OPERATION BY PROPORTION.

$100 + $75 $175; $7: $3 :: $175 $75.

=

$100-$75 = $25; the loss is therefore 25 per cent. By analysis. If $7 give $ 175, $ 3 will give of $175= $75. Therefore, for $100 there are received only $75; there fore there is $100 - $75 = $ 25, or 25 per cent. loss, answer.

If, therefore, goods be sold at a certain price, and there be gained or lost a certain per cent., and we wish to ascertain what would be gained or lost per cent. at some other price, we deduce the following

RULE. 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 assumed price.

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

EXAMPLES TO EXERCISE THE PRECEDING RULES.

9. Sold broadcloth at $ 6.124 per yard, and by so doing lost 121 per cent. What was the original cost per yard ?

Ans. $7.00. By analysis. If 124 per cent. be lost, 87 per cent. will remain. It is now required to find of what number § 6.12+ is 874. done by multiplying $6.12 by 100, and dividing by 871, and it produces the answer, $7.00.

100°

This is

10. Bought cloth at $7.00 per yard, and sold it at $6.12. What per cent. did I lose? Ans. 12 per cent. 11. Bought cloth at $7.00 per yard, and sold it for 12 per

What did I receive?

cent. less than what it cost.

Ans. $6.12.

12. Bought cloth at $3.60 per yard.

For

I sell it to gain 123 per cent.?
13. Sold cloth at $ 4.05 per yard, and by

121 per cent. What did it cost?

14. Bargained for cheese at $8.50 per cwt. must I sell it to gain 10 per cent. ?

Ans. $3.60.
For how much

Ans. $9.35 per cwt.

15. Sold cheese at $9.35 per cwt. and gained 10 What did I give for it?

cent. per Ans. $8.50 per cwt. 16. Sold cloth at $1.25 per yard, and by so doing lost 15 per cent. For what should I have sold it to gain 12 per cent.? Ans. $1.64,7 per yard. 17. Sold cloth at $1.25 per yard, and lost 15 per cent. What

રા

per

per cent. should I have gained had I sold it for $1.64,7 yard? Ans. 12 per cent. 18. Sold cloth for $1.64,7 per yard, and gained 12 per For what should I have sold it to lose 15 per cent. ? Ans. $1.25 per yard.

cent.

cent.

19. Sold cloth for $1.64,7 per yard, and gained 12 per What should I have lost had I sold it for $1.25 per yard? Ans..15 per cent. 20. A buys corn for $0.90 per bushel, and sells it for $1.20. B buys for $1.12, and sells for $1.50. Who gains the most per cent. ? Ans. both gain alike. 21. If I buy cotton cloth at 13 cents per yard, on 8 months' credit, and sell it again at 12 cents cash, do I gain or lose, and how much per cent. ? Ans. lose 4 per cent. 22. If 24 yards of cloth are sold at $2.50 per yard, and there is 7 per cent. loss in the sale, what is the prime cost of the whole ? Ans. $64.86,434. 23. Bought 24 yards of cloth for $64.86,433. For what must I sell it per yard to lost 7 per cent. ? Ans. $2.50. 24. Bought a certain quantity of cloth for $64.86,4%, and by selling it at $2.50 per yard, I lost 7 per cent. How many yards were bought? Ans. 24 yards.

25. Bought 24 yards of cloth for $ 64.86,437, and sold it at $2.50 per yard; what per cent. is lost? Ans. 7 per cent. 26. If 271cwt. of sugar be sold at $12.50 per cwt., and there is gained 17 per cent., what was the first cost?

Ans. $10.68,389. 27. Sold a horse for $75, and by so doing I lost 25 per cent.; whereas, I ought to have gained 30 per cent. much was he sold under his real value?

How

Ans. $55.00.

28. Bought a horse which was worth 30 per cent. more than I gave for him; but having been injured, I sold him for 25 per cent. less than what he cost, and thereby lost $55 on his original value. What was received for the horse? Ans. $75.00. 29. Bought molasses at 42 cents per gallon, but not proving so good as I expected, I am willing to lose 5 per cent. what must I sell it per gallon?

For

Ans. $0.39,9. 30. Bought a hogshead of molasses for $112, but 15 gallons having leaked out, I am willing to lose 5 per cent. on the cost. For how much per gallon must I sell it? Ans. $2.21,63.

31. Bought a hogshead of molasses for $112, but a number of gallons having leaked out, I sell the remainder for $2.21,63

per gallon, and by so doing I lose 5 per cent. on the cost. How many gallons leaked out? Ans. 15 gallons. 32. Bought a hogshead of molasses for a certain sum; but 15 gallons having leaked out, I sell the remainder for $2.21,63 per gallon, and thereby lose 5 per cent. on the cost.

What was the cost? Ans. $112.00. 33. Bought a hogshead of molasses for $112.00; but 15 gallons having leaked out, I sell the remainder at $2.21,63 per gallon. What per cent. is my loss? Ans. 5 per cent. 34. If I sell cloth at $ 5.60 per yard, and thereby lose 7 per cent., should I gain or lose, and how much per cent., by selling it at $6.25 per yard? Ans. 38 per cent. gain.

