480 51 6", North. Required the difference of latitude be tween these two places. Ans. 6° 57' 12". 16. The latitude of London, (St. Paul's,) is 51° 30′ 49′′, North; and of Dublin, 53° 21, North. Required their difference. Ans. 1° 50′ 11. 17. The following are the times in which the primary planets perform their revolutions about the sun; required the differences of the first and second, of the second and third, &c. 52. When the multiplicand expresses a quantity of the same kind, but of more denominations than one, the process is termed compound multiplication. • Problem 1. To multiply a number of more denominations than one, by a number not exceeding 12. RULE. Commencing with the lowest denomination, multiply successively the several numbers in the multiplicand by the multiplier, dividing, setting down, and carrying as in compound addition. Ex. 1. Multiply 37 dollars, 374 cents by 6. Here, X6 is equal to 3, because 6 halfcents are equivalent to 3 cents; then 6 times 7 is 42 and 3 are 45, set down 5 and carry 4 as in multiplication of whole numbers. $ cts, 37 37 225 25 2. Multiply £1 14s. 73d. by 9. £ s. d. 1 14 72 9. 15 11 93 In this example the farthings, pence, and shillings are multiplied successively by 4, 12, 20, (or the tens of the shillings by 2;) the several remainders are written, down, and the quotients carried. The pounds are multiplied as in simple multiplication; and the product is found to be £15, 11s. 9. 3. Multiply 25 tons, 13cwt. 3qr. 14lb. by 8. In this example, the lbs. qrs. and cats. are multiplied successively by 28, 4, 20, (or the tens of the cut. by 2,) the several remainders are written down, and the quotients carried. tons. cwt. qrs. lb. 25 13 3 14 8 205 11 0 0 4. Multiply 72 pounds, 10 ounces, 19 pennyweights, 18 Ans. 364lb. 6oz. 18dwt. 18gr. grains, by 5. 5. Multiply $187.371 by 10. Ans. $1873.75. 8. Multiply 36 pounds, 3 ounces, 3 drams, 1 scruple, 19 grains, by 2. Ans. 72lb 63 73 0 9. Multiply 75 yards, 2 quarters, 2 nails, by 3. 18gr. Ans. 226yd. 3qr. 2 nails. 10. Multiply 75 miles, 3 furlongs, 30 poles, by 4. Ans. 301 miles, 7 fur. 11. Multiply 300 acres, 3 roods, 3 poles, by 5. Ans. 1503a. 3r. 15p. 12. Multiply 35 tuns, 1 hogshead, 36 gallons, 3 quarts, pint, by 6. Ans. 212 tuns 1hhd. 32gal, 1qt. 13. Multiply 32 years, 325 days, 12 hours, by 7. 14. Multiply 33° 12 13" by 7. Ans. 230yrs. 88dys. 12hrs. PROBLEM 2. Ans. 232° 25'′ 31′′. To multiply by a number which exceeds 12, but is the pro duct of two or more factors, each less than 13. Rule. 53. By the preceding problem, multiply the given multiplicand, by one of the factors. Multiply the re sult by another. Multiply this last result by another, if there be so many; and thus proceed, whatever is their number. Ex. 1. Multiply $75.311, by 24. In this example, the multiplicand is multiplied by 4, and the product is $301.25. This again is multiplied by 6, and the product is $1807.50. The reason of the operation is sufficiently obvious, since 24 is the product of 4 and 6. The work might be proved, by multiplying the multiplicand by 6, and the result by 4. When the multiplicand contains one or more quarters, if one cts. 75 31 31/1 301 25 6 1807 50 of the factors be even, (that is, divisible by 2,) it is better to use it first, as the quarters may thus disappear, and the rest of the work be easier. 2. Multiply £756 13s. 9d. by 30. In this example, the multiplicand is multiplied by 6, and the product is £4540 2s. 6d. This again is multiplied by 5, and the product is £22700 12s. 6d. The reason of the operation is sufficiently obvious, since 30 is the product of 6 and 5. £ S. d. 756 13 9 4540 6 5 22700 12 6 3. Multiply 11 cents by 16. 4. Multiply 12 cents by 14. 5. Multiply $1.37 by 40. 6. Multiply 12 shillings and 6 pence 7. Multiply £1 1s. and 9d. by 48. 8. Multiply 3 pounds, 3 ounces, grains, by 42. Ans. 139/b. 19 pennyweights, 21 11oz. 14dwts. 18gr. Ans. 472cwt. 2qrs. 9. Multiply 7cwt. 3qr. 14lb. by 60. Ans. 204 tons. 11. Multiply 1 pound 1 ounce, 7 drams, 1 scruple, and 18 grains, by 70. Ans. 81fb 43 63 19. 12. Multiply 70 yards, 1 quarter, 1 nail, by 45. Ans. 3164yds, Oqrs. 1 nail. 13. Multiply 13 leagues, 2 miles, 3 furlongs, 30 poles, by 50. Ans. 691lea. Om. 3fur. 20p. 14. Multiply 756 acres, 3 roods, 30 poles, by 108. Ans. 81749a. Ir. 15. Multiply 13 tuns, 2 hogsheads 31 gallons, by 75.* Ans. 1021 tuns, 2hhd. 57gal. 16. Multiply 30 bushels, 3 pecks, 4 quarts, by 112. 17. Multiply 29 days, 12 hours, 30 by 128. Ans. 3458 bush. minutes, 30 seconds, 128dys. 17hrs. 4 min. Ans. 253° 24′ 0′′.. PROBLEM 3. Ans. 10yrs. 18. Multiply 1° 30′ 30′′ by 163. To multiply by a number which exceeds 12, but is not produced by factors below 13. Rule 1. 54. Use those factors whose product is nearly equal to the multiplier. Increase or diminish the result, as the case may require, by the product of the multiplicand, and the difference between the multiplier and the product of the factors employed. Ex. 1. Multiply $3 621 by 38. In this example, 38 not being the product of any two factors not exceeding 12, we multiply by 36, as before, and to the product we add twice the multiplicand, to find the product by 38. The answer would have been attained with nearly the same facility, had we multiplied by 40 (4× 10.) and subtracted twice the multiplicand; and thus the operation might be proved. 43 cts. 62/1/20 12 50 product by 12 3 130 50 36 7 25 137 75 38 181 * In this and each of the following three examples, the multiplier is the product of three factors; 75=3×5×5; 112=8×2×7, or 4X4X7; 128 8X8X2, or 4X4×8; and 168-8×3×7, or 4×6×7. Rule 2. 55. Multiply the given price or quantity by 10, which will give the price of 12, this again multiplied gives the price of 100, this again by 10 for 1000, &c. Then multiply the first line or price of 1 by the units; the second product, or the price of 10, by the tens; and proceed in like manner for the hundreds, &c. The products then added together will be the answer required. Ex. 2. What cost 2485 yards of broadcloth, at 15s. 71. per yard? In this example, we find successively the prices of 10, 100, 1000. We then multiply the price of 1000 by 2; of 100 by 4; of 10 by 8; and of 1 by 5. We have thus the prices of 2000, of 400, of 80, and of 5, the sum of which is £1941 8s. 1 d. the answer. This method is useful when the multiplier is very great, or when the factors, whose product is nearly equal to it, cannot be easily found. 3. Multiply $1.371, by 13. 4. Multiply 75 cents by 23. K Ans. $17.87 |
