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6. Sold 17 barrels of flour at $8 per barrel, for which I received a note payable in 3 months. This note I had discounted at the Granite Bank, but on examining my account, I find I have lost 10 per cent. on the flour; what was the cost of it?

Ans. $ 148.76+. Art. 258. The selling price of goods and the rate per cent. being given, to find what the gain or loss per cent. would be, if sold at another price.

Ex. 1. If I sell flour at $5 per barrel, and gain 25 per cent., what should I gain, if I were to sell it for $ 7 per barrel ?

OPERATION.

The solution of this question involves two principles : First, to find the cost of the flour per barrel, (Art. 257.) Thus, $5

$.04; $.04 x 100 $ 4.00, the cost per barrel. Second, to find the gain per cent on the cost when sold at $ 7 per barrel, (Art. 255.) Thus, $ 7-$4=$3; 4= 3.00 ; 4=.75, or 75 per cent.

• 125 =

OPERATION BY PROPORTION

$100 + $ 25=$ 125; $5: $ 7:: $125: $175;

$ 175 —$100=$ 75, that is, 75 per cent. Rule I. —Find the cost of the goods, (Art. 257,) and then the gain or loss per cent. on this cost at the last selling price. (Art. 255.) Or,

RULE II. —As the first price is to the proposed price, so is $ 100 with the profit per cent. added, or the loss per cent. subtracted, to the gain or loss per cent. at the proposed price.

Note. - If the answer exceeds $ 100, the excess is the gain per cent.; but, if it is less than $ 100, the deficiency is the loss per cent.

EXAMPLES FOR PRACTICE. 2. Sold a quantity of oats at 28 cents per bushel, and gained 12 per cent. ; what per cent. should I gain or lose, if I were to sell them at 24 cents per bushel ? Ans. Lose 4 per cent.

3. S. Rice sold a horse for $ 37.50 and lost 25 per cent. ; what would have been his gain per cent. if he had sold him for $ 75 ?

Ans. 50 per cent. 4. S. Phelps sold a quantity of wheat for $ 1728, and took

QUESTIONS. - Art. 258. What is the first rule for finding what gain or loss is made by selling goods at another price when the selling price and rate per cent. are given ? What is the second rule? If the answer exceeds $ 100, wha is the excess? If it is less than $ 100, what is the deficiency?

a note payable in 9 months without interest, and made 10 per cent. on his purchase; what would have been his gain per cent. if he had sold it to James Wilson for $ 2000 cash ?

Ans. 33+ per cent.

MISCELLANEOUS EXERCISES IN PROFIT AND LOSS.

1. A horse that cost $ 84, having been injured, was sold for $ 75.60; what was the loss per cent. ? Ans. 10 per cent.

2. Sold a horse for $ 75.60, and lost 10 per cent. on the cost, but I ought to have sold him for $97.44 to have made a reasonable profit; what per cent. did I lose on the price for which I ought to have sold the horse ?

Ans. 16 per cent. 3. M. Star sold a horse for $ 97.44, and gained 16 per cent. ; what would have been his loss per cent. if he had sold the horse for $ 75.60, and what his actual loss?

Ans. Loss 10 per cent. $ 8.40 loss. 4. If I buy cloth at $5 per yard, on 9 months credit, for what must I sell it per yard for cash to gain 12 per cent. ?

Ans. $5.35+: 5. A. Pemberton bought a hogshead of molasses, containing 120 gallons, for $ 40; but 20 gallons having leaked out, for what must he sell the remainder per gallon to gain 10 per cent. on his purchase ?

Ans. $ 0.44. 6. H. Jones sells flour, which cost him $5 per barrel, for $ 7.50 per barrel; and J. B. Crosby sells coffee for 14 cents per pound, which cost him 10 cents per pound; which makes the greater per cent ? Ans. A. Jones makes 10 per cent. most.

7. J. Gordon bought 160 gallons of molasses, but having sold 40 gallons, at 30 cents per gallon, to a man who proved a bankrupt, and could pay only 30 cents on the dollar, he disposed of the remainder at 35 cents per gallon and gained 10 per cent. on his purchase; what was the cost of the molasses ?

Ans. $ 41.45+. 8. D. Bugbee bought a horse for $ 75.60, which was 10 per cent. less than his real value, and sold him for 16 per cent. more than his real value ; what did he receive for the horse, and what per cent. did he make on his purchase ?

Ans. Received $ 97.44, and made 285 per cent. 9. A merchant bought 70 yards of broadcloth, that was 13 yards wide, for $ 4.50 per yard, but the cloth having been wet, it shrunk 5 per cent. in length and 5 in width; for what must the cloth be sold per square yard to gain 12 per cent. ?

Ans. $ 3.19+ 22 *

$ XXXVII. DUODECIMALS. Art. 259. Duodecimals are a kind of mixed numbers in which the unit, or foot, is divided into 12 equal parts, and each of these parts into 12 other equal parts, and so on indefinitely ; thus, ih, , &c.

Duodecimals decrease from left to right in a twelvefold ratio, and the different orders, or denominations, are distinguished from each other by accents, called indices, placed at the right of the numerators. Hence the denominators are not expressed.

Thus,

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1'".
ini.

1 inch or prime, equal to it of a foot, is written 1 in. or 1' 1 second

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1". 1 third

1728 1 fourth

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TABLE. 12 fourths

make 1". 12 thirds

1". 12 seconds

1'. 12 inches or primes

ift.

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ADDITION AND SUBTRACTION OF DUODECIMALS.

Art. 260. Duodecimals are added and subtracted in the same manner as compound numbers.

