(5) 8 bu. 3 pk. 7 qt. 1 pt. to pints Reduce to higher denominations (10) 2,456 inches (11) 5,937 square inches (14) 4,213 gills (15) 4,683 pints Ans. 575 pt. Ans. 7,156 sq. in. Ans. 39,4804 sq. yd. Ans. 197,706 cu. in. Ans. 12 rd. 2 yd. 8 in. Ans. 4 sq. yd. 5 sq. ft. 33 sq. in. Ans. 4 cu. yd. 17 cu. ft. 840 cu. in. Ans. 2 T. 19 cwt. 83 lb. Ans. 2 hhd. 5 gal. 2 qt. 1pt. 1 gi. Ans. 73 bu. 5 qt. 1 pt. 33. RULE. To change a compound number to a simple number of a lower denomination : Multiply the number of the highest denomination by the number of units it takes of the next lower denomination to make one of that higher, and to the product add the given number of the next lower denomination. Multiply this sum in the same way, and proceed in this manner until the given number is changed to units of the required denomination. RULE. To change a simple number to a compound number of higher denominations: Divide the given number by the number of units it takes of its denomination to make one of the next higher. Set aside the remainder and divide in the same manner the quotient thus obtained, and proceed in this way until the required denomination is reached. The last quotient of the several remainders is the result sought. OPERATIONS WITH DENOMINATE NUMBERS. 34. There are two methods by which to add, subtract, multiply or divide denominate numbers. One is to reduce the given numbers to the lowest denominations given in the example, and then perform the required process, and reduce the result to higher denominations. The other is to perform the process on the numbers as they stand, making the necessary reductions during the operation. The following furnish examples of the two methods. 35. Addition. Find the sum of 3 m. 182 rd. 4 yd. 2 ft., 304 rd. 1 ft., and 5 m. 76 rd. 4 yd. 2 ft. 3 m. 182 rd. 4 yd. 2 ft. = 15,840 +3,003 + 12 + 2 = 18,857 ft. 304 rd. 0 yd. 1 ft. = 5,016+ 1 = 5,017 ft. 26,400+ 1,254 +12+2 = 27,668 ft. 51,542 ft. 9 m. 243 rd. 4 yd. 0 ft. 6 in. 51,542 ft. NOTE. We have ft. or 6 in. left because the odd number of rods (each 16 ft.) leaves ft. i.e. we divide an integral number of feet by the mixed number 163. As the quotient is an odd number, 243, we have a mixed number, 121, as a remainder, 12 ft. equal 4 yd. 6 in. The other method is the following: The sum of the right hand column (ft.) is 5 ft. which equals 1 yd. 2 ft. We write down the 2 and add the 1 into the next column. The sum of the next column (yd.) is, with the 1 carried over into it, 9 yds. Since 5 yd. make a rod we put down the difference between 9 and 51 which is 31, and add 1 into the rd. column. The sum of the rd. column with this 1 added is 563 rd. Of this 563 rd. we take 320 rd. to add a mile into the m. column, which thus becomes 9 m. Our result is, therefore, 9 m. 243 rd. 31 yd. 2 ft. We do not want the fraction of a yard in our answer, so we reduce it to feet. As a yard is 3 ft., yd. equals 1 ft. This gives 3 in the ft. column. But 3 ft. equal 1 yd., so we write 4 in the yd. column, and, instead of writing in the ft. column, we write 6 in. in a new