For small cattle and calves of a girth of from 3 to 5 feet, allow 16 lbs. to the superficial foot. For hogs, sheep, and all cattle of a girth less than 3 feet, allow 11 lbs. to the superficial foot. 237. To find the number of pounds of beef, veal, etc., in an animal. 1. What is the estimated weight of beef in a steer whose girth is 6 ft. 8 in., and length 5 ft. 4 in. ? SOLUTION.-6 ft. 8 in. = 80 in.; 5 ft. 4 in. = 64 in. 80 in. × 64 in. sq. in. This divided by 144 = 35 square, or superficial, feet. 35§ 8177 lbs. = RULE.-Multiply the number of superficial feet by the number of pounds allowed for animals of different girths, and the product will be the estimated weight. When the animal is very fat added, and if only half fattened greater accuracy. of the weight as found above should be of the weight should be deducted for 238. To find the cost when the price per 100 or 1000 is given. 1. What will 245 rails cost at $4.75 per hundred? $4.75 EXPLANATION.-Since 100 rails cost $4.75, 245 rails, which are 2.45 equal to 2.45 times 100 rails, will cost 2.45 times $4.75, or $11.64 $11.6375 RULE.-Reduce the quantity to hundreds by pointing off two places from the right, or to thousands by pointing off three places. Multiply the price by this result and point off the product as in multiplication of decimals. 1. In business, the mills in final results are dropped when less than five. When five mills or more, they are usually regarded as a cent. 2 The price per hundred may be expressed per C, and per thousand, per M. The character P is often used for per. 2. What will be the cost of 500 bushels of lime at $6.35 per C? 3. What will 1240 pickets cost, at $1.12 C? 4. How much will 9850 shingles cost, at $5.35 M? 5. What will be the cost of 3471 feet of boards at $15.25 M? 6. A grocer sold one month 8640 pounds of flour at $2.65 C. What was the amount of the sale? 7. Mr. Lake bought 25750 laths at $2.85 M. How much did they cost him? 8. How much must be paid for 635 feet of boards at $15.50 P M, 2540 feet of scantling at $1.35 PC, and 3764 feet of flooring at $17.50 M? 239. To find the cost of hay. 1. Find the cost of 879 pounds of hay at $12 per ton. 879 EXPLANATION.-At $12 a ton is $6 per 1000 pounds. Hence, by $6 multiplying 879 by 6 and removing the decimal three places from $5.274 the right, the cost is found to be $5.27. RULE.-Multiply the number of pounds by half the price per ton, and point off three places from the right. 2. How much must be paid for 4680 pounds of hay at $8.50 per ton (2000 pounds)? SUGGESTION.-Dividing the price of a ton by 2, gives the price per 1000 pounds. Or dividing the quantity expressed in thousands (the quantity with the decimal point moved three places toward the left) by 2 will give the number of tons. 3. What will be the cost of 5157 pounds of fertilizer at $18 a ton? 4. A farmer sold 16750 pounds of hay for $15.60 a ton. How much did he receive for it? 5. What must be paid for the shipment of 17587 pounds of merchandise at the rate of $13 a ton? 6. If a ton of bone dust is worth $25.75, what is the value of 7240 pounds? 7. A farmer received $56.76 for 9460 pounds of hay. What was the price per ton? 8. What is the value of 13760 pounds of hay when the price is $15 per ton? 9. A farmer sold 25 loads of hay each weighing 1700 pounds, at $13.50 per ton. How much did he receive for the bay? 240. To find the cost of a given number of pounds when the price per bushel is given. 1. How much should be given for 476 pounds of corn, at 40 per bushel of 56 pounds? 476 56) $190.40 ($3.40 169 EXPLANATION.-At $.40 per lb. the cost would be 476 times $.40, or $190.40. But since the price is $.40 per bu. of 56 lb, the cost will be of $190.40, or $3.40. 224 224 0 RULE.-Multiply the weight in pounds by the price per bushel, and divide the product by the number of pounds in a bushel. Parts of bushels are sometimes written without the denominator expressed, thus, 828 bu. of corn = 838 bu. = 476 lb. For table of weights of produce see page 132. 