their least common multiple. Therefore, 2 × 2 × 2 × 3 × 5×7=840, the least common multiple of 56, 280, and 840. From the preceding illustration we derive the following rule for finding the least common multiple of two or more numbers. RULE. Find all the prime factors of each of the given numbers. Then find the product of all their different prime factors, using each factor the greatest number of times it occurs in any of the given numbers; this product will be their least common multiple. 1. What is the least common multiple of 4, 6, 8, and 12? 2. What is the least common multiple of 6, 8, 12, 16, and 24? 3. What is the least common multiple of 5, 15, 36, and 72? Ans. 360. 4. What is the least common multiple of 28, 35, 42, 70, and 77 ? Ans. 4620. 5. What is the least common multiple of 2, 3, 4, 5, 6, 7, 8, and 9? Ans. 2520. 6. What is the least common multiple of 23, 46, 92, and 374? Ans. 17204. 7. What is the least common multiple of 5, 7, 11, 13, 17, and 19? Ans. 1616615. Art. 76. We may also find the least common multiple of two or more numbers by the following RULE. Write the numbers in a horizontal line; divide them by the least prime number that will measure two or more of them; write the quotients and undivided numbers in a second horizontal line; divide the numbers in this second line in the same manner, and thus continue dividing until the quotients and undivided numbers are prime to each other. Find the product of the several divisors and numbers in the last horizontal line, this product will be the least common multiple. 1. What is the least common multiple of 28, 35, 42, 70, and 77 ? OPERATION. 228, 35, 42, 70, 77 5 14, 35, 21, 35, 77 714, 7, 21, 7, 77 2, 1, 3, 1, 11 It may be observed that the several divisors and numbers in the last horizontal line comprise all the different prime factors of the given numbers; hence the rule. 2 × 5 × 7X2 × 3 × 11 = 4620, the least common multiple required. FRACTIONS. Art. 77. FRACTIONS, or fractional numbers, are the expressions or measures of the parts of a unit, or integer. There are two kinds of fractions, viz., vulgar fractions, and decimal fractions. VULGAR FRACTIONS. Art. 78. A vulgar fraction is any number of the equal parts of a unit, or integer. When a unit, or any whole number, is divided into any number of equal parts, those parts are called fractions; and the number of parts into which it is divided always indicates the name of the parts, and their name is the ordinal number of the number of parts. Thus, if any whole number is divided into two equal parts, one of the parts is called one half; if it is divided into three equal parts, one of the parts is called one third, two of them, two thirds; if it is divided into four equal parts, one of the parts is called one fourth, two of them, two fourths, three of them, three fourths; if it is divided into five equal parts, the parts are called fifths; if into ten equal parts, tenths; &c. NOTATION AND NUMERATION OF VULGAR FRACTIONS. Art. 79. Notation of vulgar fractions is writing or expressing them by figures, and Numeration is the reading of those figures. A vulgar fraction is expressed by two numbers written one above the other, with a short line between them; thus, two three four thirds fourths; fifths. be The number above the line is called the numerator, cause it indicates the number of equal parts expressed by the fraction. The number below the line is called the denominator, because it indicates the name of the number of equal parts into which the unit, or integer, is supposed to be divided. The numerator and denominator are called the terms of the fraction. Suppose we wish to write the fraction fifteen twenty-fifths. We first write the numerator, or number of twenty-fifths, over a short line; we then write the denominator, or cardinal number of twenty-fifths, under it; thus, . Hence the following rule for writing vulgar fractions. RULE. Write the numerator, or number of parts, over a short line, then write the denominator, or cardinal number of the name of the parts into which the unit or integer is supposed to be divided, under it. 1. What figures will express each of the fractions two fifths, three sevenths, five eighths, and seven ninths? 