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70 or 7, at $1 20.
20 2, 1 50.
30 3, 2 00.
60 6, 2 50.
20 or 2, at $1 20.
2 50. 2. How much rye, at 48 cents per bushel, barley at 36 cts and oats at 24 cents, will make a mixture worth 30 cts. per bushel? Ans. 1 at 48 cts. 1 at 36 cts, and 4 at 24 cts.
3. How much sugar at 4 cts. at 6 cts. and at 11 cts. per Ib. must be mixed together, so that the composition may be worth 7 cts. per lb. ? Ans. Any weight of equal quantity.
4. It is required to mix several sorts of wine, at 75 cents, $1.00 and $1.25 per gallon, with water, that the mixture may be worth 50 cents per gallon-How much of each sort must the mixture consist of? 75
50 or 1 at 75 cents. 100
50 1 100
50 1 125
25+50+75=150 3 of water.
CASE III. When the price of each simple is given, also the quantity of one of them, and the mean rate of the whole compound, to find the several quantities of the rest.
RULE. Place the several prices one under the other, and the mean rate to the left hand, and take their difference as in case 2; then,
As the difference of the same name with the quantity given
Examples. 1. Twelve bushels of wheat at $1.00, with rye at 50 cts., barley at 40 cts. and oats at 25 cts. What quantity of these must be mixed with the wheat, to rate at 60 cts. per
40 As 65 : 40 :: 12 : 713
rye. 65 : 40 :: 12 : 713
* 3 of barley. Ans. 65 : 40 :: 12 : 7 of oats. 2. How much alloy, and how much gold, of 21 and 22 carats fine, must be put to 30 ounces, of 20 carats fine, to bring it to 18 carats fine. Ans. 30 oz. of 21, 30 of 22,
and 15 oz. of alloy, 3. How much Malaga at $1.12per gallon, sherry at 81.00, and white wine at 75 cts. must be mixed with 30 gallons of canary at 87} cts., so that the mixture may stand in 931 cts. per gallon Ans. 10 gal. at $1.12), 30 gal. at $1.00,
and 10 gal. at 75 cts. per gallon. S
CASE IV. When the particular rates of all the ingredients proposed to be mixed, the sum of all their quantities, with the mean rate of that sum being given, to find the particular quantities of the mixture.
Rule. Set down all the particular rates, with the mean rate as before; find the differences, and add them all into a sum ; then,
As the sum of the differences
Examples. 1. Hiero, king of Syracuse, gave orders for a crown, to be made entirely of pure gold; but suspecting the workmen had debased it, by mixing with it silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown. Archimedes, in order to detect the imposition procured two other masses, the one of pure gold, and the other of silver or copper, and each of the same weight with the former; and by putting each, separately, into a vessel full of water, the quantity of water expelled by them determined their specific
bulks. Now, suppose the weight of each mass to have been 5 lb.; the weight of the water expelled by the alloy, 23 oz.; by the gold, 13 ounces, and by the crown, 16 ounces; that is, that their specific bulks were as 23, 13 and 16: then, what were the quantities of gold and alloy in the crown?
13- 7 of gold? And the sum of these is 7+3 = 10, 16
23– 3 of alloy) which should have been but 5. Whence by the rule,
Answer. 10 : 3 :: 5 : 1} of alloy 2. How many gallons of water must be mixed with wine, at $1.00 per gallon, so as to fill a vessel of 100 gallons, that may be afforded at 80 cents per gallon? Ans. 20 gallons. 3. A
grocer had 4 sorts of sugar, at 4 cents, 8 cents, 10 cents and 12 cents per lb. the worst would not sell, and the best was too dear; he therefore concluded to mix 90 poundsWhat quantity of each must he take, so as to sell at 9 cents per pound ?
Ans. 27 lbs. at 4,9 at 8, 9 at 10 and 45 at 12 cts.
Or, 9 lbs. at 4 cts. 27 at 8, 45 at 10, and 9 at 12.
POSITION. Position is a rule that by false or supposed numbers, taken at adventure, and worked with according to the nature of the question, discovers the true number sought.
SINGLE POSITION Teaches to resolve such questions as require only one supposed number, by the following
Rule. Take any number, and perform exactly the same operations with it, as are described to be performed in the question: then,
As the result of that operation,
Note-If the results of two or more supposed numbers be in the same proportion as the supposed number; or if, upon working with two supposed numbers, and multiplying each of them by the result of the other, the products be equal, then the question may be solved by. Single Position ; if otherwise, it cannot.
Examples. 1. A gentleman, at his decease, left $3000 to be divided amongst his three sons, whose several ages were 18, 19, and 20 years, in such a manner that their several portions, when they would arrive to the age of 21 years, should be equal, reckoning interest at 6 per cent. during their minorities, -I desire to know the sum bequeathed to each. Suppose $1000 the sum received by each at the age of 21
years ; $106, 112 and 118, the respective amounts of
S S As 106 : 1000 :: 100 : 943.396 sum bequeathed to the eldest 112 : 1000 :: 100 : 892.857
second 118 : 1000 :: 100 : 847.453
according to this 2683.711
S S S As 2683.711 : 3000 :: 1000 : 1117.86, the sum received by
[each at the age of 21. Ans. S $ S S As 108 : 1117.86 :: 100 : 1054.58 bequeathed to the eldest. 112: 1117.86 :: 100: 998.09
second. 118: 1117.86 :: 100: 947.33
3000.00 whole sum bequeathed, according to the question.
2. A gentleman bought a chaise, horse and harness, for 240 dollars; the horse came to twice the price of the harness, and the chaise came to the price of the horse and harness- What did he pay for each ?
Ans. Harness, $40; horse, $80; chaise, $120. 3. A., B. and C. bought a quantity of goods for $420, and agreed among themselves that C. should have a third part
more than A., and B. should have as much as them both-I desire to know how much each must pay.
Ans. A. $90; B. $120, and C. $210. 4. The yearly interest of a sum of money at 6 per cent. exceeds zb of its principal by 840—I wish to know the principal.
Answer, $4000. 5. A gentleman bought two pieces of cloth, containing together 60 yards; the price of one of the pieces was $3.00 per yard, and of the other $5.00 per yard, and the value of each piece was the same- -How many yards were in each piece, and what the total amount ? Ans. 371 yards in one.
221 in the other.
$225 total amount. 6. In an orchard of fruit trees, \ of them bear apples, À pears, } plums, 30 of them peaches, and 40 cherries How many trees does the orchard contain ?
Ans. 1000. 7. What is the age of a person who says that if } of the years I have lived be multiplied by 4, and of them be added to the product, the sum will be 82?
Ans. 41 years.
DOUBLE POSITION Is by making use of two supposed numbers, and if both prove false, (as it generally happens,) they are, with their errors, to be thus ordered:
RU!E 1. 1. Place each error against its respective supposed number. 2. Multiply them crossways.
3. If the errors be alike, that is, both too much or too little, take their difference for a divisor, and the difference of the product for a dividend; but if unlike, take their sum for a divisor, and the sum of the products for a dividend ; the quotient will be the answer.
Note 1.-If O be used for the first and 1 for the second supposed number, the first of the errors, divided by their difference, will, in many instances,) be the answer.
Note 2.-Multiply the difference of the supposed numbers by the least error, and divide the product by the difference of the errors, if like, or by the sum, if unlike; the quotient is