4. What is the least common multiple of 12, 15, 42, and 60? Ans. 420. 5. What is the least common multiple of 21, 35, and Ans. 210. 42? 6. What is the least common multiple of 25, 60, 100, and 125? Ans. 1500. 7. What is the least common multiple of 16, 40, 96, and 105? Ans. 3360. 8. What is the least common multiple of 4, 16, 20, 48, 60, and 72? Ans. 720. 9. What is the least common multiple of 84, 100, 224, and 300 ? Ans. 16800. 10. What is the least common multiple of 270, 189, 297, 243 ? Ans. 187110. 11. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9? Ans. 2520. 12. What is the smallest sum of money for which I could purchase an exact number of books, at $5, or $3, or $4, or $6 each ? Ans. $60. 13. A farmer has three teams; the first can draw 12 barrels of flour, the second 15 barrels, and the third 18 barrels. What is the smallest number of barrels that will make full loads for any of the teams? Ans. 180. 14. What will be the capacity of the smallest cask that will be filled to the brim by using a 2 pt., 3 pt., 5 pt., 6 pt., or 8 pt. measure to fill it? Ans. 120 pts. 15. What is the smallest sum of money with which I can purchase cows at $30 each, oxen at $55 each, or horses at $105 each ? Ans. $2310. 16. A can shear 41 sheep in a day, B 63, and C 54. What is the number of sheep in the smallest flock that would furnish exact days' labor for each of them shearing alone? Ans. 15498. ORDER OF SIGNS. 111. When we wish to indicate that several quantities are to be subjected to the same operation, we inclose them in parentheses (), in braces { }, in brackets [], or place them under a vinculum These are called the signs of aggregation. 1. A quantity in parenthesis must be changed to a single quantity by performing the operations indicated. When there is one parenthesis within another, the inside one should be first removed, and then the next until none remains. 2. Precedence is given to the signs × and ÷ over the signs + and - ; hence the operations of multiplication and division should always be performed before addition and subtraction. 1. Find the value of: I. 15[3+5{8+(9+6)÷5+(12×3)+15−3}+10−8]= OPERATION. II. 15[3+5{8+3 +36 + 12} + 10 − 8]= III. 15[3+295 +10 −8]= IV. 15 × 300 = 4500, Ans. SOLUTION. — Removing the inner parentheses ( ) and vinculum by performing the indicated operations, we have II. Removing the braces {}, we have III. Removing the brackets [ ], we have IV ; and multiplying 300 by 15 as indicated, we have the answer, 4500. RULE. Remove all the expressions in parenthesis by performing the operations indicated, beginning with the inner parenthesis. The answer to the last operation indicated will be the value of the expression. 2. 5 × [13+2 (3 + 4 × 6) + 5]. 3. 25 x (6 x 3) × 4-(9 x 8+90). 4. 2008 x 8+ (3 × 9) −8}÷5. 5. 8x (96-26) x5x6-13x30 (5×4). Ans. 360. Ans. 1638 Ans. 31. Ans. 9000. 6. 9 × [3 + 16 + 5 ÷ 3+{3+ (44 × 5) + 18÷6}+22 -15] x 8. Ans. 17496. 7. 10x {16-4+3(2+8−2)+3×6(4÷2)+8}÷10. FRACTIONS. 112. If a unit is divided into 2 equal parts, one of the parts is called one half. If a unit is divided into 3 equal parts, one of the parts is called one third, two of the parts two thirds. If a unit is divided into 4 equal parts, one of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths. If a unit is divided into 5 equal parts, one of the parts is called one fifth, two of the parts two fifths, three of the parts three fifths, etc. The parts are expressed by figures: The parts into which a unit is divided take their name, and their value, from the number of equal parts into which the unit is divided. If we divide an orange into 2 equal parts, the parts are called halves; if into 3 equal parts, thirds; if into 4 equal parts, fourths, etc. Each third is less in value than each half, each fourth less than each third, and the greater the number of parts, the less their value. 113. A Fraction is one or more of the equal parts of a unit. 114. A Fractional Unit is one of the equal parts into which a unit is divided. Thus, if a unit is divided into 5 equal parts, 1 part or is the fractional unit. To write a fraction, two integers are required, one to express the number of parts into which the whole number is divided, and the other to express the number of these parts taken. If one dollar is divided into 4 equal parts, the parts are called fourths, and three of these parts are called three fourths of a dollar. This three fourths may be written. 3 the number of parts taken. 4 the number of parts into which the dollar is divided. 115. The Denominator is the number below the line. It denominates or names the parts; and It shows how many parts are equal to a unit. 116. The Numerator is the number above the line. It numerates or numbers the parts; and It shows how many parts are taken or expressed by the fraction. 117. The Terms of a fraction are the numerator and denominator, taken together. 118. Fractions indicate division, the numerator corresponding to the dividend, and the denominator to the divisor. 119. The Value of a fraction is the quotient of the numerator divided by the denominator. DEFINITIONS, NOTATION, NUMERATION. 93 120. To analyze a fraction is to designate and describe its numerator and denominator. Thus, & is analyzed as follows: 4 is the denominator, and shows that the integer is divided into 4 equal parts; it is the divisor. 3 is the numerator, and shows that 3 parts are taken; it is the dividend, or integer divided. 3 and 4 are the terms, considered as dividend and divisor. The value of the fraction is the quotient of 3÷4, or 4. EXAMPLES. Express the following fractions by figures: 1. Seven eighths. 3. Nine one-hundredths. 2. Three twenty-fifths. 4. Sixteen thirtieths. 5. Thirty-one one hundred eighteenths. 6. Seventy-five ninety-sixths. 7. Two hundred fifty-four four hundred forty-thirds. 8. Eight nine hundred twenty-firsts. 9. One thousand two hundred thirty-two seventy-five thousand six hundredths. 10. Nine hundred six two hundred forty-three thousand eighty-seconds. Read and analyze the following fractions: 915 38065 12345 13.; 0; 82; 81; 4to; 587480. 121. Fractions are distinguished as Proper and Improper. A Proper Fraction is one whose numerator is less than its denominator; its value is less than the unit, 1. Thus, Tz, fe, fo, 4 are proper fractions. |