NOTE 1. The principle, which is the same in the two rules, is most readily perceived by the first operation.

12. What is the least common multiple of 30, 40, 45, and 75? 13. What is the smallest sum of money with which I can buy horses at $50 each, cows at $30 each, or sheep at $8 each, using the same sum in each case? Ans. $600.

14. I have 4 wine measures; the first holds 4 quarts, the second 5 quarts, the third 6 quarts, and the fourth quarts; what is the size of the smallest cask that can be exactly measured by means of each of these measures? Ans. 120 quarts. 15. What is the least common multiple of 10, 15, 45, 75, and

90?

In solving Ex. 15, it is evident that 10, 15, and 45, may at once be struck out, for each of these numbers is a measure of 90, and .. whatever multiple of 75 and 90 is found, it, certainly, must be a multiple of 10, 15, and 45; hence, the question is reduced to this: What is the least common multiple of 75 and 90?

NOTE 2. Many other abbreviations on this and other rules may be effected, but a delicate perception of the relations of numbers, and a skillful application of principles, will much more facilitate the progress of the learner than any set of formal rules.

(a) If the numbers are prime, or even mutually prime, their product is their least common multiple.

16. What is the least common multiple of 9 and 10?

Ans. 9 X 1090.

17. What is the least common multiple of 8, 9, and 25? (b) The least common multiple of two numbers is equal to their product divided by their greatest common divisor.

18. What is the least common multiple of 12 and 20? The greatest common divisor of 12 and 20 is 4, and The least common multiple is 12 × 20÷4=60, Ans. 19. What is the least common multiple of 63 and 72? 20. What is the least common multiple of 33 and 77?

128. Ex. 15, how solved? What of other abbreviations? Least common multiple of mutually prime numbers? Of two numbers?

COMMON FRACTIONS.

129. A FRACTION is one or more of the equal parts of a unit.

NOTE. A unit, or any other whole number, is often called an Integer, it is also called an Integral or Entire Number.

130. A COMMON FRACTION is expressed by two numbers, one above and the other below a line; thus (one half), % (two fifths), etc.

(a) The number below the line shows into how many equal parts the unit is divided, and is called the DENOMINATOR, because it denominates or gives name to the parts; thus, if a unit is divided into 3 equal parts, each part is one third; if into 8, each part is one eighth; etc.

(b) The number above the line is called the NUMERATOR, because it numerates or numbers the parts taken.

(c) The numerator and the denominator are the TERMS of the fraction.

131. A fraction frequently expresses nothing more nor less than division, 1st, division indicated, 2d, division performed, the numerator being the dividend, and the denominator the divisor. Hence,

(a) The value of a fraction is the quotient of the numerator, divided by the denominator; thus 42 = 12 ÷ 4 =3; and, ..,

(b) Any change in the NUMERATOR causes а LIKE change in the value of the fraction, and any change in the DENOMINATOR causes an OPPOSITE change in the value of the fraction (Art. 84).

These principles are developed in the following Problems.

129. What is a Fraction? Other names for a whole number? 130. A Common Fraction, how expressed? Number below the line, what called? Why? Number above, what called? Why? Terms of a fraction, what? 131. A fraction, what is it? Value f a fraction?

What follows?

132. A PROPER FRACTION is one whose numerator is less than the denominator; as, 3, 11, 24.

7

8

133. An IMPROPER FRACTION is one whose numerator equals or exceeds its denominator, as, 4, 7, 8, 9. An improper fraction equals or exceeds a unit; hence its name, IMPROPER fraction.

134. A SIMPLE FRACTION has but one numerator and one denominator, and is either proper or improper; as, 3, 8, 4.

135. A COMPOUND FRACTION is a fraction of a fraction; as, 3 of 71, of 3 of §.

136. A MIXED NUMBER is a whole number and a fraction united; as, 3‡, 203.

137. A COMPLEX FRACTION is one that has a fraction or a 31 3

mixed number for one or for each of its terms; as,

7' 6'

21

138. The RECIPROCAL of a number is a fraction whose numerator is 1, and whose denominator is the number itself;

thus, the reciprocals of 4, 9, and 4 are 4, 4, and.

PROBLEM 1.

139. To reduce a mixed number to an improper fraction.

Ex. 1. In 3 how many fourths?

OPERATION.

31

4

Ans. 13.

Since 4 fourths make a unit, there will be 4 times as many fourths as units, therefore, in three units there will be 4 times 3 fourths: 12 fourths, and the 1 fourth in the example added to the 12 fourths, gives 13 fourths, i. e. 12 Hence,

13, Ans.

132. A Proper Fraction, what? 133. An Improper Fraction? 134. A Simple Fraction? 135. A Compound Fraction? 136. A Mixed Number? 137. A Complex Fraction? 138. The Reciprocal of a Number? 139. Explain the Opera tion in Ex. 1.

RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum

(a) To reduce an integer to a fraction having any given denominator:

Multiply the integer by the proposed denominator, and under the product write the denominator (Art. 84, c).

27. Reduce 12 to a fraction whose denominator is 7.

28. Reduce 9 to a fraction whose denominator is 8. 29. Reduce 9 to a fraction whose denominator is 5. 30. Reduce 7 to a fraction whose denominator is 1.

Ans. 8.

Ans. 7.

31. Reduce 87 to a fraction whose denominator is 87. 32. Reduce 16 to a fraction whose denominator is 1. 33. Reduce 16 to a fraction whose denominator is 4. 34. Reduce 20 to a fraction whose denominator is 4. 35. Reduce 14 to five different fractional forms.

139. Rule for reducing a mixed number to an improper fraction? Reason? An integer, how reduced to a fractional form?

PROBLEM 2.

140. To reduce an improper fraction to a whole or mixed number.

Ex. 1. How many units in 13?

413÷4=34, Ans.

Ans. 34.

Since the numerator is a dividend and the denominator a divisor (Art. 131), the

fraction is reduced to an equivalent whole or mixed number by the following

RULE. Divide the numerator by the denominator; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quotient.

Ans. 312

Ans. 3. Ans. 219.

Ans. 26.

2. Reduce 3. Reduce

to a whole or mixed number.

to a whole or mixed number.

4. Reduce

to a whole or mixed number.

5. Reduce 56 to a whole or mixed number.

29

141. To reduce a fraction to its lowest terms.

Ex. 1. Reduce a to its lowest terms.

FIRST OPERATION.

48=18=2, Ans.

Ans 3.

Dividing both terms of a fraction by any number does not alter the value of the fraction (Art. 84, b, and 131); .. dividing each

term of 3 by 3 gives the equal fraction ; then dividing each term of this result by 4 gives 2, and as 3 and 4 are mutually prime (Art. 112), 39, in its lowest terms, equals 3.

SECOND OPERATION.

12), Ans.

In this operation both terms of the fraction are divided by their greatest common divisor, 12) Art. 119), and thus the fraction is re

duced at once to its lowest terms. Hence,

140. Rule for reducing an improper fraction to a whole or mixed number Reason?