2. The L. C. M. of 8, 12, and 45, and another number prime to each, is 2520; required the number.

Ans. 7.

3. The G. C. D. of two numbers is 5, and their L. C. M. is 30; what are the numbers? Ans. 10, 15, or 5 and 30.

4. The L. C. M. of 6, 9, 10, and a fourth number, is 630; what is the smallest number that it may be?

5. The G. C. D. of two numbers is 12, and is 72; required the numbers.

Ans. 7. their L. C. M.

Ans. 24, 36.

6. The G. C. D. of three numbers of two factors each is 7, and their L. C. M. is 210; required the numbers.

Ans. 14, 21, 35.

7. What two numbers between 13 and 78 have the latter for their L. C. M. and the former for their G. C. D.?

Ans. 26, 39.

8. What three numbers of two factors each between 17 and 510 have the former for their G. C. D. and the latter for their L. C. M.? Ans. 34, 51, 85.

9. Find a number between 209 and 247 which has with each of them the same G. C. D. that they have with each other. Ans. 228.

10. Find 3 numbers between 161 and 1265 which have the same L. C. M. as these numbers. Ans. 253, 385, 805.

11. Find 3 numbers between 119 and 187 which have the same G. C. D. as these numbers. Ans. 136, 153, 170.

12. Required three numbers between 119 and 374 which have with these numbers the same L. C. M. as the numbers themselves. Ans. 154, 187, 238. 13. The G. C. D. of four composite numbers of two factors each is 11, and their L. C. M. is 2310; what are the numbers? Ans. 22, 33, 55, 77. 14. Required all the numbers whose G. C. D. is 45 and L. C. M. is 4680.

Ans. 45, 90, 180, 360, 585, 1170, 2340, 4680.

CANCELLATION.

190. Cancellation is the process of abbreviating arithmetical operations by rejecting equal factors from both dividend and divisor.

191. The Symbol of Cancellation is an oblique line drawn across a figure; as 4, 5, 6, etc.

PRINCIPLES.

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

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

OPERATION.

3 2 25 21×24×73 14×38

=75

1. Divide 21 × 24× 75 by 14 × 36. SOLUTION.-We cancel the common factor 12 from 24 and 36, writing 2, the other factor of 24, above 24, and 3, the other factor of 36, below 36; then cancel the common factor 7 from 21 and 14, writing 3, the other factor of 21, above 21, and 2, the other factor of 14, below 14; the common factor 2 is then cancelled, and since 3 is contained in 75, we cancel the 3 and 75, writing 25 above 75. Multiplying together 3 and 25, the remaining factors, we have 75.

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.

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.

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

2. Divide 24× 9×10 by 6×4 × 5.

Ans. 18. Ans. 671⁄2.

3. Divide 42× 18× 60 × 4 by 7× 24 × 8 × 2. 4. Divide 5×16× 81 × 63 by 8×79× 45. 5. Divide 100 × 33× 250 by 125 × 150.

Ans. 18.

Ans. 44.

6. Divide 225 × 65 × 320 by 26 × 150 × 16.

Ans. 75.

7. Divide 16× 40 × 60 × 28 by 80 × 24 × 7.

Ans. 80.

8. Divide 231× 95 × 384 × 150 by 24× 38× 21 × 112. Ans. 589.

9. Divide 432 × 529 × 441 by 27× 23× 7× 9.

Ans. 2576.

10. Divide 9801 × 2025 × 2401 by 891 × 45 × 77.

PRACTICAL PROBLEMS.

Ans. 15435.

1. How many yards of alpaca, at 48 cents a yard, can be obtained for 36 bushels of corn at 84 cents a bushel?

SOLUTION.-If one bushel of corn is worth 84 cents, 36 bushels are worth 36× 84 cents; for 36x84 cents at 48 cents a yard, we can get as many yards of alpaca as 48 is contained times in 36×84, which we find by cancellation to be 63.

OPERATION.

3 21 38×84

63 Ans.

48
4

2. How many barrels of pork, at $16 a barrel, can be obtained for 64 tons of hay, at $23 a ton?

Ans. 92.