EXAMPLES FOR PRACTICE. 1. Add together 12ft. 6'9", 14ft. 7' 8", 165ft. 11' 10".

Ans. 193ft. 2' 3". 2. Add together 182ft. 11' 2'' 4"", 127ft. 7'8" 11", 291ft. 5' 11" 10"".

Ans. 602ft. 0' 11" 11". 3. From 204ft. 7' 9" take 114ft. 10' 6".

Ans. 89ft. 9' 3". 4. From 397ft. 9'6" 11"" per take 201ft. 11'7" g'" 10"".

Ans. 195ft. 9' 11" 2' 9".

QUESTIONS. - Art. 259. What are duodecimals ? Into how many parts is the unit or foot divided ? In what ratio do duodecimals decrease from left to right? How are the different denominations distinguished from each other ? Are the denominators of duodecimals expressed ? . Repeat the table ? - Art. 260. How are duodecimals added and subtracted ?

MULTIPLICATION OF DUODECIMALS. Art. 261. To find the denomination of the product of any two numbers in duodecimals, when multiplied together. Ex. 1. What is the product of 9ft. multiplied by 3ft. ?

Ans. 27ft.

OPERATION.

9ft. x 3ft. 27ft. 2. What is the product of 7ft. multiplied by 6'? Ans. 3ft. 6'.

OPERATION.

6'=of a foot; then 7ft. X ft. 1 3 2 = 42'; 42' ;

12=3ft. 6'. 3. What is the product of 5' multiplied by 4'? Ans. 1'8''.

OPERATION.

5' =*;, and 4'=15; then = tie = : 20''; 20"

• =1'8". 4. What is the product of 9' multiplied by 11"' ?

Ans. 8'' 3"".

OPERATION

9' : is, and 11" = Tir; then 9 X 1728 2094956

99" ; 99' • 12=8" 3. It will be observed, in the examples above, that feet multiplied by feet produce feet; feet multiplied by primes produce primes ; primes multiplied by primes produce seconds, &c.; and that the several products are of the same denomination as denoted by the sum of the indices of the numbers multiplied together. Hence,

When two numbers are multiplied together, the sum of their indices annexed to their product denotes its denomination.

Art. 262. To multiply duodecimals together.
Ex. 1. Multiply 8ft. 6in. by 3ft. 7in. Ans. 30ft. 5' 6".

We first multiply each of the terms in the 8ft. 6' multiplicand by the 3ft. in the multiplier ; thus, 3ft. 7 3ft. into 6' = 18' = ift. and 6'. We write the

6' under the primes, and add the 1ft. to the pro25ft. 6

duct of the 3ft. into 8ft., making 25ft. We 4ft. 11' 6" then multiply by the 7'; thus, 7' into 6' : 42" 3 Oft.

=3' and 6". Placing the 6" at the right of the 5' 6''

primes, we add the 3 to the product of 7' into

OPERATION.

QUESTIONS. Art. 261. How is the denomination of the product denoted when duodecimals are multiplied together? - If feet are multiplied by feet, what is the product? What, if feet are multiplied by primes? If primes are multiplied by primes? Is it absolutely necessary to commence the multiplication with feet?

8ft. = 59'=4ft. and 11', which we write under the feet and inches, and the two products being added together we obtain 30ft. 5' 6" for the answer.

RULE. — 1. Under the multiplicand write the same names or denominations of the multiplier ; that is, feet under feet, inches under inches, gc. Multiply each term in the multiplicand, beginning at the lowest, by the feet of the multiplier, and write each result under its respective term, observing to carry a unit for every 12 from each denomination to its next superior.

2. In the same manner multiply the multiplicand by the inches of the multiplier, and write the result of each term one place further towards the right than the corresponding terms in the preceding product.

3. Proceed in the same manner with the seconds and all the rest of the denominations, and the sum of the several products will be the produci required.

EXAMPLES FOR PRACTICE. 2. Multiply 8ft. 3in. by 7ft. 9in. Ans. 63ft. 11' 3". 3. Multiply 12ft. 9' by 9ft. 11'. Ans. 126ft. 5' 3''. • 4. Multiply 14ft. 9' 11" by 6ft. 11' 8".

Ans. 103ft. 4' 5" 8" 4". 5. Multiply 161ft. 8' 6" by 7ft. 10'. Ans. 1266ft. 8' 7". 6. Multiply 87ft. 1' 11" by 5ft. 7' 5".

Ans. 489ft. 8' 0" 2" 7'". 7. What are the contents of a board 18ft. long and ift. 10in. wide ?

Ans. 33ft. 8. What are the contents of a board 19ft. 8in. long and 2ft. 11in wide ?

Ans. 57ft. 4' 4'. 9. What are the contents of a floor 18ft. 9in. long and 10ft. bin. wide ?

Ans. 196ft. 10' 6''. 10. How many square feet of surface are there in a room 14ft. 9in. long, 12ft. 6in. wide, and 7ft. 9in. high? Ans. 791ft. 1'6”.

11. John Carpenter has agreed to make 12 shoe-boxes of boards that are one inch thick. The boxes are to be, on the outside, 3ft. Sin. long, lft. 9in. wide, and ift. 2in. high. How many square feet of boards will it require to make the boxes, and how many cubic feet will they hold ?

Ans. 280 square feet; 66 cubic feet, 864 inches. 12. My garden is 18 rods long and 10 rods wide; a ditch is dug, round it 2 feet wide and 3 feet deep; but the ditch not being of a sufficient breadth and depth, I have caused it to be dug 1 foot deeper and lft. 6in. wider. How many solid feet will it be necessary to remove ?

Ans. 7540 feet.

QUESTION. — Art. 262. What is the rule for multiplication of Duodecimals ?

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