« PreviousContinue »
RULE. Link the several prices and place their differences as before ; then,
As the sum of the differences,
EXAMPLES. 1. How much sugar at 10 cents, 12 cents, and 15 cents, per Ib. will be required to make a mixture of 20 lb. worth 13 cents per lb. ?
2 As 8:20 :: 2 : 5lb.at 10 cts. 13 12
8:20 :: 4 :,10 lb. at 15 cts. 15.
3+1 4 8: 20 :: 2 : 5 lb. at 12 cts.
8 Sum of differences. 2. A brewer has three sorts of beer, viz. at 10d. 8d. and 6d. per gallon; how much of each sort must he take to make a mixture of 30 gallons, worth 7d. per gallon?
Ans. 5 gals. at 10d., 5 gals. at 8d. and 20 gals. at 6d. 3. A goldsmith has gold of 15, 17, 20, and 22 carats fine, and would melt together of each of these so much, as to make a mass of 40 oz. of 18 carats-fine ; how much of each sort is necessary?
16 oz. of 15 carats, 8 oz. of 17 carats, 4 oz.
of 20 carats, and 12 oz. of 22 carats fine. 4. How many gallons of water must be mixed with wine,. at 4s. per gallon, so as to fill a vessel of 80 gallons, that may be afforded at 2s. 9d. per gallon?
Ans. 25 gallons of water, with 55 of wine,
Fosition is a rule for finding an unknown number, by one or more supposed numbers. It is divided into two parts, single and double. SINGLE POSITION.
. * Single Position teaches to resolve such questions as require only one supposition.
RULE. Suppose any number to be the true one, and proceed with it agreeably to the tenor of the question; then,
As the result of the operations
PROOF. Work with the answer according to the tenor of the question, and the result inust equal the given number.
EXAMPLES. 1. A, B, and C, bought a quantity of wine for 340 dollars, of which sum A. paid three times more than B, and B. four times more than C; how much did each pay ?
A. paid 240
B. paid 80 Ans.
C. paid 20
340 Proof. As 51 : 36 :: 340 : 240 suin paid by Å. 2. A person after spending and of his money had 60 L. left; how much had hę at first ?
Ans. 144L: 3. What number of dollars is that, of which the }, and , make 74 ?
Ans. 120. 4. A person having about him a certain number of crowns, said, if a third, a fourch, and a sixth of them were added together, the sum would be 45; how many crowns had he?
Ans. 60. 5. What is the age of a person who says, that, if of the
а years I have lived, be multiplied by 7, and of them be added to the product, the sum will be 292? Ans. 60 years.
6. A schoolmaster being asked how many scholars he had, answered, if to double the number I add }, }, and of them, I shall have 333; how many had he? Ans. 103.
7. A certain sum of money is to be divided among 4 persons, in such a manner that the first shall have of it, the second ļ, the third &, and the fourth the remainder, which is 28 dollars ; what is the sum ?
Ans. 112 dols.
8. What sum, at 6 per cent. per annuin, will amount to 860 L. in 12 years ?
Ans. 500L. DOUBLE POSITION. Double Position teaches to find the true numbers by making use of two supposed numbers.
RULE. Suppose two numbers, and work with each agreeably to the tenor of the question, noting the eribrs of the results : multiply the errors of each operation into the supposed number of the other; then,
If the errors be alike, i. e. both too much, or both too little, take their difference for a divisor, and the difference of the products for a dividend: but if the errors be unlike, take their sum for a divisor, and the sum of the products for a dividend.
EXAMPLES. *. A, B, and C, would divide 80 dollars among them in such a manner, that B. may have 5 dollars more than A. and C. 10 dollars more than B. required the share of each? 8
$ Suppose A's share 10
Suppose A's share 15
65 80 - 50 = 30 error too little.. 80 -65 = 15 error too little. Errors.
30 x 15 = 450
Ans. B 25 15 diff. of er. 15)300 diff. of prod. C 35
20 A's share.
80 2. D, E, and F, would divide 100L. among them, so as that E. may have 3 L. more than D., and F. 4 L. more than E.; what is the share of each?
Ans. D's share 30L. E's 33L. F's 37 L.
3. A, B, and C, owe 1000 L. of which B, is to pay 100 L. more than A, and C. is to pay as much as both A. and B.; how much is each man's share of the debt?
Ans. A's 'share is 200L. B's 300 L. and C's 500L. 4. Bought linen at 4s. per yard, and muslin at 2s. per yard; the number of yards of both was 8, and the whole cost 20$.: how many yards were there of each?
Ans. 2 yards of linen, and 6 yards of muslin. 5. The head of a certain fish is 9 inches long; its tail is as long as its head and half of its body; and the length of its body is equal to the length of its head and tail: what is the whole length ?
Ans. 6 feet. 6. A labourer hired for 40 days upon this condition, that he should receive 20 cents for every day he wrought, and should forfeit 10 cents for every day he was idle; at settlement he received 5 dollars. "How many days did he work, and how many days was he idle ?
Ans. Wrought 30 days, idle 10. 7. A father dying, left to his three sons A, B, and C, his estate in money, dividing it as follows, viz. to A. he gave half the estate, wanting 44 L. to B. he gave a third of it, and 14 L. over; and to C. he gave the remainder, which was 82 L. less than the share of B. - What was the whole sum left, and what was each son's share ?
The sum left was 588 L. of which A. had Ans.
250 L. B. 210L, and C. 128 L. 3. Two persons, A. and B. have both the same income; A. saves one fifth of his every year; but B. by spending 150 dollars per annum more than A, at the end of 8 years finds himself 400 dollars in debt: What is their income, and what does each spend per annnm ?
A. spends 400 dols. and B. 550.
OR THE RAISING OF POWERS.
A power is the product arising from multiplying any given number into itself continually a certain number of times; thus,
INVOLUTION OR THE RAISING OF POWERS.
2x2 = 4 the second power or square of 2. 2 X 2 X 2 = 8 the third power or cube of 2. 2 X 2 X 2 X 2 16 the fourth power of 2, &c. The number denoting the power is called the index or exponent of that power.
If two or more powers are multiplied together, their product is that power whose index,is the sum of the expo nents of the factors; thus,
2 x2 = 4 the square of 2; 4 X 4 = 16 = 4th power of 2; and 16 x 16 = 256 = 8th power of 2, &c.
1.11 11 1 11 11 11
11 2 | 41 8 161 321 641
128 256 512 3 | 9| 271 811 2431 7291 2187
65611 19683 4 1161 641 256) 10241 40961 16384 655361 262144 5 125 125) 625| 3125/ 15625 781951 3906251 1953125 6 136/216|1296) 77761 466561 279936 1679616] 10077696 7 149|343|2401|16807|1176491 823543) 5764801| 40353607 8 164151214096|32768|26214412097152|16777216|134217728 9 181729|6561|59049|531441 4782969 43046721|387420489
EXAMPLES. 1. What is the cube or third power of 4 ?
Ans. 64. 4 X 4 X 4 = 64. 2. What is the fifth power of 7 ?
Ans. 16807. 3. What is the cube or third power of 35? Ans. 42875. 4. - What is the fourth power of ?
736 5. What is the cube or third power of .13?
Ans. .002197 6. What is the sixth power of 5.03?