3. A merchant sold 18 hhd. of molasses, each containing 75 gal., at 64 cents a gal., and received in payment a number of chests of tea, each containing 24 pounds, at 90 cents a pound; how many chests were there? Ans. 40.

4. Multiply 45 by 6 times 25 and divide by 91; multiply the quotient by 13 times 63 and divide by 81; multiply this result by 12 times 19 and divide by 6 times 95. Ans. 300.

5. A dealer exchanged Minnesota extra flour, at $9.50 per barrel, for 19 cases of children's shoes, each containing 60 pairs, at $1.25 a pair; how many barrels of flour were exchanged? Ans. 150 barrels.

6. A commission merchant sold 21 bales of "middling upland” cotton, each containing 400 pounds, at 16 cents a pound, and received in payment 16 hogsheads of molasses, containing 120 gallons each; what was the cost of the molasses per gallon? Ans. 70 cents.

7. A grocer bought 7 chests of souchong tea, containing 24 pounds each, at $1.05 per pound; how many firkins of butter, at 35 cents a pound, will be required to pay for it, each firkin containing 56 pounds? Ans. 9 firkins.

4*

SECTION IV.

COMMON FRACTIONS.

192. A Fraction is a number of the equal parts of a unit; as 3 fourths.

193. A Fractional Unit is one of the equal parts of the Unit. A Fraction is a number of fractional units.

194. Similar Fractional Units are those which are alike; as 2 fourths, 3 fourths.

195. Dissimilar Fractional Units are those which are unlike; as 3 fourths, 4 fifths.

196. Fractions are divided into two classes; common fractions and decimal fractions.

197. A Common Fraction is one in which the unit is divided into any number of equal parts.

198. A Decimal Fraction is a number of the decimal divisions of the unit.

units.

NOTES.-1. Units are distinguished as Integral units and Fractional The word Unit, without any qualifying word, means the Integral unit. When the term fraction is used without any qualifying word, the common fraction is meant.

2. A fraction implies three things: 1st, a thing to be divided; 2d, equal parts of the thing; and 3d, the number of parts taken—that is, the integral unit, the fractional unit and its relation to the integral unit, and the number of fractional units taken.

3. The primary conception of a fraction is that it is a number of equal parts of a unit. It may, however, be regarded as a number of equal parts of one thing, or one equal part of a number of things. Thus, four fifths may be regarded as four-fifths of one or one-fifth of four.

199. A Common Fraction is expressed by two numbers, one written above the other, with a line between them. Thus, expresses 4 fifths.

200. The Denominator denotes the number of equal parts into which the unit is divided; it is written below the line.

201. The Numerator denotes the number of equal parts which are taken; it is written above the line.

202. The Terms of a fraction, called respectively the Numerator and the Denominator, are the two numbers by which it is expressed.

CLASSES OF COMMON FRACTIONS.

203. Common Fractions consist of three principal classes; namely, Simple, Compound, and Complex.

204. A Simple Fraction is a fraction having a single integral numerator and denominator; as,, .

205. A Proper Fraction is a simple fraction whose value is less than a unit; as, 3, 4.

206. An Improper Fraction is a simple fraction whose value is equal to or greater than a unit; as, 5, 7, 12, etc. 207. A Compound Fraction is a fraction of a fraction; as, of, of of, etc.

208. A Complex Fraction is one whose numerator or 4 3 3 of 3 denominator, or both, are fractional; as, '5' of 209. A Mixed Number consists of an integer and a fraction; as, 21, 7, etc.

210. An Integer may be expressed fractionally by writing 1 under it as a denominator; as, 6={.

211. The Reciprocal of a quantity is a unit divided by that quantity; thus the reciprocal of 5 is .

NOTES.-1. A fraction means primarily a part, hence only a proper fraction is properly a fraction. The improper fraction is not properly a fraction according to the primary signification of the term.

2. The complex fraction is not properly a fraction, according to the definition of a fraction, or the functions ascribed to the terms. Thus, if the denominator is it indicates that the unit is divided into & equal parts, which is impossible.

PRINCIPLES OF THE TERMS.

212. The Principles of the terms state the use and relation of the terms of a common fraction.

1. The numerator expresses the number of fractional units taken.

2. The denominator expresses

a. The number of equal parts into which the unit is divided.