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Rule.-I. Find the greatest common divisor of two numbers, divide one number by it, and multiply the other number by the quotient.

II. When there are more than two numbers, find the least common multiple of two of the numbers, and then of this number and the third number, etc.

WRITTEN EXERCISES. Find the least common multiple of 2. 671 and 793.

Ans. 8723, 3. 3503 and 4859.

Ans. 150629. 4. 6527 and 7597.

Ans. 463417 6. What is the smallest sum of money for which I could hire workmen for one month, paying either $16, $20, $28, or $35 a ; month?

Ans. $560. 6. A can dig la cods of ditch in a week, B can dig 18 rods, C 22 rods, and D 24 rods; what is the least number of rods that would afford an exact number of weeks' work for each one of them?

Ans. 5544 rods. 7. What is the smallest number of bushels of corn that will All a number of barrels containing 3 bushels each, a number of sac ks containing 5 bushels each, a number of casks containing 14 bushels each, or a number of bins containing 48 bushels each ?

Ans. 1680 bu.

SUPPLEMENTARY PROBLEMS,

To be omitted unless otherwise directoa. 8. 7205 and 9432.

Ans. 618760. 9. 11183 and 15403.

Ans. 816359. 10. 357, 612, and 663.

Ans. 55692. 11. 4141, 6161, and 7171.

Ans. 17934671. 12. Four men start at the same place to walk around a gardon; A can go around in nine minutes, B in 10 minutes, C in 12 minutos, aud D in 15 minutes; in what time will they all meet at the start ing point ?

Ans. 180 minutos. 13. A, B, C, and D start from the same point, A traveling a mile in 18 minutes, B in 24 minutes, C in 30 minutes, and D in 85 minutos; what is the least whole number of miles each may travel that they may return to the starting point at the samo moment ?

Ans. A, 140; B, 105; C, 84; D, 72.

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INTRODUCTION TO CANCELLATION.

NENTAL EXERCISES. 1. If we omit the factor 2 from 12 and 6, what factors will remain ! 2. Divide 24 by 6. Divide 24 by of 6. Divide ; of 24 by 6.

3. Divide 5 of 24 by 5 of 6. Divide 38 by 18, first taking out the common factor 6.

4. Is there any difference in the quotient of 48 divided by 12, and of 48 divided by 1 of 12?

5. Divide 72 by 48, first omitting common factors. Divide 90 by 80 in the same way ; 144 by 96.

6. Divide 2x2x2 by 2x2; 3x3x4 by 2x 3; 3x4x6 by 3x8.

7. Divide 2x3x7 by 2x7; 2x3x4 by 3x3; 3x5x8 by 3x8; 8x 6 x 8 by 3x6; 2x7X9X 10 by 9x2.

CANCELLATION. 131. Cancellation is a process of abbreviating arithmetical operations by rejecting common factors in both dividend and divisor.

PRINCIPLES, 1. The cancelling of a factor from any number divides the number by that factor.

DEM.—Thus if we take the factor 3 out of 24 we shall divide 24 by 3.

2. The cancelling of a factor in both dividend and divisor will not change the quotient.

DEM.-Cancelling a factor in both dividend and divisor is the same as dividing them both by the same number, which, by the principles of division, does not change the quotient.

1. Divide 84 x 60 by 24 x 63.

SOLUTION.- We cancel the common factor 12 from 60 and 24, writing 5, the other factor of 60,

2 above 60, and 2, the other factor of 24, below 24; we then cancel the common factor 21 from 84 and 63,

84 x 10 writing 4, the other factor of 84, above 84, and 3,

=31 the other factor of 63, below 63; we then cancel 2

24x833 fronz 2 and 4, writing 2 above the 4; the product

2 3 of the remaining factors of the dividend is 10, the product of the remaining factors of the divisor is 3; hence the quotient is 10 divided by 3, or 3j.

Rule.-I. Cancel the common factors from the dividend and divisor.

II. Then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor.

OPERATION.

Ans. $

OPERATION.

NOTES.-1. The unit 1 takes the place of a cancelled factor, but need not be written, except in the dividend of the quotient, when there are no other factors of the dividend remaining.

2. A factor in one term will cancel two or more factors in the other tarm, when their product is equal to the former.

3. Some prefer to place the dividend upon the right and the divisor upon the left, of a vertical line. 2. Divide 12x 14 x 16 by 6 x7x8.

Ans. 8. 3. Divide 20 x 32 x 35 by 4x5x16.

Ans. 70. 4. Divide 125 X 250 by 15 x 50 x 75. 6. Divide 180 x 270 by 45 x 108.

Ans. 10 6. Divide 120 x 140 by 60 x 350.