2. How much will 1560 lbs. of oats cost, at $.25 per bu. of 32 lbs. ? 3. Find the cost of 2150 lbs. of wheat, at $.80 per bu. of 60 lbs. 4. What will 1728 lbs. of barley cost, at $. 65 per bu. of 48 lbs. ? 5. How much must be paid for 2347 lbs. of corn, in the ear, at $.40 per bu. of 70 lbs. ? 6. I bought 360 lbs. of timothy seed at $2.25 per bu. of 45 lbs. How much did I pay for it? It is well always to observe the relation of numbers for exact multiples or aliquot parts. In problem 6, 360 lbs. is just 8 times 45 lbs. or 8 bu. 7. How much will 1400 pounds of shelled corn cost at $.65 per bu. of 56 lbs. ? 8. A load of potatoes which weighed 1860 pounds was sold at $.75 per bushel of 60 lbs. How much was received for the ioad? 9. Find the cost of 3500 pounds of turnips selling at 25 per bu. of 55 lbs. 10. How much will 336 lbs. of buckwheat cost at $.70 per bushel of 48 lbs. 11. A farmer sold 340 lbs. of timothy seed at $2.35 per bu. of 45. lb. How much did he get for it? PROOFS OF FUNDAMENTAL PROCESSES. 241. The proofs given under addition, subtraction, multiplication, and division are the most reliable that can be given. Other methods are employed, among which perhaps the one given below is as reliable a test as any of them. 242. Method by casting out the nines. It was probably the Arabians who discovered centuries ago that when the number of 9's in a number is found, the remainder is equal to the sum of the digits of the number, or to the sum with the 9's cast out. Thus, take any number, as 65, and dividing by 9, we have 7 (nines) and 2 remaining. Adding the digits 6 + 5 = 11, and dividing by 9, we have 1 (nine) and 2 remaining as in dividing the number. Similarly 426 ÷ 9 = 47 with 3 remaining, and adding the digits 4+2 + 6 = 12, which divided by 9 gives a remainder of 3. Similarly 7135 ÷ 9 792 with 7 remaining, and adding the digits 7+1+3+5=16, which divided by 9 gives a remainder of 7. = PROOF OF ADDITION. 243. 1. Prove that 283 + 462 + 375 + 859 = 1979. = 4 283 4623 375 = 6 = 4 = 859 1979 = 8 EXPLANATION.-The remainder or excess of 9's, in the first addend is 4 units; in the second, 3 units; in the third, 6 units; in the fourth, 4 units. The sum of the units remaining is 17, which divided by 9 gives a remainder of 8. The remainder or excess of 9's in the sum 1979 is also 8. Hence, the result is probably correct. It should be remembered that this method of proof is not infallible, for the figures might be transposed and yet the same excess of nines would appear. = : 1189. = = 1562. = 2093. 2. Prove that 276 +420 + 158 +335 3615. 116 PROOFS OF FUNDAMENTAL PROCESSES. PROOF OF SUBTRACTION. 244. 1. Prove that 8352 - 3416 = 4936. EXPLANATION.-The excess of 9's in the minuend is 0. The excess of 9's in the subtrahend is 5. 5 subtracted from the radix, 9, leaves a remainder of 4. The excess of 9's in the remainder 4936 is also 4. Hence, the result is likely to be = 38924. EXPLANATION.-The excess of 9's in the multiplicand is 4; in the multiplier 2. The product of these two remainders is 8, and the excess of 9's in the product 38924 is also 8. Hence, the result is believed to be correct. 7 PROOF OF DIVISION. 246. 1. Prove that 9350 ÷ 34 8 5 = = 275. EXPLANATION.-The excess of 9's in the divisor and 34) 9350 (275 quotient are respectively 7 and 5. This product is 35 or an excess of 8, which corresponds to the excess of 9's in The work is therefore presumed to be correct, since the the dividend. product of the divisor and quotient equals the dividend. 6 2. Prove that 516725 ÷ 357 = 1447, and rem. 146. 8 7 2 357) 516725 (1447 Rem. 146 EXPLANATION.-The product of the divisor and quotient, plus the remainder, equals the dividend. The excess of 9's in the divisor, quotient, and remainder are respectively 6, 7, and 2. The product of 6 and 7 is 42, to which is added the excess of the remainder 2, making 44, or an excess of 8. The excess of 9's in the dividend is also 8. Hence, the result is presumed to be correct. 4. Prove that 875342 ÷ 426 = 2054, and rem. 338. |