234. To reduce money, weights, and measures, to decimals. RULE I. Reduce the given number to a fraction of the inte ger of which it is to be made a decimal, by Art. 185. II. Reduce this fraction to a decimal by the last rule. 13. Reduce 2s. 6d. to the decimal of a pound. 1 Thus, 2s. 6d. = -L. Then 8)1.000 .125L. the answer. 14. Reduce 1gr. 14lb. to the decimal of a cwt. 15. Reduce 5bu. 3pks. to the decimal of a chaldron. Thus, 5bu. 2pks.= .152777 &c. chald. Ans. 11 72 a The foundation of this rule is sufficiently obvious without explanation. 16. Reduce 5s. to the decimal of a pound. Ans. .251. 17. Reduce 3s. 6d. to the decimal of a guinea. Ans. .1666 &c. guinea. 18. Reduce 3oz. 5dwts. to the decimal of a lb. troy. Ans. .27083 &c. lb. 19. Reduce 2r. 20p. to the decimal of an acre. Ans. .625 acre. 235. When the given quantity is of several denominations. RULE I. Write all the given denominations in a line under each other, beginning with the least, and proceeding in order up to the greatest. II. On the left of each, place that number for a divisor which will reduce it to a decimal of the next superior denomination. III. Divide each of the denominations, together with the decimals which arise, by its proper divisor, and the last quotient will be the decimal required. 20. Reduce 4s. 6d. to the decimal of a pound. OPERATION. 43.00 12 6.75 20/4.5625 .2281251. Ans. I put down Explanation. farthings first, 6d. next, and 4s. last; opposite these on the left I place 4, 12, and 20, being respectively the divisors which will reduce each to decimals of a superior denomination; I then divide each line, ciphers being subjoined where they are required. 21. Reduce 53 43 29 1gr. to the decimal of a lb. 20 1.00 32.05 84.68333333 125.58541666 .46545158 lb. Answer. By this process the least denomination is reduced to a decimal of the next superior denomination, as is obvious from the foregoing rule; this latter denomination, with the said decimal subjoined, is reduced to a decimal of the next superior denomination; this again with its decimal to a decimal of the next; and so on to the highest: for instance, in ex. 20, is reduced to the decimal of a penny, by dividing the 3 by the 4; then 6d., with this decimal, (viz. 6.75,) is reduced to the decimal of a shilling, by dividing it by 12; lastly, the whole number, with this decimal, (viz. 4.5625,) is reduced to the decimal of a pound; which reductions are true, according to Art, 233; and the same of other examples: wherefore the rule is manifest. 22. Reduce 3s. 2d. to the decimal of a pound. Answer .1593751. Answer 23. Reduce 7d. to the decimal of a shilling. Ans. .625s. 24. Reduce 1s. 2d. to the decimal of a crown. .245833cr.* 25. Reduce 2qr. 3n. to the decimal of a yard. Ans. .6875yd. 26. Reduce Ir. 14p. to the decimal of an acre. Ans. .3375a. 27. Reduce 4bu. 2pks. 1gal. to the decimal of a quarter. Ans. 578125qr. 28. Reduce 1gr. 2lb. 3oz. 4dr. to the decimal of a cwt. Ans. .2696707 &c. cwt. 29. Reduce 3w. 4d. 5h. 6m. 7" to the decimal of a month. Ans. .9004493 &c. month. 236. To reduce a decimal to its proper quantity. RULE I. Multiply the given decimal by that number which will reduce it to the next lower denomination, and mark off as many decimals from the product as there are decimal places in the given number. II. Multiply these decimals by the number which will reduce them to the next lower denomination, and mark off decimals as before. III. Proceed in this manner until you have reduced the decimals to the lowest denomination possible; then all the whole numbers being collected, and placed in order, will be the answer f 30. Reduce .228125 of a pound to its proper quantity. OPERATION. .228125 L. 20 4.562500 s. 12 6.750000 d. 4 3.000000 qrs. Explanation. I multiply the given decimal (of a pound) by 20, and mark off 6 places; these I multiply by 12, and mark off 6 places; these I multiply by 4, and again mark off 6 places. Then of the whole numbers, the 4 will evidently be shillings, the 6 pence, and the 3 farthings, which together are the answer. Ans. 4s. 6d.4. f This rule has its origin in the nature of Compound Division; for if the given decimal be considered as a remainder, the successive multiplications, and marking off decimals, are equivalent to the successive reducing and dividing in that rule. 31. Reduce .7583 of a lb. troy to its proper quantity. 32. Reduce.159375 of a pound to its proper quantity. Ans, 3s. 2d.4. 33. Reduce .625 of a shilling to its proper quantity. Ans. 7d.. 34. Reduce .05854 of a guinea to its proper quantity. Ans. 1s. 2d.4. 35. Reduce .6875 of a yard to its proper quantity. Ans. 2gr. 3n. 36. What is the value of .625 cwt.? Ans. 2qr. 14lb. 37. Reduce .3375 of an acre to its proper quantity. Ans. 1r. 14p. 38. What is the value of .461 of a chaldron? Answer 16bu. 2pks. 39. What is the value of .85714 of a month? Ans. 3w. 2d. 23h. 59m. 53". 237. PROMISCUOUS EXAMPLES FOR PRACTICE. 1. What is the value of .135 + .243 x .312? Ans. .210816. 2. What is the value of .0098.00098 x .54? Answer .0092708. 3. Required the value of 3.74 × 2.35 · 1.23. Ans. 7.559. 4. Required the value of 1.35 × 3.8 + .019. Ans. 5.15565. 5. What is 2.04 x 4.5 2.95 equal to? Ans. 3.162. a pound, to be reduced to its proper quantity. Ans. 131. 19s. 9d. .006. 238. PROPORTION, OR, THE RULE OF THREE IN DECIMALS. RULE. Reduce the first and third terms to decimals of the same denomination, and the second to a decimal of the greatest denomination mentioned; then state the question, and proceed as in the Rule of Three in Vulgar Fractions: the result will be of the same denomination with the second term, and (if a decimal) must be reduced to its proper quantity". 3 EXAMPLES. 1. If of a cwt. of pimento cost 31. 2s. 6d. what is the value 8 of 1cwt. 2qr. 14lb. ? 3 8 Reduction of the terms. -cwt.=.375cwt. 31. 2s. 6d.=3.1251. 1cwt. 2qr. 14lb.=1.625cwt. = 13.54166 = 137. 10s. 9d. .99 &c. Answer. Explanation. Having reduced the first and third terms to decimals of a cwt. and the second to the decimal of a pound, I state the question, and, finding that the answer 8 The rules of Proportion in whole numbers, vulgar and decimal fractions, depend all on the same principles; they differ only in the different modes of operation peculiar to each of these three kinds of numbers. |