« ΠροηγούμενηΣυνέχεια »
If £100 in twelve months gain £6, in what time will £75 gain £3 7s. 6d.?
Ans. 9 months. If £75 in 9 months gain £3 78. 6d., what will £100 gain in 12 months ?
Ans. £6. If a regiment of soldiers, consisting of 1360 men, consume 351 quarters of wheat in 108 days, how much will 11232 soldiers consume in 56 days ?
Ans. 1503 35 quarters. If 50 men can do a piece of work in 100 days, working 8 hours per day, in what time will 120 do it, working 6 hours per day?
Ans. 55, days. Note.—The two statements are in inverse proportion ; in one, more requires less, and in the other, less requires more.
The pupil having now learned to state with readiness sums of five given numbers to find a sixth, and being told that inverse statements require the right, and direct statements the left, figures, to be taken for a divisor, will, I trust, fully understand the statements of the few sums it will be necessary for me to insert in the Chain Rule; and for the purpose let us take, as our example, the last sum of five numbers, and add a couple more terms to it, by giving dimensions to the work supposed to be done, and that which is to do.
If 50 men can build a wall 180 yards long in 100 days, working 8 hours per day, in what time will 120 build a wall 205 yards long, working 6 hours per day? Supposition.
Demand. 50 men : 100 days :: 120 men * 8 hours :
:: 6 hours * * 180 yards :
:: 205 yards.
The first and second statements, as before described, are inverse; the third is direct, as it will require longer to build 205 yards than 180 ; the answer is 6371 days.
We can increase it two terms more, by taking the height of the wall : thus
If 50 men can build a wall 180 yards long, and 7 feet high, in 100 days, working 8 hours per day, in what time will 120 men build a wall 205 yards long and 10 feet high, working 6 hours per day? Supposition.
Demand. 50 men : 100 days :: 120 men. 8 hours :
:: 6 hours.* * 180 yards :
:: 205 yards. * 7 feet :
:: 10 feet.
The last statement is direct, as in proportion it must take longer to build ten than seven feet. Ans. 90966
If we add two more terms, the principle is still the
If 50 men can build a wall 180 yards long, 7 feet high, and 13 inches thick, in 100 days, working 8 hours per day, in what time can 120 men build a wall 205 yards long, 10 feet high, and 10 inches thick, working 6 hours per day ?
The last state- 50 men : 100 da. :: 120 men. ment is direct pro
8 hrs. :
6 hrs. *180 yds. : : : 205 yds. portion; as less
10 ft. requires less; it *13 in.
10 in. will take less time to build a ten inch wall, than a thirteen inch one.
Having now shewn the pupil how in one statement eleven terms are easily worked, I wish to remark the expedition that is acquired by this rule. The above sum worked by proportion, as it must be if not done by the chain rule, would require five statements, three direct
and two inverse, the fourth proportional of one becoming the centre term of the next; and each centre term having a fraction, the work would necessarily be complicated. And I wish here to impress upon the pupil's mind that Arithmetic cannot be kept too simple; for a momentary dulness, or a stray thought crossing the mind, will often lead the cleverest calculator into error. Tradesmen knowing this have adopted a rule, which, from being almost always in use, and from the readiness with which it can be applied to almost every calculation, is called Practice.
PRACTICE. For Practice there can be no settled rule, as every person that has goods or any other commodity calculates their amount in the shortest possible manner; and the more extensive the person's business is, the less time he can afford to lose, and the shorter his method is likely to be. And I would here take an opportunity to advise the pupil to follow an example so good, and use all diligence to attain the knowledge useful to his station in life, and never to neglect the duties of that station to acquire knowledge which perhaps can never be useful to him.
But to shew the great dispatch that this rule, if I may so call it, enables us to use in calculation, we will work the following sum.
Bought 267 stone of oatmeal at 2s. per stone, how much did it cost? The person knowing that part of
267 Arithmetic already taught; would
2 proceed as I have done in the sum 2,0) 53,4 on the right, viz. multiply it by 2,
£26. 14s. and divide the product by 20; but the person knowing Practice, at once knows that Os.
is the io of a pound, and that to divide by ten we have only to cut off the right-hand figure, thus; 26,7. He then has £26, and he knows that the seven cut off is seven stone; at 2s. is fourteen shillings.
If such is the expedition that is gained in calculation by this Rule, I am certain the pupil will, to make himself master of a rule so useful, study with attention the fol. lowing tables, as an hour or two will make him sufficiently acquainted with them.
When the Price is less than a Penny.
Here, if the price was ld.,
1 66 found by the example to be 12 ) 198d. 16s. 6d.
The pupil will observe that the farthing may be taken either as the quarter of the first or penny line, or the half of the second or half
2 } 132
Bought 149 yards of ribbon at 4d. per yard, what did it cost?
Ans. 3s. 14d. Bought 168 yards of rope at d. per yard, what did it cost?
Ans. 7's. What will 240 red herrings cost at id. each?
When the Price is less than a Shilling. What will 2004 lbs. of lard cost at 3 d. per pound? Three pence is the į of a shilling, and the first line 501 is
2004 the amount in shillings at 3d.
501 per pound. In the next line i = 833 d. is the of three-pence,
2,0) 58,43 and the second line 831 shil
£29 4s. 6d. lings is the amount at įd. per pound. Both sums added together, and the total divided by twenty, is the answer in pounds and shillings.
What will 3257 ounces of coffee cost at 4d. per ounce?
Ans. £54. 55. 8d. Bought 5752 yards of rope at 4 d., what is the amount ?
Ans. £107. 17s. What is the price of 5272 lbs. of bees-wax at 9d.
Ans. £197 14s. Bought 7921 loaves of bread at 10 d. per loaf, what did it amount to ?
Ans. £326. lls. 6 d. What is the price of 3752 lbs. of meat at 4 d. per lb. ?
Ans. £70. 7s.
per lb. ?