them first to a common denominator; for example, by . Thus 3-5-32-32=24=24÷20=1}. A. 1. 60. This process, however, involves no new principle, for it is in reality the same as that of the general rule. For, by inverting the divisor, the same terms are multiplied together in both cases. 3 × 8 24 = 61. Thus, taking the last example, 4x5 20' 62. When the divisor is a whole number, and the dividend consists of a large integer and a fraction-Divide each separately, and if the whole number leave a remainder, multiply it by the denominator of the fraction, and add the product to the numerator, for a new numerator, which divide as before. 63. For every unit of the whole number is equal to as many parts of the fraction as are indicated by the denominator. 3) 4 6 8 248 4) 1 5 6 0 8 2 2) 3 9 0 2 0 3 5) 195 10 31 64. Divide 468,248 by 3, 4, 2, 5, and 6. These operations involve all the variations that can possibly take place in dividing a fraction by an integer. The 1st rem. is : 2nd, 23=÷4=3: 3d, 2: 4th, 5: 5th, 2 21÷6=3. 137 A. 65,73042 A. 13,725800 65. Divide 47,325,737 by 6, 5, 4, 3, and 2. 66. Divide 76,543,210 by 8, 7, 5, 4, and 3. 67. Divide 37,754,276 by 5, 4, 7, 6, and 5. 68. Divide 76,864,207 by 8, 7, 5, 5, and 4. 69. Divide 42,680,960 by 12, 2, 6, 2, and 7. 70. Divide 98,765,432 by 9, 8, 7, 6, 5, 4, 3, and 2. A. 272,7758 71. Find by the preceding rules what number multiplied by will make 153. 72. What part of 108 is 5 of an unit? 45360. A. 21. A. 1298 5 73. What number is that which, if multiplied by of will produce only of an unit? of 15 MISCELLANEOUS EXAMPLES. LXVII. 1. A gentleman has two sons; the age of the elder added to his, make 126 years, and the age of the younger son is equal to the difference between the age of the father and the elder son. Now if the father be 80 years of age, how old is each of his sons? A. 34 years, and 46 years. 2. What number is that, from which, if a twelfth part of 1,728 be Q. How may a mixed number be divided wlthout any reduction, by a whole number? 62. Why is the remainder, if there be any multiplied by the denomi nator of the fractional part 63 deducted, and the remainder increased by the ninety-fifth part of 82175, the sum will be 1185? A. 464. 3. What number divided by 1185 will give 497 for the quotient and leave just a fifth part of the divisor remaining? A. 589,182. 4. A merchant, having bought at one time 25bl. 27gal. 2qt. 1pt. 2gi. of molasses, and at another 5 times as much, sold of the whole; how much has he on hand, unsold? A. 116bl. 14gal. 1qt. 3gi. 5. "At the Clinton works in Scotland, a sheet of paper has beenrecently [1839] produced, which, though but 50 inches wide, measures a mile and a half in length." How many square feet does this mammoth sheet contain? How many smaller sheets 1 feet long and 1 foot wide will it make? How many quires will these sheets make? What will be their value at $4 per ream? A. 33,000 sq. ft.; 26,400 sheets; 1,100 quires; $247. 6. How many acres of land are there in a piece 80 rods long and 36 rods wide? A. 18A.—In a piece 802 rods long and 367 rods wide? A. 18A. 2R. 173rd. 7. If a talent of silver be worth £357. 11s. 10 d. sterling, what is the value of a shekel, of which 300 make a talent; and what is the weight of a talent, a shekel weighing 9dwt. 3gr. A. £1. 3s. 103d.; 136oz. 17dwt. 12gr. 8. If London was built 1108 years before Christ's nativity, how many hours is it since, to Christmas, 1835, allowing 365 days to the year? A. 25,798,338 hours. 9. A father gave of his property to his daughter, to his son, and the rest, being $1,200, to his nephew; what was the value of the father's estate? A. $8,400. 10. "The tail of the comet of 1811 was no less than one hundred and thirty-two millions of miles in length. Now, allowing the earth to be 25,000 miles in circumference, and the tail of the comet a bandage, how many times would it enwrap the earth?" A. 5,280 times. 