containing each an exact number of acres, into which the whole can be divided? Ans. 4 A. lots.

4. A farmer has 12 bu. of oats, 18 bu. of rye, 24 bu. of corn and 30 bu. of wheat. What are the largest bins of uniform size, and containing an exact number of bushels, into which the whole can be put, each kind by itself, and all the bins be full?

Ans. 6 bu. bins.

5. A has a four-sided field whose sides are 256, 292, 384, and 400 feet respectively; what is the length of the rails used to fence it, if they are all of equal length and the longest that can be used? Ans. 4 ft.

6. In a triangular field whose sides are 288, 450, and 390 feet respectively, how many rails will it require to fence it, if the fence is 5 rails high, and what must be the length of the rails if they lap over one foot? Ans. Length of rail, 7 ft. No. 940.

CHAPTER IX.

LEAST COMMON MULTIPLE.

1. A Multiple of a number is a number that will exactly contain it; thus, 24 is a multiple of 6.

2.

A Common Multiple of two or more numbers is a number that will exactly contain each of them.

3. The Least Common Multiple of two or more numbers is the least number that will exactly contain each of them. Find the L. C. M. of 30, 40, 50.

I.

(4. L. C. M.=2×2×2×3×5×5=600. III... L. C. M. of 30, 40, 50-600.

Explanation.-The L. C. M. must contain 2 three times, or it would not contain 40; it must contain 5 twice, or it would not contain 50; it must contain 3 once, or it would not contain 30. Since all the factors of the numbers, 30, 40, 50, are contained in the L. C. M., it will contain each of them without a remainder. Find the L. C. M. of 2310, 210, 30, 6.

I.

Explanation.-2 and 3 must be used, else the L. C. M. would not contain 6. 2, 3, and 5 must be used, else the L. C. M. would not contain 30. Hence 5 must be taken with the factors of 6. In like manner 7 must be taken with the factors already taken, else the L. C. M. would not contain 210. The factor 11 must be taken with those already taken, else the L. C. M. would not contain 2310. Hence 2, 3, 5, 7, and 11 are the factors to be taken and their product 2310 is the L. C. M.

I. The product of the L. C. M. of three numbers between 1 and 100 is 6804; and the quotient of the L. C. M. divided by the G. C. D. is 84. What are the numbers?

1. L. C. M.XG. C. D.-6804, and

III. . 36, 54, and 63=the numbers.

Explanation.-Since 9 is the G. C. D., each of the numbers contains the factors of 9. Since there are two 2's in the L. C. M., one of the numbers must contain these factors. In like manner one of the numbers must contain three 3's; one of them must also contain 7. .. We write two 3's for each of the numbers, two 2's to any set of these 3's, and 3 and 7 with either of the remaining sets, observing that the product of the factors in any set does not exceed 100. If we omit 2 in step 9, the product of the factors is 27. Hence 27, 36, 63 are numbers also satisfying the conditions of the problem.

EXAMPLES.

1. What is the L. C. M. of 13, 14, 28, 39, and 42?

2. What is the L. C. M. of 6, 8, 10, 18, 20, 36, and 48? 3. What is the L. C. M. of 18, 24, 36, 126, 20, 48, 96, 720, and 84?

4. What is the smallest sum of money with which I can purchase a number of oxen at $50 each, cows at $40 each, or horses at $75 each? Ans. $600.

42.

5. Find three numbers whose L. C. M. is 840 and G. C. D. Ans. 84, 210, and 420.

6. their G. C. D. and 2772 for their L. C. M. ?

What three numbers between 30 and 140 having 12 for

Ans. 36, 84, and 132.

7. At noon the second, minute, and hour hands of a clock are together; how long after will they be together again for the first time?

8. J. S. H. has 5 pieces of land; the first containing 3 A. 2 rd. 1 p.; the second, 5 A. 3 rd. 15 p.; the third 8 A. 29 p.、 the fourth, 12 A. 3 rd. 17 p.; and the fifth, 15 A. 31 p. Required the largest sized house-lots, containing each an exact number of square rods, into which the whole may be divided. Ans. 1 A. 21 p.

