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il : 16. of pure siber in the mass.
expel 4 lb. of water; required the quantity of gold and silver in the said mass ?
in Hence is solved the famous question with respect to the crown of Hiero, King of Syracuse.-He had ordered a crown to be made of pure gold : but suspecting that the founder had mixed silver or copper with the gold, he desired Archimedes. to examine, and tell him how much gold and how much alloy was in the crown. Archimedes by immersing the crown, and the like quantities of pure gold and silver, in water seva erally, and observing the respective quantities of water expelled, found an answer to the question.
POSITION. Position is a rule that by false or supposed numbers taken at pleasure, discovers the true ones required, it is divided into two parts single and double.
SINGLE POSITION. Single position by supposing one number taken at adventure, and working with it, according to the nature of the question, as if it were the true number, the true nuinber is discovered.
RULE-As the result of the position : is to the position :: so is the given number : to the number required.
EXAMPLES 1. What number is
being increased by 1, , and 1 of itself, the sum/shall be 125 ? Ans. 60.
2. Five-sixths of a certain number exceed three-fourths by 5; what is the number?
Ans. 60 3. One being asked his age, replied, if of the years I have lived be multiplied by 7, and of this product be divided by 3, the quotient will be 20: what was his age ?
Ans. 4. A General after sending out a foraging and fof his meng had yet 700 remaining: what number had he in command ?
Ans. 4200 DP A gentleman distributed 78 cents among a number of poor people, consisting of men, women and children ; to each man he gave 6 cents, to each woman 4, and to each child 2: moreover there were twice as many women as men, and thrice as many children as women : How many were there of each?
Ans. 3 men, 6 women, and 18 children. 6. A Father divided his fortune among his sons, giving A. 7, as often as B. 4 ; to C. he gave as often 2, as B. 5, and yet
the dividend of C. came out 21661. 7s, 6d. what was the val. ue of the whole fortune ?
Ans. 170601. 45. Od. 7. There is a cistern with 3 unequal pipes, containing 600 gallons of water; and if the greatest pipe be opened the cistern will be empty in one hour ; if the second pipe be opened it will be empty in two hours; if the third be opened it will be empty in three hours : required the time it will take to em. ty if all run together?
min. v 8. Peter drinks a barrel of beer containing 32 gallons in 24 days, and Charles when he goes about it, does it in 16 days ; now if they should drink together, in what time will they make an end of it ?
Ans. 9 days 14 hours. 9. Suppose I lend at interest a certain sum of money, at the rate of 8 per cent. per annum, simple interest, and that at the end of 10 years, I received both for principal and interest $1539. I demand the principal sum lent ?
DOUBLE POSITION. Double Position, is by making use of two supposed numbers, and if both prove false (as generally happens) proceed with them and their errors according to the following
Rule.--Place each error against its respective position, and multiply them cross-wise. If the errors are alike, that is, both greater (marked+) or both less, (marked-) than the given number, take their difference for a divisor, and the difference of their products for a dividend. But if unlike, that is, one too much, and the other too little, then take their sum for a divisor, and the sum of their products for a dividend, the quotient will be the answer.
EXAMPLES. 1. What number is that which being increased by 30 is equal to 6 times the same lessened by 45 ?
Ans. 15. 2. Three persons discoursed together concerning their ages; says A. I am 20 years of age ; says B. I am as old as A. and half C ; and says C. I am as old as you both: I demand each of their ages ? Ans. A. was 20, B. 60, C. 80 years old.
3. A. B. and C. found a bag cortaining a certain number of dollars, and when they divided the booty, A and B's share amounted to $47 ; B. and C's to $88 ; and A. and C's to $71 required the amount of the sum found, and each man's share thereof? Ans. found $103, A's share $15, B's $32, C's $56.
4. A certain man having driven his swine to market, viz. hogs, sows, and pigs, received for them all $250, being paid for every hog $41, for every sow $4, and for every pig 50 cents-there were as many hogs as sows, and for every sow
there were three pigs : I demand how many there were of each?
Ans. 25 hogs, 25 sotus, and 75 pigs. 5. A. and B. settling accounts found, that if 61. were ad. ded to şof A's bill, and the same sum taken from of B's, the sum would be of the remainder ; and that the sum and remainder added together made 721. I demand each person's bill?
