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MULTIPLICATION OF DUODECIMALS.
599. The index of the unit of a product of any two duodecimal orders is equal to the sum of the indices of those factors. That is, feet multiplied by a number denoting feet produces feet; feet by a number denoting primes produces primes; primes by a number denoting primes produces seconds, &c.
Note. — In multiplication of duodecimals, or in other multiplication, the multiplier is always regarded as an abstract number, though the notation of feet, primes, &c. is usually retained, in order the better to note the different orders of units. For the same reason, in division of duodecimals, the divisor usually retains the notation of feet, primes, &c.
600. To multiply one duodecimal by another.
Ex. 1. Required the number of square feet in a platform 6 feet 8 inches long, and 4 feet 5 inches wide.
Ans. 29 sq. ft. 5' 4".
riRsT Operation. We first multiply each of the terms fn the multU 6ft. 8' plicand by the 5.in the multiplier; thus, 8'X5 =40" tjz = 3. and 4". Writing the 4" under the multiplier,
we reserve the 3' to add to the next product.
2 9' 4" Then 6ft. X 5. = 30'; and 30' + 3' = 33' = 2ft. 26 8' and 9', which we write in their order beneath the
multiplier. We next multiply by the 4ft., thus:
29sq.ft. 5' 4" 8' X 4 feet = 32' = 2ft. and 8.. We write the 8' under the primes in the other partial product, and reserve the 2ft. to add to the next product; and 6ft. X 4ft. = 24ft.; 24ft. -\- 2ft. = 26ft., which we write under the feet in the other partial product. The two being; added together, we have 29sq. ft. 5' 4"; or (the primes and seconds being changed to a fraction of a foot), 29T«^sq. ft.
■ ECOND OPERATION.
In the second operation the work 6.g is performed as in pure duodecimals . - (Art. 594). The point (•) separates lower duodecimal orders from those be
2 9 4 ginning with feet. As to the number
2 2 8 of feet m tne multiplicand and multi
plier, no change is required in that 2 o,o 4 = 29sq. ft. 5' 4". part of either of the given numbers in
expressing them according to the duodecimal scale. In performing the multiplication, we make the several reductions required according to the radix 12; and have, after pointing off, 25-54 square feet, expressed according to the duodecimal scale. The units, or feet, at the left of the point, are readily changed to the decimal scale by multiplying the left-hand figure, 2, by 12, the number of units in the radix, and adding the right-hand figure, 5, and giving the figures to the right of the point their proper notation, we have then 29sq. ft. 5' 4" for the answer, as before.
Rule. — Write the multiplier under the multiplicand, so that units of the same orders shall stand in the same column.
Beginning at the right, multiply each term in the multiplicand by each term of the multiplier, and write the first term of each partial product under its multiplier, observing to carry a unit for every twelve from each lower order to the next higher.
The sum of the partial products will be the product required.
2. How many square feet in a floor 48 feet 6 inches long, and 24 feet 3 inches broad? Ans. 1176sq. ft. 1' 6".
3. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches, how many square yards of painting are in it, deducting a fireplace of 4 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 2 inches?
Ans. 73^- square yards.
4. Required the solid contents of a wall 53 feet 6 inches long, 10 feet 3 inches high, and 2 feet thick.
5. There is a house with four tiers of windows, and four windows in a tier; the height of the first is 6 feet 8 inches; the second, 5 feet 9 inches; the third, 4 feet 6 inches; the fourth, 3 feet 10 inches; and the breadth is 3 feet 5 inches; how many square feet do they contain in the whole?
Ans. 283sq. ft. 7'.
6. How many cords in a pile of wood 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high? Ans. 10£^£ cords.
7. Required the number of cords of wood in a pile 100 feet long, 4 feet wide, and 6 feet 11 inches high. Ans. 211|.
DIVISION OF DUODECIMALS.
601. To divide one duodecimal by another.
Ex. 1. A board in the form of a rectangle, whose area is 27 sq. ft. 8' 6", is lft. 7in. wide; what is its length?
Ans. 17ft. 6in. First Operation. We find how many times 27
lft. 7') 27sq. ft. 8' 6" ( 17ft. 6' square feet contains the divisor, 26 11 and obtain 17 feet for the quo
—- tient, we multiply the entire di
"" visor by the 17ft., and subtract
9 6 the product, 26ft. 11', from the
corresponding portion of the dividend, and obtain 9., to which remainder we bring dewn the 6", and, dividing, obtain 6' for the quotient. Multiplying the entire divisor by 6", we obtain 0ft. 9' C", which, subtracted as before, leaves no remainder. Therefore 17 feet 6 inches is the length required.
