In finding the least common multiple, all numbers that are factors of other given numbers may be disregarded. Thus, the multiples of 8, 16, 32, 64, 80, and 128 are the same as the multiples of 80 and 128.

29. What is the smallest number that will exactly contain 16, 24, and 30?

30. How long must a box be that no room may be lost in packing in it books 6 inches, 8 inches, or 12 inches long?

31. A lady desires to purchase a piece of cloth that can be cut without waste, into parts 4, 5, or 6 yards long. How many yards must the piece contain?

32. I have a certain number of pennies which I can arrange in either 4, 6, 8, 10, or 12 equal piles. What number of pennies have I, if it is the least number that admits of such arrangement?

33. How many bushels will the smallest bin contain that can be emptied by taking out either 7 bushels, 10 bushels, or 30 bushels at a time?

34. Four agents start from New York at the same time. The first makes his trip in 8 weeks, the second in 9 weeks, the third in 15 weeks, and the fourth in 20 weeks. How many weeks will pass by before they will again start out from New York together?

35. Three men walk around a circular island, the circumference of which is 360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a day. If they start together and walk in the same direction, how many days will elapse before they will be together again?

36. Divide 5 X 15 X 80 × 56 × 81 by 10 × 5 × 16 × 78.

37. If a man buys a lot whose sides measure respectively 48 feet, 60 feet, 96 feet and 108 feet, what will be the length of the longest boards which he can use to fence all the sides without cutting?

38. Find the greatest common divisor of 1744, 9564 and 8524.

39. What is the smallest number which can be divided by 250, 350, and 525 respectively, and leave a remainder of 25? 40. What is the greatest common divisor of 1728, 6912, and 8640?

41. A stock buyer wishes to invest the same amount of money in sheep at $3 each, hogs at $14 each, and cows at $21 each, as he does in beef cattle at $48 each. What is the smallest possible amount which he can invest in each?

42. Jones Brothers & Co., of Cincinnati, O., received an order for a number of Lyman's Historical Charts. It was found that if the charts were packed in boxes containing either 24, 28, 32, or 36 charts each, there was a remainder of 9 each time, but if packed in boxes containing 25 each, there was no remainder. How many charts were ordered?

43. A dealer in real estate purchased 3 lots of land whose width on the street were respectively 152 rods, 288 rods, and 184 rods. What is the width of the largest lots of equal size which can be formed from them?

44. Divide 3× 5 × 20 × 10 × 3× 13 by 26 × 9 × 3 × 4. 45. Find the greatest common divisor of 2219, 4501, and 5964.

46. Divide 5× 8 × 3 × 7 × 28 × 99 by 11 × 4×7 × 5 × 4.

are two of the parts called? What are three of the parts called?

4. When any thing is divided into five equal parts, what is each part called? What are three parts called? What are four parts called?

5. When things are divided into 6, 7, 8, 9, 10, 15 equal parts respectively, what is each of the parts called? What are four of them called?

6. How many halves are there in any thing? How many thirds? Fourths? Fifths? Sixths? Tenths?

7. If 10 marbles are separated into 5 equal groups, what part of the marbles will be in each group? 8. How many are one-fifth of 10? fifths?

Two-fifths? Three

9. How many are one-sixth of 12? Two-sixths?

DEFINITIONS.

141. A Fraction is one or more of the equal parts of a unit.

142. The Unit of a Fraction is the unit which is divided into equal parts.

A fraction in which the unit has been divided into any number of equal parts is called a Common Fraction.

A fraction in which the unit has been divided into tenths, hundredths, thousandths, etc., is called a Decimal Fraction.

143. A Fractional Unit is one of the equal parts into which a unit is divided.

144. Since a fraction is one or more of the equal parts of any thing, to express a fraction two numbers are necessary, one to express the number of equal parts into which the unit has been divided, the other to express how many make the fraction. These numbers are written one above the other

with a horizontal line between them.

145. The Denominator is the number which shows into how many equal parts the unit is divided.

It is written below the line.

Thus, in the fraction, 7 is the denominator. It shows that the unit of the fraction has been divided into 7 equal parts.

146. The Numerator is the number which shows how many fractional units form the fraction.

It is written above the line.

Thus, in the fraction, 5 is the numerator and shows how many fractional units form the fraction.

147. The numerator and denominator are called the Terms of a Fraction.

148. Fractional units are named from the number of parts into which the unit is divided. Thus, & is read one-sixth; +, one-seventh.

Fractions are read by naming the number and kind of fractional units. Thus, & is read five-sixths;, five twenty-firsts; 18, thirteen thirty-fifths.

1. Three elevenths. Five thirteenths. Eight twentyfirsts.

2. Forty-eight fiftieths. Twenty-seven eighty-fifths.

3. Sixty forty-eighths. Fifty-seven ninety-ninths.

4. Forty-two eighty-sevenths. Thirty-nine ninety-thirds. 5. Seventy-four one-hundredths. Ninety-seven one-hundred-fifths.

6. Fifty-two seventy-eighths. Thirty-six eighty-fourths. 7. Two hundred three-hundred-ninetieths.

8. Seven hundred seventy-one eight-hundred-sixtieths. 9. Two hundred forty-nine three-hundredths. 10. Five hundred sixty-six seven-hundred-fiftieths. 11. One hundred eleven two-hundredths.

12. Four thousand six hundred thirty five-thousandths.

Fractions are classified with reference to the relation of numerator and denominator thus:

150. A Proper Fraction is one in which the numerator is less than the denominator.

Thus,,,, etc., are proper fractions.

The value of a proper fraction is therefore less than 1.