column. Take the following example: Add 35 bu. 3 pk. 7 qt., 9 bu. 2 pk. 3 qt., 48 bu. 3 pk. 7 qt., and 2 bu. 2 qt. The sum of the qt. column is 19 qt. or 2 pk. 3 qt. in the 2 pk. the sum of the pk. column is 10 pk. or 2 Adding in the 2 bu. we have 96 in the bu. column. result is 96 bu. 2 pk. 3 qt. Adding bu. 2 pk. Thus the 36. Subtraction. The same methods are here used except that in the second method we take away from the higher denominations to increase the lower ones. Subtract 3 pk. 1 qt. 1 pt. from 1 bu. 2 pk. 4 qt. By first method: 1 bu. 2 pk. 4 qt. = 64 +32 + 8 = 104 pt. 53 pt. = By second method: 48+2+1 = 51 pt. NOTE. The upper row shows how we took 1 qt. to make 2 pt. leaving 3 qt., and how 1 bu. was taken in order to add 4 pk. to 2 pk. so as to make 6 and allow the subtraction of 3 pk. This line of figures was inserted only to show the method, and it should ordinarily be left out. 37. Multiplication. Multiply 14 gal. 3 qt. 1 pt. by 7. By first method: 14 gal. 3 qt. 1 pt. = 112 + 6 + 1 = 119 pt. 119 pt. X 7833 pt. 416 qt. 1 pt. 104 gal. 1 pt. Ans. By second method: 14 gal. 3 qt. 1 pt. multiplicand. 7 multiplier. 104 gal. 0 qt. 1 pt. product. By multiplying 14, 3, and 1 by 7 we obtain the following: unreduced product with pints reduced 98 gal. 21 qt. 7 pt. 98 gal. 24 qt. 1 pt. entirely reduced (product as before) 104 gal. 0 qt. 1 pt. These figures show the process of reduction which is carried on when obtaining the answer. 38. Division. Divide 14 gal. 3 qt. 1 pt. by 4. 293 pt. = 14 qt. 13 pt.= 3 gal. 2 qt. 13 pt. Hence the quotient is 3 gal. 2 qt. 13 pt. NOTE. After dividing 14 gal. by 4, the remainder is 2 gal. This remainder is added to 3 qt. making 8+ 3 or 11 qt. Dividing 11 by 4 the remainder is 3. As there are 2 pt. in 1 qt., we have 3 x 2 + 1 or 6+1 or 7 pt. Dividing by 4 we obtain 13 pt. NOTE. The second method is generally the easier. 39. RULE. Reduce the given numbers to the lowest denomination stated in the example. Perform the required operation, (addition, subtraction, multiplication or division,) and then if required reduce the result to a higher denomination. Or: Perform the required operation on the numbers as they stand, reducing the number to other denominations whenever necessary. EXAMPLES FOR PRACTICE. Find the sum of: (1) 10 yd. 2 ft. 10 in.; 15 yd. 1 ft. 9 in.; 8 yd. 2 ft. 7 in.; 18 yd. 1 ft. 11 in.; 16 yd. 2 ft. 8 in. (2) 12 A. 35 sq. rd.; 14 A. 110 11 A. 96 sq. rd.; 25 A. 100 sq. rd. Ans. 70 yd. 2 ft. 9 in. sq. rd.; 15 A. 132 sq. rd.; Ans. 79 A. 153 sq. rd. (3) 5 T. 6 cwt. 14 lb. 10 oz. ; 7 T. 15 cwt. 36 lb. 15 oz.; 17 T. 5 cwt. 84 lb. 12 oz.; 70 T. 9 cwt. 94 lb. 11 oz. From Ans. 100 T. 17 cwt. 31 lb. 15 oz. (4) 12 gal. 2 qt. 1 pt. 2 gi. take 6 gal. 3 qt. 1 pt. 3 gi. Ans. 5 gal. 2 qt. 1 pt. 3 gi. Ans. 10 yd. 2 ft. 9 in. (5) 15 yd. 2 ft. 7 in. take 4 yd. 2 ft. 10 in. (6) 25 T. 8 cwt. 75 lb. 10 oz. take 10 T. 11 cwt. 35 lb. Ans. 14 T. 17 cwt. 39 lb. 11 oz. Multiply Ans. 16 hr. 42 min. 55 sec. (7) 3 hr. 20 min. 35 sec. by 5. Ans. 18 T. 3 cwt. 91 lb. 8 oz. (9) 12 cu. yd. 15 cu. ft. 1115 cu. in. by 6. |