Ans. 7. Divide 45 x 49 x 81 by 35 x 84 x 63.

Ans. 47 8. Divide 60 x 77 x 320 by 25 x 42 x 33.

Ans. 421. 9. Divide 75 X 42 X 99 by 125 x 63 x 33.

Ans. 13. WRITTEN EXERCISES. 1. How many yards of muslin, worth 12 cents a yard, may be bought for 16 pounds of butter, worth 15 cents a pound?

SOLUTION.-If one pound of butter is worth 15 cents, 16 pounds are worth 16X15 cents; for 16x15

5 4 cents, at 12 cents a yard, we can get as many yards of

13 x 18 muslin as 12 is contained times in 15x16, which we

-20

12 find, by cancellation, to be 20.

2. How many bushels of corn, worth 45 cents a bushel, must be exchanged for 125 pounds of butter, at 18 cents a pound?

Ans. 50. 3. A exchanged rye, wortb 84 cents per bushel, for 78 busbels of wheat, worth 98 cents per bushel ; required the number of bushels of rye.

Ans. 91. 4. How many bushels of corn, at 42 cents a bushel, must be given in exchange for 7 pieces of cloth, each containing 40 yards, at 36 cents a yard ?

Ans. 240. 6. How many boxes of tea, each containing 24 pounds at 90 cents a pound, must be given for 27 firkins of butter of 56 pounds each, at 20 cents a pound?

Ans. 14. 6. A farmer sold a grocer 9 loads of apples, each load containing 18 bags, and each bag 2 bushels, at 35 cents a bushel, and received in payment 12 boxes of sugar, each con. caining 135 pounds; what was the sugar worth a pound?

Ans. 7 cents

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PRIME NUMBERS. 132. No general method of determining prime numbers, beyond a certain limit, has yet been discovered, although much time has been spent in the investigation.

133. We give the following practical method, which consists in writing a series of numbers, and sifting out those which are composite.

METHOD.-Since the even numbers after 2 are composite, we write the series of odd numbers; thus,

3 5,7 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41. 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99.

Now, commencing at 3, since every third term is divisible by 3, every third number is composite, which we indicate by putting the figure 3

Commencing at 5, every fifth number is divisible by 5, and is therefore composite, hence we place a figure 5 over every fifth number. Proceed in the same manner with 7, and the numbers unmarked will be the prime numbers up to 100.

This method was discovered by Eratosthenes, a Greek mathematician, He inscribed the series of odd numbers on parchment, and then cutting out the composite numbers, his parchment with its holes resembled å sieve; hence the method has been called Eratosthenes' Sieve.

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NOTB.-This page will be of interest to the student to read, but is not to bo recited.

INTRODUCTION TO FRACTIONS.

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MENTAL EXERCISES. 1. If an apple is divided into two equal parts, what is one of these parts called ? amin

2. What are two of these parts called? How many halves in any. ding?

3. What is of 6? of 47 of 12 ? of 16? of 10? of 187 of 207 of 34 ? of 287 of 36?

4. If I divide an apple into 3 equal parts, what is one of these parts caller ?

5. What are 2 and 3 of these parts called ? How many thirds in anything?

6. The number of equal parts into which a unit may be divided is represented by a figure below the line ; thus I represents lualves ; 1, thirds ; , fourths, etc.

7. The number of fractional parts taken may be represented by a figure above the line; thus, f represents 2 thirds; }, 3 fourths; £, 5 sixths, etc.

8. What is of 6? of 9? of 127 of 18? of 152 of 21? of 27? What are of 127 of 157 of 21 ? of 187 of 247 of 33 ?

9. If I divide an apple into 4 equal, arts, what is one of these parts called? If I divide in 5, or 6, etc., equal parts ?

10. How many fourths make a whole? How many fifths? Sixths ? Sevenths ? Eighths? Ninths? Tenths ?

11. What is of 12? $ of 20 ? ; of 24? of 16? f of 307 of 289 of 40? \ of 35?

12. What is of 20? of 15? of 12? 1 of 24? $ of 279 $ of 80 ? Å of 227 of 64 ?

13. If & yard of muslin cost 24 cents, what will of a yard cost ? What will of a yard cost ?

14. Henry's age is 36 years, and his wife's age is as much ; what is bis wife's age?

15. If 5 melons cost 60 cents, what will 7 melons cost at the same rate ?

16. What will 4 yards of satin cost at the rate of $6 for f of a yard ?

17. What must I pay for three-fourths of a ton of hay if five-sixths of a ton cost $20?

18. What will f of a ton of coal cost at the rate of $4.50 for f of a ton ?

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