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EXAMPLES. 1. A merchant has several kinds of tea; some at 8 shillings, some at 9 shillings, some at 11 shillings, and some at 12 shillings per pound; what proportions of each must he mix, that he may sell the compound at 10s. per pound?
The pupil will perceive, that there may be as many different ways of mixing the simples, and consequently as many different answers, as there are different ways of linking the several prices.
8 -2+1=3 -1
1 =1 10s.
11 -1+2=3 12s. -2
-2 =2 Here the prices of the simples are set one directly under another, in order, from least to greatest, and the mean rate, (10s.) written at the left hand. In the first way of linking, we take in the proportion of 2 pounds of the teas at 8 and 12s. to 1 pound at 9 and 1ls. In the second way, we find for the answer, 3 pounds at 8 and Ils. to 1 pound at 9 and 12s.
2. What proportions of sugar, at 8 cents, 10 cents, and 14 cents per pound, will compose a mixture worth 12 cents per pound?
Ans. In the proportion of 2 lbs. at 8 and 10 cts, to 6 lbs. at 14 cents.
Note. As these quantities only express the proportions of each kind, it is plain, that a compound of the same mean price will be formed by taking 3 times, 4 times, one half, or any proportion, of each quantity. Hence,
When the quantity of one simple is given, after finding the proportional quantities, by the above rule, we may say, As the PROPORTIONAL quantity : is to the GIVEN quantity : : so is each of the other PROPORTIONAL quantities : to the REQUIRED quantities of each.
3. If a man wishes to mix 1 gallon of brandy worth 16s. How can the required quantities of each of the simples be obtained ?
with rum at 9s. per gallon, so that the mixture may be worth 11s. per gallon, how much rum must he use ?
Taking the differences as above, we find the proportions to be 2 of brandy to 5 of rum ; consequently, 1 gallon of brandy will require 2} gallons of rum.
Ans. 24 gallons. 4. A grocer has sugars worth 7 cents, 9 cents, and 12 cents per pound, which he would mix so as to form a compound, worth 10 cents per pound; what must be the pro. portions of each kind ? Ans. 2 lbs. of the first and second, to 4 lbs of the 3d kind.
5. If he use 1 lb. of the first kind, how much must he take of the others ? if 4 lbs., what?
- if 6 lbs., what? if 10 lbs., what ? if 20 lbs., what ?
Ans. to the last, 20 lbs of the 2d, and 40 of the 3d. 6. A merchant has spices at 16d. 20d. and 32d. per pound; he would mix 5 pounds of the first sort with the others, so as to form a compound worth 24d. per pound; how much of each sort must he use ?
Ans. 5 lbs of the second, and 74 lbs. of the third. 7. How many gallons of water, of no value, must be mixed with 60 gallons of rum, worth 80 cents per gallon, to reduce its value to 70 cents per gallon ? Ans. 84 galls.
8. A man would mix 4 bushels of wheat, at $1,50 per bushel, rye at $1,16, corn at 75 c. and barley at 50 c. so as to sell the mixture al 84 c. per bushel ; how much of each may he use ?
When the quantity of the compound is given, we may say, As the sum of the PROPORTIONAL quantities, found by the above rule, is to the quantity REQUIRED, so is each PROPORTIONAL quantity, found by the rule, to the REQUIRED quantity of EACH.
9. A man would mix 100 pounds of sugar, some at 8 cents, some at 10 cents, and some at 14 cents per pound, so that the compound may be worth 12 cents per pound; how much of each kind must he use?
We find the proportions to be, 2, 2, and 6. Then, 2+2 +6=10, and
2 : 20 lbs. at 8 cts. 10: 100 :: 2: 20 lbs. at 10 cts. Ans.
6: 60 lbs. at 14 cts. 10. How many gallons of water, of no value, must be
mixed with brandy at $1,20 per gallon, so as to fill a ves. sel of 75 gallons, which may be worth 92 cents per gal. ?
Ans. 17. gallons of water to 571 gallons of brandy. 11. A grocer has currants at 4d., 6., 9d., and 11d. per Ib.; and he would make a mixture of 240 lbs., so that the mixture may be sold at 8d. per lb.; how many pounds of each sort may he take ?
Ans. 72, 24, 48, and 96 lbs., or 48, 48, 72, 72, &c. Note. This question may have five different answers.
Duodecimal is derived from the Latin word duodecim, signifying twelve.
They are fractions of a foot, which is supposed to be divided into twelve equal parts called primes, marked thus, (). Each prime is supposed to be subdivided into 12 equal parts called seconds, marked thus, ("). Each second is also supposed to be divided into twelve equal parts called thirds, marked thus ("'), and so on to any extent.
It thus appears that
Whenever therefore any number of seconds, (as 5'') are mentioned, it is to be understood as so many idā of a foot, and so of the thirds, fourths, &c.
Duodecimals are added and subtracted like other com. pound numbers, 12 of a less order making 1 of the next higher, thus,
12 fourths make 1 third 1".
The addition and subtraction of Duodecimals is the
are called indices.
What are duodecimals ?
MULTIPLICATION OF DUODECIMALS. Duodecimals are chiefly used in measuring surfaces and solids.
How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide ?
NOTE. The square contents of any thing are found by multiplying the length into the breadth.
The following example is explained above.
It is generally more convenient to multiply by the higher orders of the multiplier first.
Thus we begin and multiply the multiplicand first by the 1 foot, and set the answers down as above.
We then multiply by the 3' or of a foot. 16 is changed to a fraction, thus y, and this multiplied by is, or 48', which is 4 feet, (for there are 12 in every foot) and is set under that order.
We now multiply 7' (or 1) by 3' (or ) and the answer is 22 or 21".
This is l' to set under the order of twelfths, and 9" (144 to be set under the order of seconds.
The two products are then added together, and the answer is obtained, which is 20 feet 8 primes 9 seconds.
Another example will be given in which the cubic con. tents of a block are found by multiplying the length, breadth and thickness together.
How many solid feet in a block 15 ft. 8' long, 1 ft. 5 wide, and 1 ft. 4' thick ?
When are duodecimals used ?
4" Let this example be studied and understood before the rule is learned. If any difficulty is found, let both multiplier and multiplicand be expressed as Vulgar Fractions, and then multiply.
In duodecimals it is always the case that the product of two orders, will belong to that order which is made by adding the indices of the factors.
Write the figures as in the addition of compound num. bers. Multiply by the higher orders of the multiplier first, remembering that the product of two orders belongs to the order denoted by the sum of their indices.
If any product is large enough to contain units of a higher order, change them to a higher order, and place them where they belong.
How many square feet in a pile of boards 12 ft. 8' long, and 13' wide ?
What is the product of 371 ft. 2' 6" multiplied by 181 ft. 1' 9!!?
Ans. 67242 ft. 10° 1" 4" 6".
What is the rule ?