35. Sold a watch which cost me $30 for $ 35, on a credit of 8 months. What did I gain by the bargain? Ans. $3.65,314. 36. When tea is sold at $1.25 per lb. there is lost 25 per cent.; what would be the gain or loss per cent. if it should be sold at $1.40 per lb. ? Ans. 16 per cent. loss.

37. A exchanges with B 50lbs. of indigo at $1.00 per lb. cash, and in barter $1.35; but he is willing to lose 12 per cent. to have one third ready money. What is the cash price of 1 yard of cloth delivered by B at $5.00 per yard to equal A's bartering price reduced 12 per cent., and how many yards were delivered? Ans. $4.20 cash price of 1 yard; 73 yards delivered by B.

SECTION LIX.

DUODECIMALS.

DUODECIMALS are so called because they decrease by twelves from the place of feet towards the right.

Inches are called primes, and are marked thus '; the next division after is called seconds, marked thus"; the next is thirds, marked thus ""; and so on.

Duodecimals are commonly used by workmen and artificers in finding the contents of their work.

EXAMPLES.

1. Multiply 6 feet 8 inches by 4 feet 5 inches.

OPERATION.

68' 4 5 26 8'

29 4"

29 5' 4"

RULE.

-

As feet are the integers or units, it is evident that feet multiplied by feet will produce feet; and as inches are twelfths of a foot, the product of inches by feet will be twelfths of a foot. For the same reason, inches multiplied by inches will produce twelfths of an inch, or one hundred and fortyfourths of a foot.

Under the multiplicand write the same names or denominations of the multiplier; that is, feet under feet, inches under inches, &c. 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. 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. 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 product required.

The denomination of the particular products will be as follows.
Feet multiplied by feet give feet.

Feet multiplied by primes give primes.
Feet multiplied by seconds give seconds.
Primes multiplied by primes give seconds.
Primes multiplied by seconds give thirds.
Primes multiplied by thirds give fourths.
Seconds multiplied by seconds give fourths.
Seconds multiplied by thirds give fifths.
Seconds multiplied by fourths give sixths.
Thirds multiplied by thirds give sixths.
Thirds multiplied by fourths give sevenths.
Thirds multiplied by fifths give eighths, &c.

2. Multiply 4ft. 7' by 6ft. 4'.
3. Multiply 14ft. 9' by 4ft. 6'.
4. Multiply 4ft. 7' 8" by 9ft. 6'.
5. Multiply 10ft. 4' 5" by 7ft. 8' 6".

Ans. 29ft. 0' 4".

Ans. 66ft. 4' 6". Ans. 44ft. 0' 10".

Ans. 79ft. 11' 0'' 6''' 6''''.

6. Multiply 39ft. 10' 7" by 18ft. 8' 4".

Ans. 745ft. 6' 10" 2" 4"".

7. How many square feet in a floor 48 feet 6 inches long, 24 feet 3 inches broad? Ans. 1176ft. 1' 6". 8. What are the contents of a marble slab, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches?

Ans. 10ft. 2' 10".

9. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches, how many yards of painting are in it, deducting a surplus of 4 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 2 inches ?

Ans. 73 yards. 10. Required the solid contents of a wall 53 feet 6 inches long, 10 feet 3 inches high, and 2 feet thick.

Ans. 1096ft. 9'.

11. There is a house with four tiers of windows, and 4 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. 283ft. 7in.

12. How many square feet of paper would it require to line 15 boxes, each of which is 7 feet 9 inches long, 3 feet 4 inches wide, and 2 feet 10 inches high; and how many cubic yards would the boxes contain? Ans. 1717ft. lin. 40315 cubic yds.

13. 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 there are to be 90 yards deducted for doors, windows, &c. How many yards must he be paid for?

Ans. 251yd. 1ft. 6in.

14. How many feet in a board which is 17 feet 6 inches long, and 1 foot 7 inches wide? Ans. 27ft. 8' 6". 15. How many feet in a board 27 feet 9 inches long, 29 inches wide? Ans. 67ft. 0' 9". 16. How many feet of boards will it take to cover the side of a building 47 feet long, 17 feet 9 inches high?

Ans. 834ft. 3'.

NOTE. A board to be merchantable should be 1 inch thick; therefore to reduce a plank to board measure, the superficial contents of the plank should be multiplied by its thickness.

17. How many feet, board measure, are in a plank 18 feet 9 inches long, 1 foot 6 inches wide, and 3 inches thick ?

Ans. 84ft. 4' 6".

18. How many feet, board measure, are in a plank 20 feet long, 1 foot 6 inches wide, and 2 inches thick? Ans. 75ft. 19. How many feet in a plank 40 feet 6 inches long, 30 inches wide, and 23 inches thick ? Ans. 278ft. 5' 3".

NOTE. A pile of wood that is 8 feet long, 4 feet high, and 4 feet wide, contains 128 cubic feet, or a cord, and every cord contains 8 cord-feet; and as 8 is of 128, every cord-foot contains 16 cubic feet; therefore, dividing the cubic feet in a pile of wood by 16, the quotient is the cord-feet; d if cord-feet be divided by 8, the quotient is cords.