11. A general distributed £307. 17s. among 4 captains, 5 lieutenants, and 60 common soldiers; to every lieutenant he gave twice as much as to a common soldier, and to every captain three times as much as to a lieutenant; what did each receive? A. Soldier £3. 51s.; lieut. £6. 11s.; capt. £19. 13s. 12. What number is that from which if you subtract of of a Lit, and to the remainder add 3 of 3 of a unit, the sum will be 9? 9 139 A. 82081 3960 13. Suppose a quotient to be 31⁄2 times, and a dividend 5 times what will be the divisor? 179 A. 512 14. A merchant invested $8,300 in a certain bank, being just of its capital; what was the capital of the bank? A. $190,900. 15. A merchant gave $1,956 for 4 of a sloop, and of the value of the sioop for its entire cargo; what was the estimated value of both sloop and cargo A. $4,794.032. 16. Suppose a man's family expenses are $730 annually, being only of his profits in trade; how much then can he save every year? A. $1,217 inches in circumference, 17. Suppose a carriage wheel to be 15 how many times would it turn round in going 3673 miles? A. 1,530,53443 times. 18. Divide 12 sq. rd. 30 sq. yd. 6 sq. ft. by 7, and multiply 1 sq. rd. 25 sq. yd. 1233 sq. in. by 7. 19. Suppose a pile of wood to be 200 feet long, 63 feet high, and 4 feet wide; how many solid feet does the pile contain, and how many cords? A. 6,1943 s. ft.; 48C. 503 s. ft. 20. How many cubic feet in a box 93 feet and 5 feet wide? long, 83 feet high, A. 411 s. ft. 21. Suppose a room to be 10 feet between floors and 18 feet square, what will be the expense of plastering it at 8 cents per square yard? There are 4 sides each 10 ft. by 18, and the surface over head is 18 by 181. A. $9.4417. 22. A wealthy merchant, on retiring from business, invested of all his property in banks, in private loans, in real estate, and the rest, which was $3,000, he reserved for repairs on his estate; how much must he have accumulated? A. $216,000. 23. Suppose a man buys of a ship, which is valued at $63,000, and divides it equally among his sons, giving to each of his part; how many sons has he, and how many dollars does each receive? A. 7 sons; $6,750 each. 24. Divide 11⁄2 of a roll of broadcloth, which contains 32 yards. into equal pieces, each to contain of the whole roll. 25. Suppose two boys, having bought a kite together, one paying of a dollar and the other of a dollar, sell it for of a dollar more than they paid for it; what did they pay for the kite? What did they get for the kite? What is each one's part of the kite? What is each one's share of what it sold for? is the value of both ship and cargo? 27. A person left of his property to A, A. $11. A. $1. A. ; . A. $19; $35. A. $14; $113. of her cargo, what A. 841,7773. to B, to C, to D, to E, to F, and the rest, which was $800, to his executor; what was the value of the whole property, and of each person's share? A. A's $4,000; B's $3,000; C's $1,250; D's $500; E's $250); F's $200 DECIMAL FRACTIONS. LXVIII. 1. A DECIMAL FRACTION1 differs from a vulgar fraction only in respect to its denominator, being uniformly either 10 or 100 or 1000, &c., and therefore it need not be, and seldom is, expressed. 2. The numerator then is written alone with a point before it, to distinguish it from whole numbers; this point is thence called separatrix, and sometimes the decimal point. 3. Thus .3 is; .34 is 34; .345 is 345; ; .3456 is 3456 4. UNITY then in decimals is first divided into 10 equal parts, which are therefore called TENTHS. 345 10000 5. The TENTH is divided into 10 other equal parts, making 100 equal parts of unity, which are thence called HUNDREDTHS. 