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5 T. 8 cwt. Ans. 5 T. 8 cwt. 3.qrs. 24 lbs. 13 oz. 14 drs
65 pks. 544 qts.
4 Reduce 2 cwt. 3 qrs. 17 lbs. to pounds. Ans. 325 lbs. 5 Bring 16 bushels i peck to pecks. 6. Reduce 2 hhds. 10 gallons to quarts. 7. Reduce 18 years, 6 months, to months.
222 mo. 8. Reduce 15 yards, 2 feet, to inches.
564 in. 9. Reduce 3 leagues, 2 miles, 7 furlongs, to furlongs 95 fur. 10. Reduce 11 acres, 2 roods, 19 poles, to poles. 11. In 12 cords of wood how many solid feet and inches ?
Ans. 1536 ft. ; 2654208 in. 12. In 21 cords of wood how many solid feet ? 2688 ft. 13. In 24796800 seconds, how many weeks? 14. In 20692 square rods, how many acres ? 129 A. 1 R. 12 po. 15. In 15840 yards how many miles and leagues ? 9 in. 31. 16. In 51 miles how many furlongs and poles ?
Ans. 408 fur. ; 16320 po 17. In 3783 nails how many yards ? 236 yds. 1 qr. 3 na. 18. Bring 892245 ounces into tons.
Ans. 24 T. 17 cwt. 3 qr. 17 lb. 5 oz. 19. In 12 hhds. of sugar, each 11 cwt. 25 lb., how
many ounces ?
Ans. 241344 oz. 20. How many revolutions will the wheel of a steamboat, 27 feet in circumference make, in navigating the Hudson from New York to Albany, supposing the distance 160 miles, with out making any allowance for current or tide ? Ans. 31288
21. How many inches between the earth and the sun, the distance being 95,000,000 miles ? Ans. 6019200000000.
22. If a cannon ball move at the rate of 1 mile in 7 seconds, how long would it be in passing from the earth to the sun ? (Multiply by 7, divide by 60, &e.) Ans. 21yr. 31da. 18h. 13m. 20s.
23. Admitting your age to be 16 years, 1 week, 17 days, 20 minutes, 45 seconds, how many seconds old are you? (52w.=ly.)
Ans. 5052684458. 24. Reduce 184320 grains to pounds.
Ans. 32 lbs. 25. In 16 tons how many hundred weight ? quarters ? pounds ? ounces? and drams?
Ans. 320cwt.; 1280qrs.; 32000lbs.; 51200002.; 8192000drs. 26. In 1332005 grains (apoth.) how many pounds?
Ans. 231 lb. 3 oz. 5 gr. 27. In 9173 nails how many yards ? Ans. 573 yds. I qr. 1 na. 28. In 3 ells French, how
Ans. 162. 29. In 31594550 inches how many miles ?
Ans. 546 m. 30. What is the circumference of the earth in yards ?
Ans. 44035200 yds. 31. Reduce a sular year to seconds: Ans. 31556937 sec. 32. In 622080 solid inches how many tons of round timber?
Ans. 9 T. 33. In 5529600 cubical inches how many cords. Ans. 25.
34. How many minutes and seconds in one complete revolution of a planet ? Ans. 21600 minutes; and 1296000 sec. 35. In 89763 square yards how many acres ?
Ans. 18 A. 2 R. 7 po. 101 st. 36 in. 56. In 5054 pints how many bushels ? Ans. 78 b. 3 pk. 7 qts.
37. How many quarters of rice, at 6c. per pound, may bought for D3.36 ?
33. How many tuns of wine, at 6 cents a gill, may be bought for D483.84.
Ans. I tun. 39. What is the value of a silver cup weighing 10 oz. 5 pwt. 18 gr. at 5 mills per grain ?
Ans. 24D. 69c. 40. At 9 cents an diour what will 2 yrs. 6 mo. 3 w. 6 da. 1? h. labor be ?
Ans. D1873.80. 41. At 320 cents a yard, what will 64 nails of cloth cost?
Ans. D 12.80. 42. In 85400 grains how many scruples? drams? and ounces?
Ans. 4320 sc.; 1440 dr. ; 180 oz 43. In 890330 1 barleycorns how many leagues ?
Ans. 15 l. 1 m. 6 fur. 28 po. 4 yus. 44. In 20 cords of wool how many solid inches ?
Ans. 4423680 in.
be Ans. 2 qrs. 45. In 20 tuns of wine how many gills ? Ans. 161280 g 46. In 11378955037 seconds, how many years?
Ans. 360 yrs. 300 da. 20 ho. 50 m. 37 sec. 47. How many barley-corns will reach round the globe, it being 360 degrees?