9. The product of the L. C. M. of three numbers by their G. C. D.=864, and the L. C. M. divided by the G. C. D.—24; find the numbers. Ans. 12, 18, and 48.

CHAPTER X.

FRACTIONS.

1. A Fraction is a number of the equal parts of a unit.

2. Fraction.

A Common Fraction, or Vulgar Fraction, is one in which the unit is divided into any number of equal parts; and is expressed by two numbers, one written above the other, with a horizontal line between them. Thus, & expresses fivesixths.

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

5. A Complex Fraction is a fraction whose numer4 2/1/21/100 ator, or denominator, or both, are fractional; as, 31' 31' 5

A Compound Fraction is a fraction of a fraction; as, of .

7. A Proper Fraction is a simple fraction whose numerator is less than its denominator; as, .

8. An Improper Fraction is a simple fraction whose numerator is greater than its denominator; as, 4.

9. A Mixed Number is a whole number and a frac tion; as, 34.

10. A Decimal Fraction is a fraction whose denominator is ten, or some power of ten; as, 10, 100, 180. The denominator of a decimal is usually omitted and the point (.) 18 used to determine the value of the decimal expression. Thus, 18=3, +27=.027.

11. A Pure Decimal is one which consists of decimal figures only; as, .375.

12. A Mixed Decimal is one which consists of an integer and a decimal; as, 5.25.

13. A Circulating Decimal, or a Circulate, is a decimal in which one or more figures are repeated in the same order; as, .2121 etc. When a common fraction is in its lowest terms and the denominator contains factors other than 2 or powers of 2, and 5 or powers of 5, the equivalent decimal fraction will be 7 circulating. Thus, T500 will, when reduced to a 22×3×58 decimal, be circulating because the denominator contains the factor 3.

The repeating figure or set of figures is called a Repetend, and is indicated by placing a dot over the first and the last figure repeated.

14. A Pure Circulate is one which contains no figures but those which are repeated; as, .273.

15. A Mixed Circulate is one which contains one or more figures before the repeating part; as, .45342.

16. A Simple Repetend contains but one figure; as, .3. 17. A Compound Repetend contains more than one figure; as, 354.

18.

Similar Repetends are those which begin and end at the same decimal places; as, .3467, and .0358.

19. Dissimilar Repetends are those which begin or end at different decimal places; as, .536. .835, and 3567.

20. A Perfect Repetend is one which contains as many decimal places, less 1, as there are units in the denominator of the equivalent common fraction; thus, 4.142857.

21. Conterminous Repetends are those which end at the same decimal place; as, .4267, 3275, and .0321.

22. Co-originous Repetends are those which begin at the same decimal place; as, .378, .5624, and 3.623.

I. Reduce to its lowest terms.

Explanation.-Dividing the numerator 9, by 3, without changing the denominator, the value of the fraction is diminished as many times as there are units in the divisor 3. Dividing the denominator 12, by 3, without changing the numerator 9, the value of the fraction is increased as many times as there are units in the divisor 3. Hence, if we divide both terms by 3, the increase by dividing the denominator will be equal to the decrease by dividing the numerator, and the value of the fraction will remain unchanged.

I. Reduce to a higher denomination.

Explanation. Multiplying the numerator 2, by 4, without changing the denominator, the value of the fraction is increased as many times as there are units in the multiplier 4. Multiply. ing the denominator 3, by 4, without changing the numerator, the value of the fraction is decreased as many times as there are units in the multiplier 4. Hence, if we multiply both terms by 4, the increase by multiplying the numerator is equal to the decrease by multiplying the denominator, and the value of the fraction remains unchanged.

I. Reduce 97 to an improper fraction.

Solution: II.

1. 97=9+7.

2. 1-8-8-eighths.

3. 9-9X8-2-9X8-eighths-72-eighths. 4. 72+7=2=79-eighths.

2.

II.

8=24, or 8-eighths=24-twenty-fourths.

3

3

1=1 of 24-24, or 1-eighth=} of 24twenty-fourths-3-twenty-fourths.

3. §5 times, or 5-eighths 5 times 3-twenty-fourths 15-twenty-fourths.

III. ...=1-15-twenty-fourths.