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1. $ d 2. Divide 432 12 1. by 3. 3. Divide 507 3 5 by 4. 4. Divide 632 7 64 by 5. 5. Divide 690 14 31 by 6. 6. Divide 705 10 2 by 7. 7. Divdie 760 5 6 by 8. 8. Divide 761 5 73 9. 9. Divide 829 17 10 by 10. 10. Divide 937 8 8. by 11. 11. Divide 1145 11 41 by 12.
2 8 a
1. If the divisor exceed 12, find what simple numbers, inultiplied together, will produce it, and divide by them separately, as in Simple Division, as below.
1. What is Cheese per cwt, if 16 cwt cost 251 148 8d?
d 14 8
II. If the divisor cannot be produced by the multiplication of small numbers, divide by the whole divisor at once, after the manner of Long Division, as follows.
k. Divide 17 Ib 9 oz 0 dwts 2 gr by 7.
Ans. 2 lb 6 oz 8 dwts 14 gr. 2. Divide 17 lb 5 oz 2 dr 1 scr 4 gr by 12.
Ans. 1 lb 5 oz 3 dr 1 scr 12 gr. 3. Divide 178 cwt 3 qrs 14 lb by 53. Ans. 3 cwt I qr 14 lb. 4. Divide 144 mi 4 für 2 po 1 yd 2 ft 0 in by 39. 3
Ans. 3 mi 5 fur 26 po 0 yds 2 ft 8 in. 5. Divide 534 yds 2 qrs 2 na by 47. Ans. Il yds 1 qr 2 na. 6. Divide 71 ac 1 ro 33 po by 51. Ans. I ac 2 ro 3 po. 7. Divide 7 10 0 hhds 47 gal 7 pi by 65 Ans. 27 gal 7 pi, 8. Divide 387 la 9 qr by 72.
Ans. 5 la 3 qrs 7 bu. 9. Divide 206 mo 4 da by 26. Ans 7 mo 3 we 5 ds,
THE GOLDEN RULE, OR RULE OF THREE.
THE RULE OF THREE teaches how to find a fourth proportional to three numbers given : for which reason it is sometimes called the Rule of Proportion. It is called the Rule of Three, because three terms or numbers are given, to find a fourth. And because of its great and extensive usefulness, it is often called the Golden Rule. This Rule is usually considered as of two kinds, namely, Direct, and Inverse.
The Rule of Three Direct is that in which more requires more, or less requires less. As in this ; if 3 men dig 21 yards of trench in a certain time, how much will 6 men dig in the same time? Here more requires more, that is, 6 men, which are more than 3 men, will also perform more work in the same time. Or when it is thus : if 6 men dig 42 yards. how much will 3 men dig in the same time? Here then, less requires less, or 3 men will perform proportionably less work than 6 men, in the same time. In both these cases then, the Rule, or the Proportion, is Direct; and the stating must be
thus, As 3:21 :: 6 : 42,
or thus, As 6 : 42 :: 3:21. But the Rule of Three Inverse, is when more requires less, or less requires more. As in this : įf 3 men dig a certain quantity of trench in 14 hours, in how many hours will 6 men dig the like quantity ? Here it is evident that 6 men, being more than 3, will perform an equal quantity of work in less time or fewer hours. Or thus : if 5 men perform a certain quantity of work in 7 hours, in how many hours will 3 men perform the same? Here less requires more, for 3 men will take more hours than 6 to perform the same work. In both these cases then the Rule, or the Proportion, is Inverse ; and the stating must be
thus, As 6 : 14 :: 3: 7,
or thus, As 3: 7:;6: 14. And in all these statings, the fourth term is found, by multiplying the 2d and 3d terms together, and dividing the product by the 1st term.
Of the three given numbers ; two of them contain the supposition, and the third a demand. And for stating and working questions of these kinds observe the following general Rule :
STATE the question, by setting down in a straight line the three given numbers, in the following manner, viz. so that the 2d term be that number of supposition which is of the same kind that the answer or fourth term is to be; making the other number of supposition the 1st term, and the demanding number the 3d term, when the question is in direct proportion; but contrariwise, the other number of supposition the 3d term, and the demanding number the 1st term, when the question has inverse proportion.
Then, in both cases, inultiply the 2d and 3d terms together, and divide the product by the ist, which will give the answer, or 4th term sought, viz. of the same denomination as the second term.
Note, If the first and third terms consist of different denominations, reduce them both to the same : and if the second term be a compound number, it is mostly convenient to reduce it to the lowest denomination mentioned.--If, after di. vision, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last denomination. Proceed in the same manner with all the remainders, till they be reduced to the lowest denomination which the second admits of, and the several quotients taken together will be the answer required.
Note also, The reason for the foregoing Rules will appear, when we come to treat of the nature of proportions. Someumes iwo or more statings are necessary, which may always be known from the nature of the question.
1. If 8 yards of Cloth cost 12 48, what will 96 yards cost?
yds I s yds 1 s
Ex. 2. An engineer having raised 100 yards of a certain work in 24 days with 5 men; how many men must he employ to finish a like quantity of work in 15 days?
ds men ds men
15) 120 (8 Answer.
3. What will 72 yards of cloth cost, at the rate of 9 yards for 5l 128 ?
Ans 441 68. 4. A person's annual income being 1461 ; how much is that per day?
Aps, 8s 3. If 3 paces or common steps of a certain person be equal to 2 yards, how many yards will 160 of his paces make ?
Ans. 106 yds 2 ft. 6. What length must be cut off a board, that is 9 inches broad, to make a square foot, or as much as 12 inches in length and 12 in breadth contains ?
Ans 16 inches. 7. If 750 men require 22500 rations of bread for a month; how many rations will a garrison of 1200 men require ?
Ans. 36000. 8. If 7 cwt I qr of sugar cost 262 108 4d; what will be the price of 43 cwt 2 qrs?
Ans 1591 28. 9. The clothing of a regiment of foot of 750 men amounting to 28311 58 ; what will the clothing of a body of 3500 men amount to ?
Ans. 132121 108. 10. How many yards of matting, that is 3 ft broad, will cover a floor that is 27 feet long and 20 feet broad?
Ans. 60 yards. 11. What is the value of 6 bushels of coals, at the rate of 12 14s 6d the chaldron ?
Ans. 58 9d, 12. If 6352 stones of 3 feet long complete a certain quan. lity of walling; how many stones of 2 feet long will raise a like quantity ?
Ans 9528. 13. What must be given for a piece of silver weighing 73 lb 5 oz 15 dwts, at the rate of 58 9d per ounce ?
Ans. 2531 108 0 d. 14. A garrison of 536 men having provision for 12 months; how long
will those provisions last, if the garrison be increased to 1124 men ?
Ans 174 days and it 15. What will be the tax upon 7632 158, at the rate of 38 6d per pound sterling?
Ans. 1334 138 11d.