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12. If A and B travel to the city of Washington from the same place, and the road on which A travels, be 24 m.8 r. 16 ft. 4 in., and that which B travels, be 21 m. 3 fur. 38 r. 14 ft. 6 in., how much farther does A travel than B 2
Operation 1. Operation 2. JM. fur. r. ft. in. JM. fur. r. ft. in. 24 O 3 16 4 23 7 48 15 16 21 3 38 14 6 21 3 38 14 6 2 4 10 1 10 2' 4 10 1 10
Illustration.—When the upper number in any denomination is less than the one below it, by adding as many to the upper number as it takes of that particular denomination to make 1 in the next higher, and subtracting the under number from the sum, we subtract the under number from the number above it and 1 taken from the upper number in the next higher denomination. Now if we were to take away 1 from the upper number in the next higher denomination, and subtract the under number from the remainder, we should certainly obtain the difference between the under and upper numbers of both denominations; but as we do not generally take 1 from the upper number, we may suppose that we have added 1 to the upper number, after we suppose 1 carried from it to the other denomination, and by adding 1 to the under number, there will be no difference between
the 1 added to the upper and the 1 added to the under ; consequently the difference must be between the two numbers as they were given, after supposing 1 taken from the upper number, and carried to the next right hand denomination.* An example will make this principle plain. In operation 1, of the last question, we subtract in the common method, supposing 12 in. added to the 4 in., 40 r. to the 8 r., and 8 fur, set in the place of the cipher because these numbers are just equal to 1 in the next higher denomination, and carrying 1 in each instance to the next higher denomination. In operation 2, instead of supposing a number added to the upper number, when it is the smallest, we actually separate the numbers. As we cannot take 6 from 4 in. we take 1 foot from the 16 ft. and add 12 in. (which are equal to 1 ft.) to the 4, and subtract the 6 from the 16. As we have taken 1 ft. from the 16, we shall have more to add to the 14. By examining operation 1, it may not at first appear from what place we are to obtain the 40 r. which are to be added to the 8, as there are no fur. in the minuend. These really come from 1 of the miles in the 24, as will be seen in operation 2... We take 1 m. from the 24, which is equal to 8 fur., and set 7 of the fur. in the place of furlongs, and for the other furlong, we add 40 r. to the 8 r., which makes all the upper numbers the largest. Had we not actually taken I of the miles from the 24, we should have supposed 40 r. added to the 8, and 1 fur. added to the 3 fur.; in that case, we should have supposed 8 fur. added in the place of fur., and 1 m. added to the 21, to balance the 8 fur. By the 1st operation, we suppose 9 fur. added to the minuend, and 1 m. and 1 fur, added to the subtrahend, but there being no difference between 1 m. and 1 fur., and 9 fur. the remainder of miles, furlongs and rods, must be the difference between the given numbers of those denominations.
* Instead of adding 1 to the next under number, it would be equally correct to take 1 from the upper, but it is not always se convenient.
The whole minuend in operation 2, is equal to the minuend in the 1st operation, for what we took from the miles and feet, we added to the other denominations, and as we find the same answer by both operations, it appears that we do essentially the same thing whether we suppose numbers added, or actually remove them from higher to lower denominations in the minuend.
13. Two roads leading to the same place, measure forty-five miles, six furlongs and thirteen feet; the shortest one is nineteen miles, two furlongs and fifteen feet; what is the length of the longest road, and how much longer is one than the other? Ans. Longest road, 26 m, 3 fur. 39 r. 144 ft. * R Difference, 7 m. 1 fur. 38 r. 16 ft.
JNote 1. When one of two numbers, which are to be added together, contains a fraction, the fraction must be set at the right hand of the sum of the other numbers. When one number is to be taken from another, which contains a fraction, the fraction must be brought down without being altered.
14. How much greater is the circumference of a circle than the diameter, if the former be 16 ft. and the latter 5 ft. 1 in.” Ans. 10 ft. 11 in.
15. Two miles and five rods of fence will enclose the whole of a certain farm, and one mile, one furlong and ten feet, one half of it. How much more fence will be required to enclose the whole than one half? - Ans. 7 fur. 4 r. 64 ft.
16. How much more land in a field, which contains 25 A. 3 R. 37 r. 160 ft. 143 in., than in another, which contains 17 A. 179 ft. 2 *
17. Subtract 1 A. 1 R. 1 r. 1 ft. 1 in. from 2 A. Rem. 2 R. 38 r. 2703 ft. 143 in. 18. A pond measuring fifteen acres, one hundred and fifteen feet, is situated in the middle of a farm, containing one hundred acres and fourteen rods. How much land in said farm exclusive of that occupied by the pond? Ans. 85 A. 13 r. 157+ ft.
19. If a man sell 6 C. 127 ft. 1006 in. from a pile of wood measuring 11 C. 120 ft. 1009 in., how much will remain Ż
20. A man agrees to chop 15 C. 11 ft. of wood; but doing his work better than was expected, his employer pays him for chopping 17 C.; for how much is he paid, that he did not chop 2 Ans. 1 C. 117 ft. 21. A man cut 49 ft. of timber from a square stick, which measured 2 T. 6 ft. ; how much remained ? Ans. 1 T. 7 ft. 22. How many more solid feet in a cask, which measures 43 ft., than in one measuring 42 ft. 75 in. ” - Ans. 1653 in.
Y. mo. we d. h... m. s. 131 00 0 0 00 00 51 65 2 3 4 20 40 00
65 10 0 2 3 20 51
24. From 15 Y. subtract 3 w. 21 h. 25 s. Rem. 14 Y. 12 mo. 6 d. 2 h. 59 m. 35 s. 25. A note was given in 1822, April 21, and paid *826, March 15; how long was it standing 2
Y. mo, d. 1826 - 3 15 Explanation.—In this 1822 4 21. example, 30 days are * ---- reckoned a month: this Ans. , 3.10 × 24 is not strictly true in all
cases, but sufficiently
mear for most practical purposes. When 30 days are reckoned a month, we must carry for the number of Calendar months in a year, which is 12. o
26. How long was that note on interest, which was given in 1821, December 29, and paid 1824, June 227 Ans. 2 Y. 5 mo. 23 d.
JNote2. In the place of months, set down the number of any given month, counting from the beginning of the year, calling January 1, February 2, March 3, &e.
27. If a note were given on interest, 1826, January 3, and paid August 1, of the same year; how long was said note on interest? Ans. 6 mo. 28 d.
Add the remainder and subtrahend together, if the ūuestion has been correctly performed, the sum will equal the remainder, for the difference of two numbers added to the less, will give a sum equal to the greater.
cir. S. o ' " 5 1 29 40 5 1 3 2 12 57 48