ALE AND BEER MEASURE. 25. What is the product of 40 gal. 3 qt. 1 pt., multiplied by 325? Ans. 246 hhd. 1 qt. 1 pt. DRY MEASURE. 26. If a field of wheat containing 20 acres, produce 16 bus. 1 pt. of wheat to the acre, how much does the whole field produce? Ans. { 320 bus. 1 pk. 2 qt. or 8 ch. 32 bus. 1 pk. 2 qt. 26. How many bushels of oats must I pay for the rent of a farm of 101 acres, if I pay 3 bus. 7 qts. an Ans. 325 bus. 3 qt. acre? When the multiplier is a compound number, and the denominations in the multiplicand and multiplier increase by the same number. It is not often necessary to multiply by a compound number, unless it be in finding the contents of those quantities whose denominations are feet, inches, and parts of an inch. The inch is variously divided;-sometimes into 8 equal parts; sometimes into 10, and sometimes 12. It will here be considered as divided into 12 equal parts. Set the multiplier under the multiplicand, so that every denomination may stand under that of its own name. Multiply all the denominations in the multiplicand by the ft. in the multiplier, if any, beginning with the least denomination. Then multiply by the inches and seconds, in the same order. Each separate product must be removed as many places to the right of the figure multiplied, as the figure multiplied by is places from the denomination of ft.; the sum of the several products will be the answer. As the different denominations increase by 12, we must carry for that number both in multiplying and adding, except when we multiply ft. by ft. where we carry for 10, ft. being the highest denomination. 27. What is the product of 2 ft. 4 in. 2", multiplied by 2 ft. 3 in. 8"? Пlustration. In this method of multiplying, a foot is considered the unit, and the inches and seconds are parts of the foot, or of the unit. By placing the product of each denomination under itself, when we multiply by ft., we place the whole multiplicand in the product as many times as the number of ft. containing an unit. As inches and seconds are only parts of a foot, or the unit, we can only place such a part of the multiplicand in the product for each inch, or each second, as 1 in., or 1", is a part of a foot. As an inch is 1 twelfth part of a foot, by removing each product when we multiply by inches, one place to the right hand, the whole product of the inches represents but 1 twelfth part as much as it would, if it had been set directly under the multiplicand, be cause it takes 12 in every column except the left hand one, to equal 1 in the next left hand denomination. One second being the 12th part of an inch, by removing the product of the second, one place further to the right than we did the product of the inches, it can represent only 1 twelfth part as much as it would, had it stood under the product of the inches. Note 4.-It may be observed, that when inches, seconds, &c. are multiplied by inches, seconds, &c. without being in connection with any other denomination, the products will be of the same name as the numbers multiplied. 28. What is the product of 29 ft. 9 in. 1 ", multiplied by 3 ft. 2 in. ? " 29. Multiply 21 ft. 0 in. 5" by 9 in. 3". Prod. 16 ft. 2 in. 6" 10" 3 """. 30. Multiply 21 ft. 5" by 9 ft. 3". Prod. 189 ft. 9 in. 1" 3""". A surface is a figure having length and breadth, but no thickness: The outside of all bodies, and the inside of hollow bodies, are surfaces. a FIG. A. b d In figure A, the space included by the 4 lines a b, b. c, cd, and da, is a surface bounded by said lines. Note 5.--The contents of all plane or flat surfaces, are found by multiplying the length by the average breadth. FIG. B. Illustration of the last note. In figure B, were we to set down 8, the number of small squares on one side, as many times as there are rows of small squares in the figure, and add them to gether, we should obtain the whole number of small squares in the figure; but as we place the multiplicand in the product once for every unit in the multiplier, we shall obtain the same number, if we multiply 8, the number of squares in one row, by 5, the number of rows; therefore, multiplying the length of any surface by the breadth, will give the contents. Every surface is supposed divided into squares of that denomination by which the length and breadth of such surface are measured; if measured by yards, into square yards; if by feet, into square feet; if by inches, into square inches, &c. 31. How many square inches in a board 11 in. long and 7 in. wide! Ans. 77. 32. How many square feet in a board 23 ft. long, and 3 ft. wide? Ans. 69. 33. How many feet in a board 20 ft. 7 in. long, 2 ft. 5 in. wide? Ans. 49 ft. 8 in. 11 " 34. How many feet in a board 19 ft. long, 2 ft. 9 in. wide? Ans. 52 ft. 3 in. 35. How many feet in a board 18 feet in length, and 10 in. in width? Ans. 15 ft. 36. How many feet in 12 boards, each 14 ft. 1 in. long, 1 ft. 2 in. wide? Ans. 197 ft. 2 in. 37. How many feet in a floor 16 ft. 4 in. long, 15 ft. 4 in. wide? Ans. 250 ft. 5 in. 4". 38. How many feet of boards will it require to lay a floor 17 ft. long, and 16 ft. 11 in. wide? Ans. 287 ft. 7 in. Note 6.-Painting and several other kinds of work are reckoned by the square yard; 9 square ft. being equal to 1 square yd. As many times as we can take 9 from the number of square feet, there will be so many square yards. 39. How many square yards in the sides and ends of a room, 15 ft. 7 in. long, 13 ft. 1 in. wide, and 8 ft. high, allowing 5 square yards for the windows? Ans. 45 yd. 8 ft. 8 in. To find the contents of timber, wood and other solid bodies, where the length in all parts is equal, and also the breadth and thickness.* RULE, Multiply the length by the breadth, and the product by the thickness. 40. How many solid feet in a square stick of timber, 16 ft. long, 4 ft. wide, 3 ft. thick? Operation. 1 6 4 6 4 3 192 Illustration. Multiplying the length by the breadth gives the number of square feet on one side, as was proved in the last illustration. If the stick were but 1 ft. thick, there would be as many solid, as square ft., for if the stick in that case were sawed into 64 equal blocks, each block would mea sure 1 ft. on every side, and contain 1 solid foot. But as the given stick is 3 ft. thick, we have what is equal to 3 sticks of the given length and breadth, and i ft. thick, consequently there will be 3 times 64 solid ft. in the given stick. The rule is equally true as it respects any other numbers, for, as the number of square ft. or in., &c. into which one side is divided, is equal to the number of solid ft. or in. on one side, when 1 foot, or 1 inch, of the thickness is considered, there will always be as many solid feet or inches for every foot, or inch, of the whole thickness, as there are feet or inches in the thickness; therefore multiplying the number of square feet, or inches, &c. on one side, by the number of feet or inches in the thickness, will give the whole solid feet, or inches, &c. in a solid body of the above form. When the length of one side is greater than that of the other and the variation is a true slope, the average length should be taken and the same of the width and thickness. |