6. The HUNDREDTH is divided into 10 other equal parts, making 1000 equal parts of unity, which are thence called THOUSANDTHS; and so on, as in the following 7. Since one decimal figure has for its denominator 1 with one cipher, as .5=15; two decimal figures, 1 with two ciphers, as .25= 25 ; three decimal figures, 1 with three ciphers, as .125-125, and so on; therefore, 1000 8. A DECIMAL FRACTION is that fraction whose denominator is always understood to be a unit, or 1, with as many ciphers annexed as the given decimal has places of figures. 125 9. Thus, .8 is; .08 is 8; .35 is 35; .0125 is 35. 10. When the numerator has not so many decimal places as the denominator has ciphers, we must prefix ciphers enough to the numerator to make as many. 11. Thus ‰ is written .05; ro‰=.0045; 10000 = .0006. 12. Since .5, .05-180, .005-Too, then .05 is 10 times less in value than .5, and .005 is 10 times less than .05: LXVIII. Q. How does a decimal fraction differ from a vulgar one? 1. How is the numerator written? 2. What does .3, .34, .345, .3456, with the point before each number, mean? 3. How is unity divided and sub-divided in decimals? 4, 5, 6. Repeat the table of these divisions. What is the denominator of one, two, or three decimal figures? 7. What then is a Decimal Fraction? 8. What is the denominator for .8? See 9. What is the denominator for .08?-for .35?-for .0125? 9. When are ciphers to be prefixed to tho numerator? 10. How do you write decimally, or 45 or 10000? 100009 1 A DECIMAL FRACTION is so called from the Latin word decimus, signifying tenta because it increases and decreases in a tenfold proportion 5 13. For 10 times 1000 is 1500 1009 5 50 100 100 5, and 10 times == 14. Hence, a cipher placed on the left of any decimal decreases its value in a tenfold proportion, by removing it farther from the separatrix or decimal point. 15. But a cipher on the right of a decimal merely changes its name without altering its value. 50 100 29 16. For .5 is; so .50 is 51, and .500 is 150001, but the first is read 5-tenths, the second, 50-hundredths, the third, 500thousandths. 17. Decimals then increase from the right to the left like whole numbers, and of course their decrease from right to left is in the same proportion. 18. Hence every removal of any figure one place further towards the right decreases its value in a tenfold proportion. 5 19. Thus 555.555 is really 500, 50, 5, 10, 180, 1000: 20. Our system of notation, then, which begins in whole numbers, is carried out by means of decimals, so as to embrace as many places below units as above or beyond them, even millions and millionths billions and billionths, as in the following TABLE II. HUNDRED THOUSANDS. TEN THOUSANDS. THOUSANDS HUNDRED-THOUSANDTHS. TEN-THOUSANDTHS HUNDREDTHS. UNITS. THOUSANDTHS. HUNDRED-MILLIONTHS .. Separatrix. TEN-MILLIONTHS. MILLIONTHS. TENS. BILLIONTHS. 5 Ascending. Descending 21. The first decimal figure is 5 tenths; the second is 5 hundredths, or the first two 55 hundredths; the third is 5 thousandths, or the first three 555 thousandths; the fourth is 5 ten-thousandths, or the first four 5555 ten thousandths, &c. RULE FOR NUMERATING AND READING DECIMALS. 22. Begin on the left and say, tenths, hundredths, thousandths, ten-thousandths, &c., as in the table. Q. What is the difference in value between .5 and .05?-.05 and .005? (See 12.) What is the effect of the cipher? 14. What is the use of a cipher on the left of a decimal? 15. How is a decimal figure affected by changing its place? 18 Why? 17. What is the value of each figure in 555.555? 19. Describe our sys tem of notation. 20. Repeat the decimal part of the table beginning with "tenths" and ending with "billionths." Suppose each decimal place to be filled with the figure 5, what would be the value of the first 5?-of the second?third ?-fourth? &c. 21. What is the rule for numerating decimals? 22 |