13. (§ 25.) is equal to how many ninths ? how many thirty-sixths ? how many seventy-fifths ? 14. (§ 26.) Reduce the two fractions, %1 and 4, to their lowest terms, and then add them together. 15. ($ 27.) Suppose a farm to contain 98/2 acres of land; how many acres are there in a of the farm ? 16. (28.) Add together, io and is and 14; then subtract is from the sum;- what is the remainder? 17. (§ 29.) If and is of a number be subtracted from itself, what part of that number is the remainder ? 18. (8 30.) If the mail-stage run 9 { miles in 1 hour, how many hours will it be in running 175 miles ? RETROSPECTIVE OBSERVATIONS. A fraction is rendered greater by increasing the numerator, and smaller by increasing the denominator. To multiply a fraction by a whole number, - Either multiply the numerator, or divide the denominator. To divide a fraction by a whole number, - Either divide the numerator, or multiply the denominator. When a number is multiplied by 1, the product is equal to the multiplicand. Therefore, when a number is multiplied by a fraction, which is less than 1, the product must be less than the multiplicand. To multiply a whole number by a fraction,-Multiply by the numerator, and divide by the denominator. Dividing a number by 1, gives a quotient equal to the dividend. Therefore, dividing a number by a proper fraction, must give a quotient greater than the dividend, because, the fraction being less than 1, is contained a greater number of times in the dividend. To divide a whole number by a fraction, --Multiply by the denominator, and divide by the numerator. To multiply a fraction by a fraction,-Multiply numerator by numerator, and denominator by denominator. To divide a fraction by a fraction,-Multiply the numerator of the dividend by the denominator of the divisor, for a numerator; and multiply the denominator of the dividend by the numerator of the divisor, for a denominator 19. Multiply the fraction, 137, by 38. 20. Divide the fraction, by 145. 21. Multiply 8706 by the fraction, 484. 22. Divide 611 by the fraction, o 23. Multiply the fraction, 265, by the fraction, 11. 24. Divide the fraction, 14 , by the fraction, 7i. 25. What is the product of 608 multiplied by 8 ? 26. What is the quotient of 453 divided by 322 SECTION 32. MISCELLANEOUS EXAMPLES. 1. Suppose a man can perform a journey in 14 days and 3 hours, travelling 9 hours a day; in what time can he perform the journey, travelling 11 hours a day? 2. A trader gave $75 for 56 gallons of wine, and lost 11 gallons by leakage. At how much per gallon must he sell the remainder, to get the whole cost? 3. Suppose a retailer to pay $165 for a ton of sugar, at what price must he sell it per pound, in order to gain 10 per cent. on the cost? 4. What quantity of salt, worth 62 cents per bushel, must be given in exchange for 258 pounds of pork, worth 9 cents per pound? 5. What is the profit on 400 hogsheads of molasses, purchased in New Orleans at 12 cents per gallon, [63 gal. in each hhd.], freighted to New York at $3.50 per hhd., and sold at 24 cents per gallon; 3gal. 2qt. having leaked from each hhd. on the passage? 6. If a pint of rum a day will kill a man in a year and a half, how many men would a cargo of 600 hogsheads kill in the same time? 7. If 11 young men can become fools by drinking 6 bottles of wine, at $3 a bottle, what would it cost a dinner party of 25, to become fools in like manner? 8. If a man's expenses be $1.40 a day, and his income $700 a year, what will he lay up in 7 years? 9. Ä and B are laborers- A earns $19.50 a month, and B earns $16.25; but A gives B is of his earnings. What will each lay up in 14 months ? 10. Find the difference between of 91, and of 91 On the opposite page, 30 cities and towns are exhib ited in their respective situations, relative to each other; and the number of miles, by mail-road from town to town, is noted in figures. 11. Find the distance from Washington, through the intermediate towns, to Augusta, Me....... from Washington to Detroit ....... from Washington to St. Louis .... from Washington to Natchez ....... from Washington to New Orleans ....... from New Orleans to Augusta, Me. 12. Suppose a citizen in each of the places on the opposite page, to start for Washington, and travel 7 miles an hour, 10 hours in each day; how long will each one be in performing his journey? 13. How long would it take you to walk from your school-room to Washington; allowing that you could walk 37 miles an hour, 7 hours in each day? 14. Two men started at the same time one of them from New Orleans, and the other from Augusta, Me.and travelled towards each other, with equal speed. Between what two towns, and what distance from each of these towns did they meet? 15. Mr. A. went from Portland to Baltimore, travelling 5 miles an hour, and 10 hours a day. Mr. B. performed the same journey; but started 1 day later, and travelled 7 miles an hour. Where did B. pass A.? 16. Divide $ 1000 among A, B, and C, giving B twice as much as A, and C twice as much as B. 17. Gunpowder is composed of 5 parts sulphur, 7 parts charcoal, and 38 parts nitre. How many pounds of each ingredient, in 100 pounds of powder ? 18. A and B purchased a cow for $16. A paid $9 of the price, and B paid $7. They sold the cow for $21. What was each one's share of the gain ? Solution. Since A paid is of the price, and B 162 A must have 1 of the gain, and B 1. 19. C and D traded in partnership; C owned $450 of the stock in trade, and D $ 290. They gained $146. What was each one's share of the gain ? 20. Suppose $1000 stock in trade to gain $ 230; what is the gain on $ 351 of that stock ? 21. E and F purchased 245 acres of land, for $ 2600 E paid $ 1200 of the money, and F paid the remainder How much land should each one have? 22. The national debt of England is not less than $1900 000 000. Allowing 5 per cent. interest to be paid on this sum, how many families would it support, each family spending $ 400 per annum? 23. If a man can dig a trench in 15 days, and a boy can dig the same trench in 18 days, in what time can they both dig it? (See example 20, Oral sec.) 24. How many days will it take 17 men to perform a piece of work, that I man can perform in 95 days? 25. How many days will it take 30 men to perform a piece of work, that 4 men can perform in 50 days? 26. How many days will it take 25 men to perform a piece of work, that 6 men can perform in 40 days? 27. If 15 yards of carpeting, which is one yard wide, will cover the floor of a room, how many yards of carpeting, 3-quarters wide will cover the same floor? Direction. Find the number of square quarters contained in 15 yards of the wider carpeting; then divide this number, by the number of square quarters contained in one yard of the narrower carpeting. 28. Suppose 31 yards of broad-cloth 5-quarters wide, to be made into a cloak; how many yards of silk 3-quarters wide, will it take to line the cloak? 29. How many yards of carpeting that is 5-quarters wide, will cover the floor of a room which is 192 feet in length, and 15 feet in width? 30. How many bricks will it take to build a wall, 1 foot thick, 5 feet high, and 24 feet long; each brick being 8 inches long, 4 inches wide, and 2 inches thick? 31. If a man can hoe iz of an acre of corn in a day, and a boy } of an acre, how much can they both hoe in a day? In what time can they both hoe 9 acres? 32. There is a cistern, having 3 pipes; the first. pipe will discharge the cistern in 4 hours, the second in 5 hours, the third in 6 hours. What part of the contents of the cistern would all the pipes together let off in 1 hour. In what time would they all discharge the cistern? |