2. Write the fractions twelve nineteenths, seventeen twentyfourths, nineteen thirty-fifths, and forty-seven sixtieths. RULE FOR READING VULGAR FRACTIONS. First read the numerator, or number of parts, then read the ordinal number of the denominator, or name of those parts. ILLUSTRATION OF VULGAR FRACTIONS. Art. 80. The origin of fractions has been shown in division; hence, any fraction may be viewed as an unexecuted operation in division, the numerator being the dividend, and the denominator the divisor. Every fraction expresses a quotient, and it is the only way we have of expressing the quotient of a less number divided by a greater; thus, 3÷4=2. Any whole number may be expressed in a fractional form by writing the number itself for a numerator, and 1 for a denominator; the whole number 5 is expressed in a fractional form thus, . There are six kinds of vulgar fractions, viz., proper, proper, simple, compound, mixed, and complex. im A proper fraction is a fraction whose numerator is less than its denominator; thus, is a proper fraction. An improper fraction is a fraction whose numerator is equal to or greater than its denominator; thus, § and are improper fractions. A simple fraction is a single fraction, in which there is but one numerator and one denominator. A simple fraction may be either proper, or improper; thus, and are simple fractions. A compound fraction is a fraction of a fraction, or it is several fractions connected by the word of; thus, † of ‡, and of of, are compound fractions. A mixed number is a fractional number composed of a whole number and a fraction; thus, 15g is a mixed number. A complex fraction is a fractional number which has a fraction either in its numerator, or in its denominator, or in 25 and 5'7' are complex fractions. both; thus, REDUCTION OF VULGAR FRACTIONS. Art. 81. Reduction of vulgar fractions is changing them from one form to another, to prepare them for the operations of addition, subtraction, multiplication, and divi sion. TO REDUCE A FRACTION TO ITS LOWEST TERMS. Art. 82. When the numerator and denominator of a fraction are prime to each other, that is, when they have no common measure greater than a unit, the fraction is in its lowest terms. As the numerator of every fraction is some proportional part of its denominator, and since the size or value of a fraction depends upon the proportional part which the numerator is of its denominator, it follows that dividing both terms of any fraction by the same number does not alter the size or value of the fraction; thus, if we divide both terms of the fraction by 6, we obtain the quotient, 1 being the same proportional part of 2 that 6 is of 12. Hence the following rule for reducing a fraction to its lowest terms. RULE. Divide both terms of the fraction by their greatest common measure; the quotients will express the fraction in its lowest terms. 1. Reduce, §, 10, 12, 15, 2. Reduce 1, 18, 21, 35, 18, 18, and 18, to their lowest, 25, 41, and 15, to their terms. 3. Reduce lowest terms. to its lowest terms. Ans.. Ans. 1. Ans.. Ans. 1. TO CHANGE AN IMPROPER FRACTION TO A WHOLE OR MIXED NUMBER. Art. 83. As the denominator of a fraction always expresses the number of parts which make a unit, there must be as many units in any improper fraction as the denominator is contained times in the numerator. Hence, the following rule for reducing an improper fraction to a whole or mixed number. RULE. Divide the numerator by the denominator; the quotient will be the whole or mixed number. 1. Change 12, 15, 18, 24, 2. Change 56, 63, 78, 11, 35, 42, and 49, to whole or 2, and 12, to whole or mixed mixed numbers. numbers. 3. Change 379 to a whole or mixed number. Ans. 25. 4. Change 19364 to a whole or mixed number. Ans. 12653. 5. Change 317925 to a whole or mixed number. Ans. 7065. TO CHANGE A MIXED NUMBER TO AN IMPROPER FRACTION. Art. 84. Since multiplying any whole number by the denominator of any fraction reduces the whole number to fractional parts of the same kind with those expressed by the denominator, we have the following rule. RULE. Multiply the whole number by the denominator of the fraction; add the numerator to the product, and write the sum over the denominator. 1. Change 3, 4, 5, 68, 2. Change 9, 104, 11g, 7, and 8, to improper frac-12, 15, and 20%, to improper tions. fractions. 3. Change 254 to an improper fraction. Ans. 372. Ans. 49,364. 5. Change 7065 to an improper fraction whose denominator shall be 45. Ans. 317925. TO REDUCE A COMPOUND FRACTION TO A SIMPLE FRACTION. Art. 85. Let it be required to reduce the compound fraction of to a simple fraction. of is, and of is , and of is three times, which is =}· If the above analysis and results be carefully examined, it will be seen that is obtained by multiplying by, thus, ×1=2. The second result is obtained by multiplying |