8.11. DISCOUNT. DISCOUNT is an allowance made for the payment of any sum of money before it becomes due, and is the difference between that sum, due sometime hence, and its present worth. The present worth of any sum or debt due some time hence, is such a sum as, if put to interest, would in that time and at the rate per cent for which the discount is to be made, amount to the sum or debt then due. RULE. As the amount of 100 dollars for the given time and rate is to 100 dollars, so is the given sum to its present worth, which subtracted from the given sum leaves the discount. EQUATION OF PAYMENTS is the finding of a time to pay at once, several debts due at different times, so that neither party shall sustain loss. RULE. Multiply each payment by the time at which it is due; then divide the sum of the products by the sum of the payments, and the quotient will be the equated time. B b 13. GUAGING. GUAGING is taking the dimensions of a cask in inches to find its contents in gallons by the following METHOD. 1. Add two thirds of the difference between the head and bung diameters to the head diameter for the mean diameter; but if the staves be but little curving from the head to the bung, add only six tenths of this difference. 2. Square the mean diameter, which multiplied by the length of the cask, and the product divided by 294, for wine, or by 359 for ale, the quotient will be the answer in gallons. EXAMPLES. 1. How many ale or beer gallons will a cask hold, whose bung diameter is 31 inches, head diameter 25 inches, and whose length is 36 inches? OPERATION. 31 Bung diam. 25 head diam. 25 Head diam. 4 Two thirds difference. 6 Difference. 29 Mean diam. 29 261 841 Square of mean diam. 5046 2523 359)30276(84 galls. 1 qts. NOTE 1. In taking the length of the casks, an allowance must be made for the thickness for both heads of one inch, 11 inches, or 2 inches according to the size of the cask. NOTE 2. The head diameter must be taken close to the chimes, and for small casks, add 3 tenths of an inch; for casks of 40 or 50 gallons, 4 tenths, and for larger casks, 5 or 6 tenths, and the sum will be very nearly the head diameter within. § 14. MECHANICAL POWERS. 1. OF THE LEVER. To find what weight may be raised or balanced by any given power, Say as the distance between the body to be raised or balanced, and the fulcrum or prop, is to the distance between the prop and the point where the power is applied; so is the power to the weight which it will balance or raise. EXAMPLE. IF a man weighing 150lb. rest on the end of a lever 12 feet long, what weight will he balance on the other end, supposing the prop 14 feet from the weight? 2. OF THE WHEEL AND AXLE. As the diameter of the axle is to the diameter of the wheel, so is the power applied to the wheel, to the weight suspended by the axle. EXAMPLES. 1. A mechanic wishes to make a windlass in such a manner, as that 1lb. applied to the wheel should be equal to 12 suspended on the axle; now supposing the axle 4 inches diameter, required the diameter of the wheel? As 1 : 4 12 48 Ans. or diameter of the wheel. 2. Suppose the diameter of the axle 6 inches, and that of the wheel 60 inches, what power at the wheel will balance 10lb. at the axle ? Ans. 1lb. 3. OF THE SCREW. The power is to the weight to be raised as the distance between 2 threads of the screw is to the circumference of a circle described by the power applied at the end of the lever. NOTE 1. To find the circumference of the circle described by the end of the lever, multiply the double of the lever by 3,14159, the product will be the circumference. NOTE 2. It is usual to abate of the effect of the machine for friction. EXAMPLES. There is a screw whose threads are an inch asunder; the lever by which it is turned, is 36 inches long, and the weight to be raised a ton, or 2240lb. What power or force must be applied to the end of the lever, sufficient to turn the screw that is to raise the weight? The lever 36x2=723,14159-226,194+the circumference. 1. The diameter of a circle being given to find the circumference, multiply the diameter by 3,14159; the product will be the circumference. 2. To find the area of a circle, the diameter being given, multiply the square of the diameter by ,785393; the product is the area. 3. To measure the solidity of any irregular body, whose dimensions cannot be taken, put the body into some regular vessel and fill it with water, then taking out the body, measure the fall of water in the vessel; if the vessel be square, multiply the side by itself, and the product by the fall of water, which gives the solid contents of the irregular body. MISCELLANEOUS QUESTIONS. 1. THE Northern Lights were first observed in London 1560; how many years since ? 2. What number multiplied by 43 produces 88150? Ans. 2050. 3. If a cannon may be discharged twice with 6lb. of powder, how many times will 7C. 3qrs. 17lbs. discharge the same piece? Ans. 295 times. 4. Reduce 14 guineas and £75 13s. 6d. to Federal Money. Ans. $317,593. 5. What is the interest of $79,47 one year and five months? Ans. $6,754. 6. A owed B $317,19, for which he gave his note on interest, bearing date July 12th, 1797. On the back of the note are these several endorsements, viz. Oct. 17th, 1797, Received in cash $61,10. March 20th, 1798, Received 17Cwt. beef, at $4,33 per cwt. What was there due from A to B of principal and interest, Sept. 18th, 1801 ? Ans. $144.363. 7. What cost 13 yards of flannel at 1s. 8d. per yard? Ans. £1 3s. Ozd.. 8. What must I give for 3Cwt. 2qrs. 13lb. of cheese at 7 cts. per lb. ? Ans. $28,35. 9. What will 35 yards of broadcloth cost at 23s. 6d. per yard? Ans. £41 2s. 6d. 10. What will be the cost of a loin of veal, weighing 16316. at 24d. per lb. ? Ans. 3s. 5d. Ans. £3 7s. 3d. 11. What will 871. of tallow cost at 94d. per lb. ? Ans. 12s. 3d. 13. What will 56 bushels of oats cost at 23. 34d. per bushel ? Ans. £6 8s. 4d. 14. At £37s. 6d. per cwt. for sugar, what is that per lb. ? 15. How much in length of a board that is 10 inches wide will it require to make a square foot ? Ans. 14 feet. 16. How many square feet in a board 1 foot 3 inches wide, and 14 feet 9 inches long? Ans. 18f. 5′ 3′′ 17. How much wood in a load 9 feet long, 31 wide, and 2 feet 9 inches Ans. 86f. 7'6" Ans. $95,76. high? 18. At $1,33 per yard for cloth, what must I give for 72 yards? |