Answer, A's 411. 8s.-B's 66). 128. 6. A son asked his father how old he was, his father re. plied, your age is now } of mine ; but 4 years ago, your age was only 1 of what mine is now: what were their ages ?
Ans. 70 years the futher, 14 the sons. 7. A gentleman finding several beggars at his door, gave to each 4 cents, and had 16 left: but if he had given to each 6 cents, he would have wanted 12: how many beggars were there?
Answer 14 8. A merchant has 100 bushels of wheat and barley, which cost him 461. 108. the wheat stands him in 12s. per bushel, and the barley 68. required the number of bushels he has of each?
Ans. 55 of wheat, 45 of barley. 9. A gentleman had two horses, Chesnut and Swift, and a saddle worth 501. which set on the back of Chesnut, makes his value double that of Swift: but the saddle set on the back of Swift, makes his value triple that of Chesnut; what was the value of each horse ? Ans. Chesnut 301. Swift 401.
10. There was a fish caught whose head was? - inches long, its tail was as long as the head and half the body, and the body was exactly the length of both head and tail : how long was the whole fish ?
Ans. 62 inches. 11. There is an army to which if you add }, } and 4 of itself and take away 5000, the sum tot al will be 100000 : what is the number of the whole army?
Ans. 50400 12. A. and B. have the same income ; B. saves } of his ; but B. by spending $75 per annum more than A. at the end of 8 years finds himself $100 in debt: What is their income, and what does each spend per annum ? Ans. Their income is $500 per ann. A. spends $4371.
B. spends $5124 per annum. 13. A. and B. laid out equal sums of money in trade; A: gained a sum equal to of his stock, and B. lost $5624, then A's money was double that of B's. What did each lay out ?
Ans. $1500. 14. If to my months you should add half this sum,
And one-eighth more, and then should subtract one,
Ans, 22 years, 8 months.
15. When first the marriage knot was tied
Betwixt my wife and me,
As three times three does three ,
We man and wife had been,
As eight is to sixteen.
Your bride no more than just fifteen. 16. A man overtaking a maid driving a flock of geese, said to her, how do you do, sweetheart! Where are you going with these 100 geese ? No Sir, said she, I have not 100 ; but if I had as many, half as many, and seven geese and a half, I should have 100 : How many had she? Ans. 37.
17. A surly old fellow being demanded the ages of his 4 sons, answered, you may go and look ; but if you must needs know, my first son was born just 1 year after I was married to his mother, who after his birth, lived 5 years, and then died in child-bed with my second ; 4 years after that I married again, and within 2 years had my third and fourth sons at a birth ; the sum of whose ages together is now equal to that of the eldest : Idemand their several ages? Ans. 22 years 1st son, 17 years 2d, 11 years each the 3d
and 4th son.
Tho' now reduc'd, a sum receiv'd,
And by your work the sum produce. Ans. 1411. 19. A young gentleman, at the age of 21 years, was told by his guardian, that his fortune consisted in cash, to the amount of $35129, and that his father died when he was but 10 years old, and the money your father left, said the guardian, I have allowed you 6 per cent. per annum for simple interest, only I have deducted $300 per annum for your education, &c. What was the son's fortune that was left by his father?
ACON | Roots,
- 7th pow.
2187 4 | 16
256 1024 4096 16384 5 | 25 | 125
3125 15625 78125 6 36 216 | 1296 7776 46656 279936 7 | 49 | 343 | 2401 16807 | 117649 823543 8 64 5.2 4096 | 32768 262144 2097152
81 729 6561 59049 | 531441 | 4782969
SQUARE ROOT. Extracting the Square Roet is to find out such a number as being multiplied into itself, the product will be equal to the given number.
RULE.- First, Point the given number, beginning at the units place, then at the hundreds, and so upon every second figure both ways, if there be an odd figure in the decimal, annex a cipher.
Secondly, Find the greatest square in the left hand period, and place the root in the quotient, subtract the square from the period, and to the remainder bring down the next period for a dividend.
Thirdly, Double the quotient, or add the last figure to the last divisor, which will be the new divisor ; and try how of. ten it will be contained in the dividend, putting the trial figure for the unit's place in the divisor, and when found nearest, put it in the quotient, then multiply and subtract the product from the dividend, bring down the next period to the remainder (if there be any more, or add ciphers) and proceed as before.