How many pounds, shillings, and pence, will 279·301 acres let for if each acre lets for £3·1076?—Answer, £867·9558, or £867. 19. 1¼.
What does ¼ of ³/₁₃ of 17 bush. cost at ⅙ of ⅔ of £17. 14 per bushel?
Answer, £2·3146, or £2. 6. 3½.
What is the cost of 19lbs. 8oz. 12dwt. 8gr. at £4. 4. 6 per ounce?—Answer, £999. 14. 1¼ ⅙.
232. It is often required to find to how much a certain sum per day will amount in a year. This may be shortly done, since it happens that the number of days in a year is 240 + 120 + 5; so that a penny per day is a pound, half a pound, and 5 pence per year. Hence the following rule: To find how much any sum per day amounts to in a year, turn it into pence and fractions of a penny; to this add the half of itself, and let the pence be pounds, and each farthing five shillings; then add five times the daily sum, and the total is the yearly amount. For example, what does 12s. 3¾d. amount to in a year? This is 147¾d., and its half is 73⅞d., which added to 147¾d. gives 221⅝d., which turned into pounds is £221. 12. 6. Also, 12s. 3¾d. × 5 is £3. 1. 6¾, which added to the former sum gives £224. 14. 0¾ for the yearly amount. In the same way the yearly amount of 2s. 3½d. is £41. 16. 5½; that of 6¾d. is £10. 5. 3¾; and that of 11d. is £16. 14. 7.
233. An inverse rule may be formed, sufficiently correct for every purpose, in the following way: If the year consisted of 360 days, or ³/₂ of 240, the subtraction of one-third from any sum per year would give the proportion which belongs to 240 days; and every pound so obtained would be one penny per day. But as the year is not 360, but 365 days, if we divide each day’s share into 365 parts, and take 5 away, the whole of the subtracted sum, or 360 × 5 such parts, will give 360 parts for each of the 5 days which we neglected at first. But 360 such parts are left behind for each of the 360 first days; therefore, this additional process divides the whole annual amount equally among the 365 days. Now, 5 parts out of 365 is one out of 73, or the 73d part of the first result must be subtracted from it to produce the true result. Unless the daily sum be very large, the 72d part will do equally well, which, as 72 farthings are 18 pence, is equivalent to subtracting at the rate of one farthing for 18d., or ½d. for 3s., or 10d. for £3. The rule, then, is as follows: To find how much per day will produce a given sum per year, turn the shillings, &c. in the given sum into decimals of a pound (221); subtract one-third; consider the result as pence; and diminish it by one farthing for every eighteen pence, or ten pence for every £3. For example, how much per day will give £224. 14. 0¾ per year? This is 224·703, and its third is 74·901, which subtracted from 224·703, gives 149·802, which, if they be pence, amounts to 12s. 5·802d., in which 1s. 6d. is contained 8 times. Subtract 8 farthings, or 2d., and we have 12s. 3·802d., which differs from the truth only about ¹/₂₀ of a farthing. In the same way, £100 per year is 5s. 5¾d. per day.
234. The following connexion between the measures of length and the measures of surface is the foundation of the application of arithmetic to geometry.
Suppose an oblong figure, a, b, c, d, as here drawn (which is called a rectangle in geometry), with the side a b 6 inches, and the side a c 4 inches. Divide a b and c d (which are equal) each into 6 inches by the points a, b, c, l, m, &c.; and a c and b d (which are also equal) into 4 inches by the points f, g, h, x, y, and z. Join a and l, b and m, &c., and f and x, &c. Then, the figure a b c d is divided into a number of squares; for a square is a rectangle whose sides are equal, and therefore a a f e is square, since a a is of the same length as a f, both being 1 inch. There are also four rows of these squares, with six squares in each row; that is, there are 6 × 4, or 24 squares altogether. Each of these squares has its sides 1 inch in length, and is what was called in (215) a square inch. By the same reasoning, if one side had contained 6 yards, and the other 4 yards, the surface would have contained 6 × 4 square yards; and so on.