Second Operatio*. In the second operation we re
1-7 ) 23-86 ( 15-6 = 17ft. 6' duce the feet of the given multi17 plicand and multiplier to the duo
decimal scale, and thus obtain
88 23-86 and 1-7. We then conduct
7 e the division with reference to the
—radix 12, as is ordinarily done with respect to 10 (Art. 594 ). The re"° suit obtained is 15-6 in the duo
decimal, which, on changing the figures to the left of the point to the decimal scale, and giving the proper notation to the figure on the right of the point, becomes transformed to 17ft. 6. = 17ft. 6in., the answer, as before.
Rule. — Find how many times the highest term of the dividend will contain the divisor. By this quotient multiply the entire divisor, and subtract the product from the corresponding terms of the dividend. To the remainder annex the next denomination of the dividend, and divide as before, and so continue till the division is complete.
2. It required 834 sq. ft. 3' of board to cover the side of a certain building. The height was 17ft. 9in.; what was the length of the side? Ans. 47 feet.
3. How many feet wide is a plank of uniform width, whose length is 18ft. 9in., thickness 3 inches, and solid contents 84ft. 4' 6"?
4. An alley has an area of 792ft. C 9" 2"'. Its width is 12ft. 7' 8". Required its length. Ans. 62ft. 8' 6".
1. A merchant engages a clerk at the rate of $ 20 for the first month, $ 25 for the second, $30 for the third, &c, thus increasing his salary by $ 5 per month. How long must the clerk retain his situation, so as to receive on the whole as much as he would have received had his salary been fixed at $ 52.50 per month? Ans. 14 months.
2. A mason has plastered 3 rooms; the ceiling of each is 20 feet by 16 feet 6 inches, the walls of each are 9 feet 6 inches high, and 90 yards are to be deducted for doors, windows, &c. For how many yards must he be paid? Ans. 251yd. 1ft. 6'.
3. A man of wealth, dying, left his property to his ten sons, and the executor of his will, as follows: to his executor, $ 1024; to his youngest son, as much and half as much more; and increasing the share of each next elder in the ratio of 1 What was the share of the eldest?
4. A butcher, wishing to buy some sheep, asked the owner how much he must give him for 20; on hearing his price, he said it was too much; the owner replied, that he should have 10, provided he would give him a cent for each different choice of 10 in 20, to which he agreed. How much did he pay for the 10 sheep, according to the bargain? Ans. $ 1847.56.
5. If 340 square feet of carpeting are required to cover the floor of a room, how many yards will be required, provided the width of the carpeting is 3 feet 9 inches? Ans. 30yd. 8in.
6. If a clergyman's salary of $ 700 per annum is 6 years in arrears, how much is due him, allowing compound interest at 6 per cent.? Ans. $ 4882.72.
7. Suppose a clock to have an hour-hand, a minute-hand, and a second-hand, all turning on the same center. At 12 o'clock all the hands are together and point at 12.
(1.) How long will it be before the second-hand will be between the other two hands, and at equal distances from each? Ans. 60T7I82°f seconds.
(2.) Also before the minute-hand will be equally distant between the other two hands? Ans. 61||f seconds.
(3.) Also before the hour-hand will be equally distant between the other two hands? Ans. 59f§ seconds.
602. A point is that which has neither length, breadth, nor thickness, but position only.
603. A line is length, without breadth or thickness. A strait/ht line is one which has the same direction
in its whole extent; as the line A B. A.
A curved line is one which continually changes its direction; as the line C D.
604. An angle is the inclination or opening of two lines, which meet in a point.
A right angle is an angle formed by a straight line and a perpendicular to it; as the angle ABC. I c
An acute angle is one less than a right angle; as the angle E B C.
An obtuse angle is one greater than a right angle; * as the angle F B C. • x
60.5. A surface is that which has length and breadth, without thickness.
A plane surface, or simply a plane, is that in which, if any two points whatever be taken, the straight line that joins them will lie wholly in it.
Every surface, which is not a plane, or composed of planes, is a curved surface.
1. The area of a figure is its quantity of surface; and is estimated in the square of some unit of measure, as a square inch, a square foot, &c.
607. A solid, or body, is that which has length, breadth, and thickness.
608. The solidity, or volume of a solid, is estimated in the cube of some unit of measure; as a cubic inch, a cubic loot, &c.
609. Mensuration is the process of determining the areas of surfaces, and the solidity or volume of solids.
MENSURATION OF SURFACES.
610. A plane figure is an enclosed plane surface; if bounded hy straight lines only, it is called a rectilineal figure, or polygon. The perimeter of a figure is its boundary, or contour.