Ans. 4755801600. 48. In running 300 miles, how many times will a wheel 9 ft 2 in. in circumference turn round ?
Ans. 172800. 49. How many seconds in 1828 years, allowing 3651 days to the year?
Ans. 57687292800. 50. What is the cost of a tun of wine at 3 cents per gill?
When is Compound Reduction used ? When several de. nominations are given to be reduced, how will you proceed ? What is the rule ? How will you prove questions in this rule ?
PROPORTION; OR, SINGLE RULE OF THREE.
The first idea that naturally presents itself to the mind in the consideration of Proportion, is Ratio, or the connexion existing between Ratio and Proportion ; for ratio is a Latin word, and means proportion, although it can only exist between quantities of the same kind. The numbers given are called terms, and may be considered as a divisor and dividend; and the ratio, the required term, as a quotient in a division sum. The terms are written thus, 5:15, ratio 3 ; 3x5=15; the ratio of 84 : 336 =4, because 84 is contained in 336, 4 times ; so may any two numbers of the same kind form a ratio. What is the ratio of 9 to 18? Ans. 2. What is the ratio of 6 to 24? What is the ratio of 11 to 13? Ans. 1. What is the ratio of 100 to 30 ? Ans. 2 : 4 :: 8:16; here the ratio is 2, because 2 is in 4=2, and 8 is in 16=2. 2 : 10 :: 12 : 60; here the ratio is 5, because 2 is in 10=5; 12 in 60=-5. 16 : 15 :: 48 : 45; here the ratio is reversed, and will stand thus, 16, first and second terms; 48=16, the third and fourth terms.
Proportion is a comparative relation of one thing, or number, to another ratio. Proportion, when complete, consists of four numbers or terms, and is a combination of two equal ratios ; and the product of the first and fourth terms (which are the extremes) is equal to the product of the second and third terms
(middle term or means.) Therefore, it is evident that any one
lbs. lbs. D. Here the price of 32 32 4 : 12 :: 8 the tea is 2 dollars
per pound ;* conse
quently, 4 pounds will 4)96 cost 8 dollars,
pounds will cost 24 D24 cost.
dollars, &c. D. lbs. D. Again : If 24 : 12 :: 8 : 4 lbs. Ans.
8 Hence we find, to obtain either the
first or fourth term, divide the product 24)96(4lb. of the second and third (middle)
terms, by the given extreme, and the quotient will be the other extreme, or term sought. And to obtain either of the middle terms, divide the product of the extremes by the given middle terms, and the quotient will be the middle term required. The first and second terms of a proportion always express quantities of the same kind, and so likewise do the third and fourth terms.
Thus:lbs. lbs. D. D.
2 12 6 36 That is, if 2 pounds cost 6 dollars, what will 12 pounds cost? Ans. 36 dollars ; it is 3 dollars per pound, and 3x12=36 dollars. This last example is the modern method of stating Proportion, and now considered the most correct. The old method was thus: If 21b. : cost 6D. :: what cost 121b. : 36D. ; 12 x 6=72:2=36 as above.
The following short rules can often be used to advantage :
1. Divide the second term by the first, multiply the quotient by the third, and the product will be the answer.
lbs. D. lbs. D.
5 10 15 30 2. Divide the third term by the first, multiply the quotient by the second, and the product will be the answer.
lbs. D. lbs. D.
30 3. Divide the first term by the second, and the third term by that quotient, and the last quotient will be the answer.
D. lbs. D. lbs.
4. Divide the first term by the third, and the second term by that quotient, and the last quotient will be the answer. lbs.
D. : 30 5
Rule for stating:-1. Place that term in the third place, which is of the same name or kind, with that in which the answer is required; and if the answer requires to be greater than the third term, set the greater of the two remaining numbers for the second term, and the other number for the first terin.
2. But if the answer requires to be less than the third term, set the less of the two remaining numbers in the second place, and the other in the first place.
Rule for working.-1. Reduce the third term to the low denomination mentioned in it.
2. Reduce the first and second terms to the lowest denomination mentioned in either of them.
3. Multiply the second and third terms together, and divide by the first, and the quotient will be the answer, in the same name to which the third was reduced; then bring this denomination into the answer required.
When 1, or unity, is one of the terms, a formal statement is not required ; it can be done by multiplication or division
1. Multiply the first and fourth terms together (extremes). Multiply the second and third terms together (means).
If the four numbers are proportional, these products will be "equal.
2. Divide the larger of the first and second terms by the less ; divide the larger of the third and fourth terms by the less, and the two ratios will be equal.
3. Invert the question.
The terms may be distinguished generally in the following manner : the first term is preceded with an if, or supposition, and the second by a demand, “what will,” “ what cost,” &c.