21. What is the smallest sum of money with which a drover can purchase cows at $32 each, mules at $88 each, or horses at $121 each?

22. What is the least number of acres that can be exactly divided into fields of 12 acres, 15 acres, or 16 acres each?

23. What is the least sum of money with which I can purchase an exact number of yards of cloth at 50 cts., $1.20, or $2.16 a yard?

24. What is the least quantity of water that can be exactly measured by vessels which hold either 9, 32, 48, or 63 gallons each?

25. Three trolley-cars make round trips in 40, 60, and 90 minutes, respectively; if they start out at the same time, after how many minutes will all three be again at their respective starting-points at the same time?

CASE II.

57. When the numbers are not readily factored.

1. Find the L. C. M. of 115 and 161.

Since the common factors are not readily seen by inspection, their product may be found by finding the G. C. D.

The L. C. M. as found by the first case is 23 × 5 × 7. But 23 × 5

Find the greatest common divisor of the two numbers.

Divide either number by the greatest common divisor, and multiply the quotient by the other number.

MISCELLANEOUS PROBLEMS.

1. Find the prime factors of 44100.

2. A and B agreed to purchase cows at a uniform price per head, provided they could invest all their money: A had $253, and B had $299; what was the average price of a cow, and how many cows did each buy?

3. A can dig 9 rods of ditch, B 12 rods, C 15 rods, and D 16 rods, in a day; what is the least number of rods that would furnish an exact number of days' labor to each man working alone, and how long would it take him to dig it?

4. Find the G. C. D. of 1829 and 2419.

5. Find the difference between the L. C. M. of 1, 3, 5, 7, 9, and the L. C. M. of 2, 4, 6, 8, 10.

6. Divide 14 × 36 × 49 × 24 × 25 × 112 by 75 × 7 × 16 × 9 × 7, and add 364 to the quotient.

7. A farmer has bought a triangular piece of land whose sides are 357, 391, and 425 feet, respectively; what is the length of the longest boards that can be used to fence it without cutting them?

8. What is the least sum of money with which I can purchase exact numbers of sheep at $6, cows at $22, or horses at $78 a head?

9. How much does the L. C. M. of 1751 and 2369 exceed their G. C. D.?

10. Divide 213 × 84 × 190 × 264 by 30 × 56 × 36, and add 161 to the quotient.

11. What is the largest-sized vessel that will exactly measure the contents of each of four cans, holding, respectively, 28, 36, 52, and 76 quarts? How many times would the measure have to be filled?

12. Four men start at the same time and place to travel around a circular island 360 miles in circumference: A goes 45 miles, B 36 miles, C 30 miles, and D 24 miles a day; in how many days will they all be again together at the startingpoint, and how often will each have gone round?

13. A manufacturer has a quantity of wire in lengths of 105, 120, 135, and 165 feet, respectively, which he wishes to cut without waste into the longest possible equal lengths; what must be the length of each of these?

14. A school building contains 175 pupils in the high school department, 420 in the grammar school department, and 595 in the primary department; into how many classes can each department be subdivided so as to have the greatest possible equal number in all the classes?

58. DRAW three square inches. Divide the first into two equal parts. What is one of these parts called? Divide the second into three equal parts. What is one of these parts called? What are two of them called? Divide the third into four equal parts. What is one of these parts called? What are two of them called? Three of them? Four of them? These expressions, one-half, one-third, two-thirds, etc., are called Fractions.

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

60. A fraction is expressed by writing one number above another with a short horizontal line between them. Thus,

One-half is written.

One-third is written

Three-fourths is written

When a fraction is thus expressed it is called a Common Fraction.

What does the number below the line show?

What does the number above the line show?

The Denominator is the number below the line. The Numerator is the number above the line.

The numerator and the denominator are called the Terms of a fraction.

61. How many thirds are there in anything? How many fourths? How many fifths?

Is greater or less than a unit? ??

?

A Proper Fraction is one whose value is less than a unit. 62. Is greater or less than a unit? ? ? ?

An Improper Fraction is one whose value is equal to or greater than a unit.

63. The sum of 6 and 3, or 6+, is written 62. What kind of number is 6? What kind of number is ? What kind of number is 6?

A Mixed Number is the sum of a whole number and a fraction.

64. What is 1 divided by 3? By 4? By 5? By 6? By 7? By 9? By 12?

The Reciprocal of a number is 1 divided by that number.

65. The fraction is the result of either of two processes: first, of taking of 1, or, second, of taking of 5. Hence a fraction may be regarded as an expression of division, in which the numerator is the dividend and the denominator is the divisor.

It is evident that an integer can be expressed in the form of a fraction by writing 1 for its denominator. Thus, 5 is the

same as .