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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 merchant has spices, some at 9d. per lb. some at 1s. some at 2s. and some at 2s. 6d. per Ib, how much of each tort must he mix, that he may sell the mixture at 1s. 8d.

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2. A grocer would mix the following qualities of sugar; iz. at 10 cents, 13 cents, and 16 cents per lb.; what quanity of each sort must be taken to make a mixture worth o cents per pound? -Ins. 5 lb. at 10cts. 21.b. at 13 cts, and 2 lb. at 16 cts, per lb. 3. A grocer has two sorts of tea, viz. at 9s. and at 15s. or lb, how must he mix them so as to afford the composiion for 12s. per lb. ? Ans. He must mir an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with wome of 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what quantity of each must he take? 4ns. 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. 's, and 9s. per gallon, with water at 0 per gallon, togeher, so that the mixture may be worth 6s, per gallon; how nuch 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 7s., 1 do. at 9s. and gal. water. 6. A grocer hath several sorts of sugar, viz. one sort at 12 its per lb. another at 11 cts, a third at 9 cts. and a fourth at 8 cts, per lb.; I demand how much of each sort he must mix together, that the whole quantity may be afforded at 10 cents per pound?

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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.

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Take the differences between each price, and the mear rate, and place them alternately as in Case 1. Then, as the difference standing against that simple whose quantity in given, is to that quantity: so is each of the other differ ences, severally, to the several quantities required.


I. A farmer would mix 10 bushels of wheat, at 70 cents her bushel, with rye at 4Scts, corn at 36 cts, and barley a * cts. per bushel, so that a bushel of the composition may be sold for 38cts.; what quantity of each must be taken?

70--> S stands against the given quan

48 2 [tity Mean rate, 38 36 10 30- 32

2 : 2; bushels of rye. As S : 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 admitofaninfinite variety of answers: for after the ntities are sound from different methods of linking; any other numbers in the same proportion betweenthemselves, as the numbers which composethanswer, win hkewise satisfy the conditions of the question.


2. How much water must be mixed with 100 gallons of rum, worth 7s.6d. per gallon, to reduce it to 6s. 8d. per gallon? Ans. 20 gallons. 3. A farmer would mix 20 bushels of rye, at 65 cents yer bushel, with barley at 51 cts, and oats at 30 cents per ushel; how much barley and oats must be mixed with the 20 bushels of rye, that the provender may be worth 41 cts. per bushel? Ans. 20 bushels of barley, and 61 or bushels of oats. 4. With 95 gallons of rum at 8s. per gallon, I mixed other ram at 6s. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per gallon; I demand how much rum and how much water I took 2

Ans. 95 gals, rum at 6s. 8d. and 30 gals. water.

when the whole composition is limited to a given quantity. RULE.

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.

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10- 3 3 : 30-10 sum, so 120

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2. How much water at 0 per gallon, must be mixed with wine at 90 cents per gallon, so as to fill a vessel of 100 gal lons, which may be afforded at 60 cents per gallon 7 Ans. 33 gals, water, and 66% gals. wine. 3. A grocer having sugars at Scts. 16 cts, and 24 cts per pound, would make a composition of 240 lb. worth 20 cts. per lb. without gain or loss; what quantity of each must be taken? Ans. 40 lb. at Scts, 40 lb. at 16 cts, and 160 lb. at 24 cts. 4. A goldsmith had two sorts of silver bullion, one of 10 oz. and the other of 5 oz. fine, and has a mind to mix a pound of it so that it shall be 8 oz. fine; how much of each sort must he take 7 Ans. 44 of 5 oz. fine, and 71 of 10 oz. fine. 5. Brandy at 3s.6d. and 5s. 9d per gallon, is to be mixed, so that a hlid. of 63 gallons may be sold for 121. 12s. ; how many gallons must be taken of each?

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ANY rank of numbers more than two, increasing by common excess, or decreasing by common difference, is said to be in Arithmetical Progression.

So | 2,4,5,8, &c. is an ascending arithmetical series:

8,5,4,2, &c. is a descending arithmetieal series:

The numbers which form the series, are called the terms of the progression; the first and last terms of which are called the extremes.”

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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 five parts, viz. the first term, last term. number of terms, common difference, and sum of the series. By having any three of these parts given, the other two o: sound which admits of a variety of Problems; but most of them are understood by analgebraic process, and are here omo-d

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