how high would a basin of 50 acres rise in 10 hours, with the same head of water? acres. inches. acres. 190: 10 :: 50 50)1900 38 inches. 12: 38: 10 hours. 12)380 Ans. 31 inches. 5. In what time would it rise to 38 inches? 6. Suppose a wall 40 feet high and 2 feet thick, has a sufficient foundation of 3 feet in width, what should be the breadth of the foundation of a wall, 50 feet high and 24 feet thick, upon a like bed of earth? Ans. 4 ft. 84 inches. 7. If a family of 9 persons spend $450 in 5 months, how much would be sufficient to maintain them 8 months, if 5 more were added to the family? Ans. $1120. 8. If 1 pound of yarn make 3 yards of cloth, 5 quarters wide, how many pounds of yarn would be wanted to make a piece of cloth 45 yds. long, and 1 yd. wide? Ans. 12 lbs. 9. A person engaged to remove 800 tons of timber from Exeter to the Navy Yard in Portsmouth if in 6 days he has removed 450 tons with 36 oxen, how many oxen would be wanted to remove the remainder in 3 days? Ans. 56 oxen 10. If 10 acres would feed 15 oxen, how many will 24 acres feed? Ans. 36 oxen. VULGAR FRACTIONS. FRACTIONS, or broken numbers, are expressions for any assignable parts of an unit; and are represented by two numbers, placed one above the other, with a line drawn between them. The number above the line is called the numerator, and that below the line the denominator. The denominator shows how many parts the integer is divided into, and the numerator shows how many of those parts are meant by the fraction. Fractions are either proper, improper, compound or mixed. 1st. A proper fraction is when the numerator is less than the denominator, as,,,, &c. 2d. An improper fraction is when the numerator is either equal to or greater than the denominator, as,, 12, 15, &c. 3d. A compound fraction is a fraction of fractions, and known by the word of, as of, 3 of fō, 13 of 21, &c. 4th. A mixed number or fraction is composed of a whole number and a fraction, as 84, 171, 291, &c. I. To reduce a simple fraction to its lowest terms. RULE. Find a common measure by dividing the lower term by the upper, and that divisor by the remainder, continuing till nothing remains; the last divisor is the common measure; then divide both parts of the fraction by the common measure, the quotients express the fraction required. NOTE. If the common measure happens to be 1, the fraction is already in its lowest term; and when a fraction has ciphers at the right hand, it may be abbreviated by cutting them off, as. Or, divide the terms of the fractions by any number that will divide them without a remainder; divide the quotients in the same manner, and so on, till no number will divide them both, and the last quotients express the fraction in its lowest terms. 2. Reduce to its lowest terms. 192 24 3 1 the answer. 576 72 9 3 Ans. 3. 3. Reduce 14 to its lowest terms. II. To reduce a mixed number to an improper fraction. RULE. Multiply the whole numbers by the denominator of the fraction, and to the product add the numerator for a new numerator, and place it over the denominator. NOTE. To express a whole number fraction-wise, set one for a denominator to the given number. EXAMPLES. 1. Reduce 5 to an improper fraction. 5X8+34 the answer. 2. Reduce 1835 to an improper fraction. III. To reduce an improper fraction to its proper terms. RULE. Divide the upper term by the lower, and the quotient will be the whole number; the remainder, if any, will be the numerator to the fractional part. 1. EXAMPLES. 2. Reduce 245 to its proper terms. Ans. 273. IV. To find the least common multiple or denominator. RULE. Divide the given denominators by any number that will divide two or more of them without a remainder, and set the quotients and the undivided numbers underneath. Divide these quotients and undivided numbers by any number that will divide two or more of them as before, and thus continue, till no two numbers are left capable of being lessened. Multiply the last quotients and the divisor or divisors together, and the product will be the least common denominator required. EXAMPLES. 7 1. What is the least common measure of 1⁄2, 1, 1, and? 3)9 1 15 2 3 1 5 2 3X5X2=30X3X8=720 Ans. 2. What is the least number that can be divided by the nine digits without a remainder ? Ans. 2520. V. To reduce vulgar fractions to a common denominator. RULE. Find a common denominator by the last case, in which divide each particular denominator, and multiply the quotient by its own numerator, for a new numerator, and the new numerators, being placed over the common denominator, express the fractions required in their lowest terms. EXAMPLES. 1. Reduce, §, and 7 to a common denominator. 36 the com. denom. The fractions will be 37, 38, 3 20 금. 2. Reduce, 4, 5, and 7, to a common denominator. 16 Ans. 1, f, H, H. VI. To reduce a compound fraction to a single one. RULE. Multiply all the numerators for a new numerator, and all the denominators for a new denominator, then reduce the new fraction to its lowest terms by Case I. EXAMPLES. 1. Reduce of of to a single fraction. 3X5X9-135 9 4X6X10-240 16 the answer. 2. Reduce of of 1 to a single fraction. Ans. . 3. Reduce of 1⁄2 of to a single fraction. Ans. 8 VII. To reduce a fraction of one denomination to the fraction of another, but greater, retaining the same value. RULE. Reduce the given fraction to a compound one, by multiplying it with all the denominations between it and that denomination, to which you would reduce it; then reduce that compound fraction to a single one. EXAMPLES. 1. Reduce of a penny to the fi action of a pound. 2. Reduce of a pennyweight to the fraction of a pound Troy. Ans.. VIII. To reduce a fraction of one denomination to the fraction of another, but less, retaining the same value. RULE. Multiply the numerator by the parts contained in the several denominations between it and that denomination to which you would reduce it for a new numerator, and place it over the denominator of the given fraction. EXAMPLES. 1. Reduce of a pound to the fraction of a penny. 1X20X12-240 960 the answer. 2 Reduce of a lb. Troy to the fraction of a dwt. Ans. §. IX. To find the value of the fraction in the known parts of the integer. RULE. Multiply the numerator by the known parts of the integer, and divide by the denominator. EXAMPLES. 2. What is the value of of a shilling? Ans. 4d. 3} qrs. 3. Reduce of a lb. Troy to its proper quantity. Ans. 7 oz. 4 dwt |