? Why? Six sevenths? Why?

13. How 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? 6? 7 ?

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

15. How is one eighth written ? Ans. 1. Why? How are two eighths written? Why? Three eighths ? Four eighths? Fire eighths? Sixeighths? Seren eightls? Why?

What part feet? 3 square

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

17. In one square yard are 9 square feet. of a square yard is a square foot ? 2

square feet? 4? 5? 6? 7 ? 8?

Then, if a thing or number be divided, foc:

18. How is one ninth written ? Ans. » Why? Two ninths? 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. 1. 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 arove, the pupil will be able to write, for himself, ai y part, or number of parts of a unit, or any single part of a higher number. For he will observe, that two num. bers aro always employed, with a line between them; and that the number, from which the name of the part is derived, is always writ. ten 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 ; if3, he writes a 3; if 4, a 4; if 5, a 5; įf 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 347, he writes 347 above the line.

EXAMPLES FOR PRACTICE. 1. Write six twentieths. Ans. 2. Write one twentieth of 6. · Ans. zero

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

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

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 be divided, &c.

2. What do you understand by one twentieth of any thing ? 6 twentieths ?

3. What do you understand by 4 fifths of any thing? 3 fifths ?

[The teacher should continue to ask questions similar to these, un:il the pupil answers without hesitation.]

4. 2 is of what number ? 3? 4? 5? 6? 7 ? 8? 9 ? 10 ? 11 ? -5. 2 is of what number ? 3? 4 ? 5? 6? 7? 8 ? 9? 10 ? 11 ? 12 ? 6. 2 is of what number ? 3? 4 ? 5? 6? 7? 8 ? 9 ? 10 ? 11 ? 12? 7. 2 is į of what number ? 3? 4? 5? 6 ? 7 ? 8 ? 9? 10 ? 11 ? 12 ? 8. 2 of of what number ? 3 ? 4 ? 5? 6 ? 7? 8 ? 9-? 10 ? 11 ? 12 ?

9. list of what number ? 1 is į of what number? 2 is of what number?

10. 1 is of what number? 2 is of what number ? 3 is of what number?

11. 1 is į of what number ? 2 is of what number ? 3 is of what number?

12. 2 is of what number ? 4 is of what number ? 6 is of what number?

13, 4 is of what number ? 6 is şof what number ? 12 is of what number?

14. 6 is of what number ? 9 is f of what number ? 12 is of what number?

15. 8 is ĝ of what number? 12 is of what number? 16 is of what number?

$ XXVII. Expressions like the above are called FRACTIONS. Then,

FRACTIONS ARE EXPRESSIONS FOR PARTS OF NUMBERS.

They are called fractions from a Latin word which means broken ; because they stand for numbers divided or broken into parts.

The term integer, a Latin word, signifying whole, is applied to the ono whole thing or unit, of which fractions are broken parts.

THE LOWER NUMBER IN A FRACTION IS CALLED THE DENOMINA. TOR; because from this, the Fraction receives its name, or denom. ination.

THE UPPER NUMBER IN A FRACTION IS CALLED THE NUMERATOR; because from this, we know the number of parts, for which the Fraction stands. The Numerator is, usually, less than the Denominator. When this is the case, the Fraction is called a PROPER FRAC

Sometimes, however, the Numerator is not less than the Denominator, but is equal to it, or greater.

When this is the case, the Fraction is called an IMPROPER FRACTION.

It has been seen, that, when the Numerator and Denominator are equal, the Fraction is equal to a whole one, or a unit, Thus, i, j,

TION.

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