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4. Then, if only one difference stands against any rate, it will be the quantity belonging to that rate, but if there be several, their sum will be the quantity.
EXAMPLES. 1. A mercle at has spices, some at 90. per lb. some at 1s. some at 25. and some at 2s. 6d. per lb. how much of e ch sort must he mix, that he may sell the mixture at is 8d. per pound? d. d. 16.
il lb. 2 go 10 at 99
r'97 d. 12 4 12 Gives the d. | 12 10 2024 8 24 | Answer, or 20 24 11 30 11 30)
8 2. A grocer would mix the following quantities of sugar; viz. at 10 cents, 13 cents, and 16 cts. per Ib. ; what
quantity of each sort must be taken to make a mixture i worth 12 cents per pound ? ns. 3lb. at 10cts. Alb. at 13cts. and alb. at 16 cts. per lb. 3. A grocer
has two sorts of tea, viz. at 9s. and at 15s. per Ib. how must he mix them so as to afford the coinposition for 12s. per lb. ?
Ans. He must mix an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats tine, so that the compound inay be 22 carats fine; what quantity of each must ire take.
Ans. 2 of each of the first three sorts, and 9 of the last.
5. It is required to mix several sorts of rum, viz. at 5s. 75. and 9s. per gallon, with water at O per gallon together, so that the mixture may be worth 6s. per gallon ; how much of each sort must the mixture consist of: Ans. 1 gal. of Rum at 5s. I do. at 7s. 6 do at 9s. and 3
gals. water. Or, 3 gals. rum at 5s. 6 do. at 75. 1
do. at 9s. and 1 gal. water, 2 6. A grocer hath several sorts of sugar, viz.. one sort s at 12 cts. per Ib. another at 11 cts. a third at 9 cts. and a | fourth at 8 cts. per lb. ; I demand how much of each sort
must be inix together, that the whole fianuity may be afirded at 10 Cats per pound?
1b. cts. 2 at 12
'1 at 12
3 at 12 1 at 11 2 at 11
2 at 11 1st. Ans. 2d Ans.
2 at 9
3 at 8 4th Ans. 3lb. of each sort. *
CASE II. ALTERNATION PARTIAL. Or, when one of the ingredients is limited to a certain quantity, thence to find the several quantities of the rest in proportion to the quantity given.
RULE. Take the difference between each price, and the mean rate, and place them alternately as in Case I. Then, as the difference standing against that simple whose quantity is given, is to that quantity : 90 is each of the other dif. ferences, severally, to the several quantities required.
1. A farmer would mix 10 bushels of wheat, at 70 cts. per bushel, with rye at 48 cts. corn at 36 cts. and barley at 30 cts. per bushel, so that a bushel of the composition may be sold for 38 cents; what quantity of each must be taken.
S stands against the given quan Nean rate, 88% 98 2
10 30 32
2 : 2 bashels of rye. As 8:10: 10 : 121 bushels of corn.
32 : 40 bushels of barley. * These four answers arise from as many various ways of linking the rates of the ingredients together.
Questions in this rule admit of an infinite variety of answers: for after the quantities are found from different methods of linking : any other numbers in the same proportion between themselves, asthe numbers which compose the answer, will likewise satisfy the conditions of the guestion
2. How much water must be mixed with 100 gallons of rum, worth 75. 6d. per gallon, to reduce it to 6s. 3d. per gallon
Ans. 20 gallons. 3. A farmer would mix 20 bushels of rye, at 65 cents per bushel, with barley at 51 cts. and oats at 30 cts. per bushel ; how much barley and oats must be mixed with the 20 bushels of rye, that the provender may be worth 41 cents per bushel?
Ans. 20 bushels of barley, and 6111 bushels of oats. 4. With 95 gallons of rum at 8s. per gallon, I mixed other rum at 6g. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per gallon; I demand how
1 much rum and how much water I took ?
