19. Find the greatest common measure, and also the least common multiple, of 74 and 126, Ans. Greatest com. meas. 2; least com. mult. 4662. 20. The junior class in a school consists of 132 students, and the senior of 99. How might each class be divided, so that the whole school should be disposed in equal sections? Ans. Into sections of 3, 11, or 33. 21. For what sum of money could a carpenter hire journeymen for one month, at 15 dollars, 21 dollars, or 24 dollars each, allowing the whole sum to be thus expended? Ans. 840 dollars, or 1680 dollars, &c. 22. What is the smallest sum of money for which I could purchase a number of plows at 14 dollars each, or a number of carts at 30 dollars each, or a number of wagons at 90 dollars each? Ans. 630 dollars. 23. A wine merchant has 111 gallons of Madeira, 185 gallons of Port, and 259 gallons of Malaga, with which he wishes to fill a number of casks, all containing the same number of gallons, and without mixing the different kinds of wine. What must be the contents of each cask? Ans. 37 gallons. 24. A has 413 dollars, B 531 dollars, and C 590 dollars; and they agree to purchase horses, at the same price per head, provided each man can thus invest all his money. How many horses could each man purchase? Ans. A could purchase 7, B 9, and C 10. 25. An island is 200 miles in circumference, and three persons, A, B, and C, start together, and travel the same way around it. A goes 20 miles per day, B 25, and C 40 miles per day. In what time would they all come together again at the same point from which they started? First find the number of days it would require each person o go around the island. Ans. 40 days. § 87. A Fraction is an expression of one or more of the equal parts into which any quantity. may be divided. One half is one of the two equal parts,-One third is one of the three equal parts,-Two thirds are two of the three equal parts, and so on, of any quantity. What is meant by one fourth of a quantity? By three fourths of a quantity? By one fifth of a quantity? By four fifths? Any quantity consists of how many halves of that quantity? Of how many thirds ? Of how many fourths? Of how many tenths? Which is the greater part, one half or one third of a quantity? fourth or one seventh? One ninth or one fifth? hundredth? One sixth or one sixtieth? One One tenth or one How many is one half of 2? One third of 3? Two thirds of 3? Three fourths of 4? One fifth of 5? Five sixths of 6? One seventh of 7 Three eighths of 8? Four ninths of 9? If the 2 halves of any quantity were each divided into 2 equal parts, into how many equal parts would the whole quantity be divided? What would each of those parts be called? One half is how many fourths? If the 3 thirds of any quantity were each divided into 2 equal parts, into how many equal parts would the whole quantity be divided? What would each of those parts be called? One third is how many sixths? If the 4 fourths of any quantity were each divided into 3 equal parts, into how many equal parts would the whole quantity be divided? What would each of those parts be called? One fourth is how many twelfths? Numerator and Denominator. $88. A fraction is denoted by two numbers, one above the other, with a line between them. The upper number is called the numerator; and the lower, the denominator. The numerator shows the number of cqual parts in the fraction; the denominator shows the number of such parts in the whole quantity divided. Thus, three fourths; 3 is the numerator, and 4 the denominator. In, which is the numerator, and what does it show? denominator, and what does it show? In 4? In 3? Which the In ? The numerator and the denominator are together called the terms of the fraction. Proper and Improper Fractions. $89. A proper fraction is one whose numerator is less than its denominator; and whose value is, consequently, less than ̄a unit or whole one. Thus,,, are proper fractions. $90. An improper fraction is one whose numerator is equal to, or greater than, its denominator; and whose value is, accordingly, equal to, or greater than, a unit or whole one. Thus is an improper fraction,-equal to a whole one; and is an improper fraction,-equal to 13, one and two thirds. ດ 4 is equal to how many whole ones? is equal to how many whole ones? ?? ? ? 101 132 212 352 453 40 100 130? Û 69 10 Fractions Express Division. § 91. Every proper fraction expresses the part that its numerator is of its denominator. For example take the fraction, four ninths. Since 1 is one ninth of 9, 4 is four ninths of 9; that is, the fraction expresses the part that its numerator 4 is of its denom inator 9. 2 is what part of 3? 4 is what part of 7? 9 is what part of 16? 11 is what part of 25? 5 is what part of 13? 17 is what part of 39' $92. Every fraction, whether proper or improper, is equal to its numerator divided by its denominator. For example, the fraction is the quotient of 49 (§ 47); being the same as of 4 (§ 51). And the fraction is 9÷4, equal to 4 of 9, equal to 24 (§ 56). Constant Value of a Fraction. $93. The value of a fraction remains the same when its numerator and denominator are both multiplied, or both divided, by the same number. Taking, for example, 4, and multiplying both its terms by 2, we have g. Now if any quantity were divided into 4 fourths, each one of these fourths, divided into two equal parts, would make 2 eighths of the quantity; then 3 fourths would make 6 eighths; that is, 3=23=9. X Prove that is equal to. Prove that is equal to 1. The truth of this principle is also evident from regarding a fraction as a quotient, its numerator being a dividend, and its denominator the divisor (§ 92 and 57). REDUCTION OF FRACTIONS. § 94. Reduction, in general, consists in changing the form or expression of a quantity, without altering its value. Thus may be changed to the form,, &c., without altering its value (§ 93). FRACTIONS REDUCED TO THEIR LOWEST TERMS. § 95. A fraction is reduced to lower terms when its numerator and denominator are diminished, without altering its value. For example, reduced to lower terms, is §, found by dividing 12 and 16 by 2 (§ 93). Reduce to lower terms. Reduce 1 to lower terms. 20 Reduce to lower terms. 16 24 Reduce 50 to lower terms. 9. A fraction is reduced to its lowest terms when its numerator and denominator are made the smallest that will express the value of the given fraction. Thus reduced to its lowest terms is 4. What is in its lowest terms? 12 What is in its lowest terms ? 24 When a fraction is in its lowest terms, its numerator and denominator will be prime to each other (§ 77). RULE XVI. § 97. To reduce a fraction to its LOWEST TERMS. 1. Divide both terms of the fraction by any common measure greater than a unit, and the quotients by any common measure greater than a unit, and so on, until the quotients become prime to each other. The last quotients will be the lowest terms of the fraction. Or, 2. Divide both terms of the fraction by their greatest common measure; the quotients will be the lowest terms. Dividing 90 and 120 both by 10, and the quotients 9 and 12 both by 3, the quotients and 4 are prime to each other, and we have the given fraction in its lowest terms equal to & (§ 93). Or, dividing 90 and 120 both by 30, which is their greatest common measure (§ 77), we at once have The advantage of reducing a fraction to its lowest terms, is, that the value of the fraction is then more readily perceived. Thus we more readily perceive the value of than of 2 EXERCISES. 1. Reduce and to their lowest terms. 360 2. Reduce 30 and 125 to their lowest terms. 150 279 303 3. Reduce and to their lowest terms. 45 50 4. Reduce and 99. 223 to their lowest terms. 5. Reduce 32 and 1 to their lowest terms. 39 84 468 120 6. At 25 dollars per acre, what quantity of land could be purchased for 15 dollars? 7. At 18 dollars per ton for hay, what be bought for 10 dollars? 8. Allowing 365 days to a year, what cluded in 146 days? 9. There being 1760 yards in a mile included in 1320 yards? Ans. of an acre. quantity of hay could Ans. of a ton. part of a year is inAns. of a year. what part of a mile is Ans. of a mile. |