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If 12 cwt. 3 qr. 12 lb. cost $601,75 what is
Ans. $11,70. Pup. I shall have no difficulty, I think, in performing any questions in these rules, and should like now to attend to Duodecimals, that I may learn what they are, for I know nothing about them, and cannot determine aby thing about them by their name.
Tut. Duodecimals are a species of compound numbers, consisting of different denominations, but resembling simple numbers in this, that they increase in a uniform ratio, from one denomination to another.
The ratio is 12, and it is this which gives the name, Duodecimals, to the rule.
Feet are the highest denomination in this ratio, there being no denomination consisting of 12 feet.
Feet are divided into 12 inches each, called primes ; an inch is divided into 12 seconds ; a second into 12 thirds ; a third into 12 fourths, and so on to as low denominations as you please, there being no limit to the division. These several denominations are distinguished from each other by the following marks ; feet are marked ft. ; inches, or primes, thus ); seconds ("); thirds (""); fourths (""); and so on with all the denominations. These marks are called the indices of the terms. The following is the rule for
Write the multiplicand, and under it write the multiplier, so that feet may stand under feet, inches under inches, &c. and draw a line underneath. Multiply the multiplicand by the several terms of the multiplier, successively, according to the rule for compound multiplication, placing the first term of each of the partial products upder its multiplier, and carrying one for every twelve, from the lower to the next higher denomination, through the whole work ; then the sum of these several products will be the answer.
The left hand number will be the feet, the next inches, and so on with the whole.
It may be proper here for me to tell you, that feet multiplied by feet, give feet for a product ; feet multiplied by inches, give inches ; feet multiplied by seconds, give seconds, &c. Inches multiplied by inches, give seconds ; inches multiplied by seconds, give thirds; the product of any two terms will be in that denomination, which the sum of their indices represent. If you multiply 2 ft. by. 6', the product will be in inches ; if 4 be multiplied by 6', the product will be in thirds, &c.
Multiply 4 ft. 2' 6" by 2 ft. 6' 4".
Here I begin with the right hand figures of both factors, and find their product is 24, and as we carry for 12, I divide 24 by 12, and find there are twice 12, and nothing over; I then multiply the next figure of the multiplicand by 4, and add 2, which were to carry from the other product, to this product.
This does not amount to 12, therefore I set down the whole, and multiply the next figure of the multiplicand by 4, and find there is once 12, and 4 over, which I set down, carrying the li one place to the left. I then multiply the multiplicand in the same manner, by the next figure of the multiplier, setting the first term of the product directly under the figure of the multiplier by which I multiply. I next multiply by the last figure of the multiplier, the product of which, by the last figure of the multiplicand is feet, because the two terms are feet. I then add the partial products together, keeping each term separate, and carrying for 12 is in multiplying ; then the sum of these products is thi answer.
Pup. I understand the manner of operation, but do not see why that feet multiplied by inches, should give inches any more than feet; or wby inches, multiplied by
seconds should give thirds for a product any more than that they should give inches or seconds.
Tut. You have not considered the nature of these terms I expect, or you would not have doubted the truth of the rule In duodecimals all the terms below feet are fractions of a foot; thus 1 inch is jy of a foot; 2 inches are of a foot; and any number of inches may be expressed fractionwise, by writing the given inches for a numerator and 12 for a denominator. Hence if you multiply 6 feet by 6' you will have 6 feet to multiply iż of a foot, and the product of the factors will be is of a foot, or 3 feet. (5. Con. III.) Again, if you are to multiply 6' by 6", you have iz of a foot to be multiplied by le of ij of a foot; the product of 6' by 6' is 36"', which is is of 1 of 1 of a foot, equal to 18 of a foot, equal to ans of a foot, (7. Con. III.) 6 inches are of a foot, consequently, if you multiply any number of feet by 6, you obtain a product of one half the value of the feet multiplied, (5. Con. III.) and the same of all the fractional parts of a foot.
Pup I now understand why the product of these frac. tions of a foot should be in a smaller denomination than either of the factors; it is exactly like the multiplication of vulgar fractions.
Tut. It is like the multiplication of vulgar fractions having a common denominator ; and you ought to have a good understanding of this or you will be very liable to make mistakes.
The manner of operating, as I have directed, is different from the usual mode ; but it is more analogous to the manner of operating in simple and compound multiplication, and is equally convenient, and more easily learned than the ordinary way. The usual method is to begin with the highest denomination of the multiplier and mul. tiply the multiplicand by it; then take the next lower term of the multiplier and proceed as with the first, carrying the excess of 12s to the right; and in this manner multiply by each term of the multiplier, when the terms of the product will stand under the corresponding terms of the multiplier. But the advantage of this is not sufficient te coinpensate for the trouble of learning it. The following example is performed in this manner.
This rule is used to find the dimensions of wainscotting, plastering, painting and paving, and the contents are given in square yards. The following is the manner in which
PAINTERS AND JOINERS find the dimensions of their work.
Take a line and fasten it to any spot on the wall of the room ; then apply the line to the wall, going round the room and keeping the line applied close to the wall, till you arrive at the spot where the line was fastened ; then find how many feet and inches the line measures, and call its length the compass, or round, which, multiplied by the height of the room, and the product, divided by 9, will be the contents of the work in yards. If the compass of a room be 64 ft
. 10', and the height 9 ft. 6', what are the contents in yards ?
Pup. After dividing there are 6 feet over, and I do not see why they could not have been reduced to yards, for 3 feet make a yard.
Tut. A yard here does not mean a yard in length, but a square yard, by which is meant so much of the wall as will measure 3 ft. in length, and 3 ft. in breadth. If you take a square piece of the wall, 3 ft. long and 3 ft. broad, it will contain 9 ft. and this is the reason why 6 ft. cannot be expressed in yards.
Pup. I now see the reason of this, and why dividing by 9 gives an answer in yards ; it is because 9 square feet make a square yard.
Questions. What is compound multiplication ? What is the rule for compound multiplication ? When the multiplier exceeds 12, how do you proceed ?
When the given multiplier cannot be resolved into exact factors, what do you do?
When the multiplier exceeds 144, how do you proceed ?
When the lower denomination can be resolved into even parts of the highest denomination, how do you obtain the price?
What is compound division ?
When the divisor is so large as to make it inconvenient dividing by it, how do you proceed ?
When the divisor cannot be resolved into factors, what is to be done ?
When the price of several hundred weight is given, how do you find the price of one hundred weight? Of one quarter? Of one pound, &c. ?
When the quantity given consists of several denominations, and the price of the whole given, how do you find the price of any particular denomination ?
What are duodecimals?
How are the several denominations distinguished from each other ? For what number do
? What is the product of feet by inches ? Inches by inches ? Inches by seconds ? Seconds by thirds, &c. ?
What is the ordinary method of performing this rule ?
How do painters and joiners find the contents of their work?