CASE II. To reduce numbers of different denominations, as money, weights, measures, &c. to their equivalent decimal values. RULE. Reduce the given numbers to the lowest denomination mentioned, for a dividend; also reduce the integer to the same denomination, for a divisor; then annex cyphers to the dividend, divide as in whole numbers, and the quotient will be the decimal required. Note. Tables of Ale and Beer Measure, Wine Measure, Dry Mea sure, and Avoirdupois Weight, may be seen in Part IV. EXAMPLES. 1. Reduce 15s. 6d. to the decimal of a pound sterling. 2. Reduce 10s. 9 d. to the decimal of a pound. Ans. .540625. 3. Reduce 9s. 34d. to the decimal of a pound. Ans. .4635416. 4. Reduce 101d. to the decimal of a shilling. Ans. .875. 5. Reduce 19s. 11d. to the decimal of a pound. Ans. .9989583. 6. Reduce 2 pints of wine to the decimal of a gallon. Ans. .25. 7. Reduce 189 gallons of wine to the decimal of a tun. Ans. .75. 8. Reduce 8 gallons of ale to the decimal of a firkin. Ans. .888888. 9. Reduce 56 gallons of beer to the decimal of a butt. Ans. .518518. 10. Reduce 3 pecks of malt to the decimal of a bushel. Ans. .75. 11. Reduce 4 quarters, 5 bushels, and 2 pecks of barley, to the decimal of a last. Ans. .46875. 12. Reduce 12 ounces avoirdupois, to the decimal of a pound. Ans. .75. 13. Reduce 22 pounds, 9 ounces of candles, to the decimal of a quarter. Ans. .8058035. 14. Reduce 3 quarters, 14 pounds, and 8 ounces of tallow, to the decimal of a hundred weight. Ans. .8794. 15. Reduce 15 hundred weight, 2 quarters, and 21 pounds of soap to the decimal of a ton. Ans. .784375. CASE III. To find the value of a decimal fraction in the known parts of an integer. Multiply the given decimal by the number of parts contained in the next inferior denomination; and, from the right of the product, point off as many figures as there are places in the given decimal. Multiply the decimals thus pointed off, by the parts in the next less denomination; reserving as many places to the right, as before. Proceed in this manner through all the denominations to the last; then the several figures on the left of the decimal points, will be the answer required. EXAMPLES. 1. What is the value of .7362 of a pound sterling? .7362 20 14.7240 12 8.6880 4 2.7520 Ans. 14s. 8d. 2. What is the value of .8649 of a shilling? Ans. 101d. 3. Find the value of .92846 of a pound. Ans. 18s. 6d. 4. Required the value of .8694 of a hogshead of wine. Ans. 54 gal. 3 qt. 5. What is the value of .73828 of a barrel of beer? Ans. 26 gal. 2 qt. 6. Required the value of .5694 of a quarter of malt. Ans. 4 bush. 2 pk. 7. What is the value of .68328 of a last of barley? Ans. 6 gr. 6 bush. 24 pk. 8. Find the value of .9326 of a hundred weight of Ans. 3 qr. 20 lb. 7 oz. tallow. RULE OF THREE. RULE. State the question as in the common Rule of Three; reduce the inferior denominations of such of the terms as are compound, to the decimal parts of their integer; multiply the second and third terms of the proportion together, and divide the product by the first term, and the quotient will be the answer required, which must, if necessary, be reduced to its integral value. Note 1. In solving questions in the Rule of Three, proper attention must be paid to the rules given in multiplication and division, for pointing off the decimals. 2. As Arithmeticians differ in their opinions with regard to the best method of stating questions in the Rule of Three, we shall here give both Rules; and the learner may use that of which he most ap. proves. RULE I. Consider which of the three given terms is of the same kind as the answer, or number sought, and put it down as the third term of the proportion. Then, if it appears, from the nature of the question, that the answer will be greater than this number, make the greater of the other two numbers the second term, and the less the first; but if it will be less, make the less number the second term, and the greater the first. Reduce the first and second terms of the proportion to the same denomination, and the third to the lowest denomination mentioned. Multiply the second and third terms together, and divide the product by the first term; and the quotient will be the answer, in the same denomination to which the third term was reduced. Note. The answer on the fourth term, must, when necessary, be brought again, to the highest denomination of which it admits, in order that it may be expressed in a proper form. RULE II. Write down that number which is of the same name as the answer, for the second term in the proportion. Observe, from the nature of the question, whether the answer will be greater or less than this number. If it appears that it will be greater, make the less of the two remaining numbers, the first term, and the greater the third; but if the answer will be less than the second term, make the greater number the first term, and the less the third. Reduce the first and third terms of the proportion to the same denomination, and the second to the lowest denomination mentioned. Multiply the second and third terms together, divide the product by the first term; and the quotient will be the answer, in the same deno. mination as the second term. Note 1. In stating questions in the Rule of Three, the word As, is generally placed before the first term, and the signs of proportion between each of the other terms, thus, As 4 lb.: 12 lb. :: 5s.: 15s. by the first Rule. Or, As 4 lb. : 5s. :: 121b.: 15s. by the second Rule. 2. Some persons object to the second Rule, on this ground, that no proportion whatever can subsist between 4 lb. and 5s. or between 12 lb. and 15s. The first Rule is, in our opinion, more scientific; but we are inclined to think, that the second will be more easily comprehended by learners; and, according to it, the numbers are always alternately proportional. 3. Questions in the Rule of Three are of two kinds, namely, direct and inverse; but both the foregoing Rules are general. 4. Direct Proportion, is when more requires more, or less requires less, as in the following examples: If 6 men can dig a trench 48 yards in length, in a certain time; how many yards can 12 men dig in the same time? Here it is obvious, that the more men there are employed, the more work will they perform; and, therefore, in this case, more requires more. Again, If 6 men can dig 48 yards, in a given time; how many yards can 3 men dig in the same time? Here less requires less; for the less number of men there are employed, the less work will there be performed by them. All questions that are of this class, are said to be in the Rule of Three Direct. 5. Inverse Proportion, is when more requires less, or less requires more; thus, If 6 men can dig a certain quantity of trench in 12 hours; how many hours will it require 12 men to dig the same quantity? Here more requires less: for 12 men being more than 6, it is manifest, that they will require less time to perform the same work. Again, If 6 men perform a piece of work in 10 hours; how many hours will 3 men be in performing the same work? Here less requires more; for the number of men being less, they will require more time to do the same quantity of work. All questions of this kind are said to be in the Rule of Three In verse. EXAMPLES. 1. If 5.75 lb. of candles cost 4.25s. what will be the price of 17.25 lb.? |