Ans. 95 gals. rum at 6s. 8d. and 30 gals. water.
When the whole composition is lintited to a given quantity.
Place the difference between the mean rate, and the several prices alternately, as in CASE I.; then, As the sum of the quantities, or difference thus determined, is to the given quantity, or whole composition : so is the difference of each rate, to the required quantity of each rate.
1. A grocer lad four sorts of tea, at 1s. Ss. 6s. and 10s. per lb. the worst would not sell, and the best were too dear; he therefore mixed 120 lb. and so much of each sort, as to sell it at is. per lb.; how much of each sort did he take?
2. How much water at o per gallon, must be mixed with wine at 90 cents per gallon, so as to fill a vessel of 100 gallons, which may be afforded at 60 cents per gallon ?
Ans. 33} gals. water, and 663 gals. wine. S. A grocer having sugars at 8 cts. 16 čts. and 24 cis. per pound, would make a composition of 240 lb. wort! 20 cts. per lb. without gain or loss; what quantity of eacı must be taken?
Ans. 40 lb. at 8 cts, 40 at 16 cts, and 160 at 24 cts. 4. A goldsmith had two sorts of silver bullion, one of 10 oz. and the other vi 5 oz. fine, and has a mind to mi. a pound of it so that it shall be 8 oz fine; how much oi each sort inust he take?
Ans. 44 of 5 oz. fine, and 7 of 10 oz. fine. 5. Brandy at 3s. Ed. and 5s. 9d. per gailon, is to be mixed, so that a hlid. of 6s gallons may be sold for 1%. 12s.; how many gallons must be taken of each ?
Jns. 14 gals. at 5s. 9d. and 49 gals. at 5s. 6d.
ARITHMETICAL PROGRESSION. ANY rank of numbers more than two, increasing by common excess, or decreasing by common difference, is said to be in Arithmetical Progression.
52, 4, 6, 8, &c. is an ascending arithmetical series : So
28.6, 4, 2, &c. is a descending arithmetical series : The numbers which form the series, are called the terms of the pragression; the first and last terms of which are called the extremes. *
PROBLEM I. The first term, the last term, and the number of terms being given, to find the sum of all the terms.
* A series in progression includes jive zutris, viz. the first terom, last term, number of terns, common difference, and sum of the series.
By having any three of these parts given, the other tito may be found, which admits of a variety of Problems ; oui most of them are best widerstood by catehraic procesa, and are here oritieke
RULE. Multiply the sum of the extremes by the number of terms, and half the product will be the answer.
1. The first term oi an arithmetical series is 3, the last term 23, and the number of terms 11 ; required the sum of the series
23+3=26 sum of the extremes.
Then 26x11--9=143 the Auswer. 2. How many strokes does the hammer of a cock strike, in twelve hours:
fis, 73 9. A merchant sold 100 vids of clothi. viz. the first vard for 1 ct. the second 107ects. the third for Scts. &c. I demand what the cloth came io at that rate?
Ans. $501. 4. A man boucht 19 yards of linen in arithmetical progression, for the first yard he gave is. and for the last yd. 1l. 17s. what did the wirele come to ? Fins. £18 1s.
5. A draper sold 100 yıls. of broadcloth, at 5 cts. for the first yard, 10 cts. for the second, 15 for the third, &c. increasing 5 cents for every yard : What did the whole amount to, and what did it average per yard ?
Ans. Imout, $221, nu ine arerage price is $2,52cts. 5 mills per yard.
6. Suppose 144 oranges were laid 2 yards distant from each other, in a right line and a basketplaced two yarıs from the first orange, what lectie of ground will that boy travel over, whes gathers them up sing's, returning with them one by one to the basket:
Ars..25 miles, 5 furlong's, 190 yds.
PROBLEMI II. 'The first term, the last term, and the number of terms given, to find the common difference.
RULE. Divide the difference of the extremes by the number of terms less 1, and the quotient will be the common disference.