Complex Fractions. 145. Complex fractions are reduced to their simplest form, by the operations of Reduction and Division. Hence, for the reduction of a complex fraction to its simplest form, we have the following Rule. Reduce each term to a simple fraction, and then perform the division. 145. What is a complex fraction? How are complex fractions reduced to their simplest forms? What is the rule for Reduction? Miscellaneous Examples. 1. A man, having 9 dollars, paid 3 dollars for boots, and 4 dollars for a hat: how much had he left? 2. A retailer gave his customer 17 dollars in change, which, he afterwards found, was of a dollar too much : what was the exact amount of change due ? 3. A young clerk, having charged of a dollar too much for some cloth, gave in change, 15 dollars: what was the exact amount that he ought to have given? 4. A bank of issue failed, and was able to redeem its notes by paying of a dollar on a dollar: how much would he who has a 10 dollar bill, receive from the bank? 5. The sum of two numbers is 12%; one of the numbers is 74 what is the other? 6. James, Joseph, and Daniel owned three farms, whose total area was 475 acres. Daniel had 15 acres more than Joseph, and Joseph 247 acres more than James: how many acres had each in his farm? 7. A housekeeper bought 6 mahogany chairs, at 3 dollars each, and gave for them, 2 ten-dollar and 1 five-dollar bill: what change ought she to receive? 8. A mechanic that was fond of reading, wished to buy Macaulay's History, worth 6 dollars, Irving's Columbus, worth 4 dollars, and Prescott's Philip II., worth 5 dollars; his daily wages were 13 dollars a day: how many days' wages would pay for the books? 9. If 12 barrels of flour were given for a piece of cloth, measuring 31 yards, and valued at 23 dollars a yard, what would be the value of one barrel? 10. A grocer having of a barrel of sugar, sold of it for 4 dollars: what was the value of the barrel, at the same rate? 11. The product of of 23, by 2 of 3 of 9, is how much greater than the quotient of 7 divided by 7 of 64? dollars, and it will worth 13 barrels 12. The cost of a barrel of flour is 6 buy 2 barrels of apples, each of which is of potatoes: how many pounds of butter, at of a dollar for 3 pounds, would pay for a barrel of potatoes? 13. The product of 3 numbers is two of the numbers are 2 and what is the third ? 14. A father and son, working an equal number of days, earned 547 dollars: the father received 13 dollars, and the son of a dollar, a day: how many days did they work? 15. A regiment lost in battle 250 men, which was of the regiment: what was the number of men before the battle? 16. A merchant owning for 1640 dollars: what was rate? of a vessel, sold of his share the value of the ship, at that 6 25 17. How many lemons, at of a dollar a dozen, will pay for 81 oranges at 2 cents each? 5 18. A lad, multiplying by instead of obtained for a result: what result ought he to have obtained? 19. Reduce 7 of 24 X to a simple fraction. of 20. If of a yard of cloth cost of a dollar, what will be the cost of 25 yards? 21. If 231 dollars are required to pay 18 men for 1 day's wages, how much would be required to pay 33 men for 153 days' labor? 22. If A. can mow an acre of ground in 3 days, and B. in 2 days, how long would it take them both to mow it? 23. If A. and B. can do a piece of work in 10 days, and A. alone can do it in 16 days, in what time can B. do it? 25. In a piece of cloth there were 36 yards. The piece cost 65 dollars. For what must the cloth be sold at per yård, that there may be a gain of 1825 dollars ? DECIMAL FRACTIONS. 146. There are two kinds of Fractions: Common Frac tions, and Decimal Fractions. 147. A COMMON FRACTION, is one in which the unit is divided into any number of equal parts. 148. A DECIMAL FRACTION, is one in which the unit is divided into 10 equal parts, then each of these parts is again divided into 10 equal parts, and so on, using 10 constantly as a divisor. When the unit is divided into 10 equal parts, there are 10 such parts of the unit, and each part is called, one-tenth. If each tenth be divided into 10 equal parts, there will be 100 equal parts in the unit, and each part will be of = 100. If each hundredth be divided into 10 equal parts, there will be 1000 equal parts in the unit, and each part will be to of 100 = 1000; and smaller parts may be obtained, by still dividing by 10. Notation and Numeration. 149. A period (.), called the decimal point, written before a figure, denotes that its unit is 1 tenth : 146. How many kinds of Fractions are there? What are they? 147. What is a Common Fraction? 148. What is a Decimal Fraction? When the unit is divided into 10 equal parts, what is each part called? What is each part called, when it is divided into 100 equal parts? The second place from the decimal point, is the place of hundredths: The third place is the place of thousandths: Thus, .001 is read, 1 thousandth = тобо 4 thousandths = 1000. 7 thousandths = 1000. The fourth place is the place of ten-thousandths; the fifth, of hundred-thousandths; the sixth, of millionths, &c. Thus, 4, written in the different places, is read, 150. We numerate from the decimal point to the right, and read in the lowest fractional unit of the decimal. Thus, we numerate, tenths, hundredths, &c.; and read, 4 tenths, 4 hundredths, &c. 151. From the nature of decimals, and the manner of writing them, we see, 1st. That the denominator belonging to any decimal fraction, is 1, with as many ciphers annexed as there are places of figures in the decimal. 149. What is the decimal point? Where is it written? What does it denote? What is the first place to the right called? The second? The third? 150. How do you numerate decimals? How do you read them? |