10. If cloth, sold at 4 s. per yard, be 10 £. per cent. profit, what shall I gain or lose per cent. if sold at 3 s. 6 d. per yard? Ans. 3££ loss. 11. A gentleman bought 10 tons of steel for 200 pounds, the freight and duties came to 25 pounds, and his own charges to 8 pounds 6 shillings, 8 pence; how must he sell it per pound to gain 20 pounds per cent. by it? Ans. 1£. 3 d. per pound. 12. If I sell a gallon of brandy for $1.50 c. and by the sale 'lose 12 per cent. ; what shall I gain or lose per cent. if I sell 4 gallons of the same brandy for $6.75 c. ? Ans. I per cent. loss. SECTION X X V. INVOLUTION AND EVOLUTION. INVOLUTION is the raising of Powers from any given number—which is called a root. A Power is a quantity produced by multiplying any number into itself a certain number of times. Thus, 2=2, is the root, or 1st power of 2. 3+3=9, is the 2nd power, or square of 3. 4+4+4=64 is the 3rd power, or cube of 4. Evolution, is the opposite of Involution, or the extracting the root of any given power. The Root of any power is the number which being multiplied into itself a certain number of times will produce that power. Thus, 2 is the square root of 4. 2 is the fourth root of 16. 1. What is the second power of 45? Ans. 2025. 2. What is the square of s? Ans. $ 3. Involve 8 to the 3rd power ? Ans. 512. 4. What is the fourth power of 1. Ans. 5. What is the second power of 351? Ans. 123201. 6. What is the third power of 19. Ans. 1900. 1 16 6359 SECTION X X VI, SQUARE ROOT. 1. What is the square root of 2035 ? 2. What is the square root of 16? Ans. 45. root of 875 501? 3. What is the square Ans. s 4. What is the square root of 721? Ans. 81. 5. Extract the second root of 302500. Ans. 550. 6. What is the square root of 550 ? Ans. 23,4520788. 7. What is the square root of 173 ? Ans. 4,168333. 8. Extract the second root of ? Ans. 0.645497. 9. What is the square root of 000729 ? Ans. 027. 10. Extract the square root of 902500. Ans. 950. 11. A certain square pavement contains 889249 square stones, all of the same size; how many are contained in one of its sides? Ans. 943. 12. Suppose an army of 20449 soldiers, are to be drawn up in an exactly square column; how many men will be required to occupy one side of the square ? Ans. 143. 13. The wall of a town which is besieged is 18 feet high, and is surrounded by a ditch 28 feet in width; what must be the length of the scaling ladders, in order to have them reach from the outside of the ditch to the top of the wall ? Ans. 33.2+ CUBE ROOT. 1. What is the cube root of 64 ? Ans. 4. 2. What is the third root of 2803221 ? Ans. 141. 3. Extract the cube root of 262144. Ans. 64. 4. What is the cube root of 224755712? Ans, 608. 5. What is the cube root of 5737? Ans. 81. 6. Extract the third root of 13 ? Ans. 1.1447. 7. What is the cube root of .0001357 ? Ans. .05138+. 8. What is the cube root of 1321 ? Ans. 2.3908. 9. There is a stone which contains 68921 cubic inches ; what is the length of one of its sides? Ans. 41 in. 10. Suppose a stone of a cubic form contains 474552 solid inches ; what is the superficial contents of one of its sides? Ans. 6084 inches. 11. The diameter of a bushel measure being 184 inches, and the height 8 inches; what will be the length of the side of a cubic box which shall contain the same quantity ? Ans. 12.908. 12. What is the cube root of 200 ? Ans. 5.848035. VARIOUS ROOTS. 1. What is the 3d root of 2 ? 2. What is the 4th root of 2 ? Ans. 1.259921. 3. What is the 4th root of 97.41 ? Ans. 3.1415999. 4. What is the 5th root of 2 ? Ans. 1.148699. 5. Extract the 6th root of 21035.8. Ans. 5.254037. 6. What is the 6th root of 2 ? Ans. 1.122462. 7. What is the 7th root of 21035.8 ? Ans. 4.145392. 8. What is the eighth root of 21035.8 ? Ans. 3.470323. 9. What is the 9th root of 2? Ans. 1.0580059. SECTION XXVII. a ARITHMETICAL PROGRESSION. 1. The first term of a series is 1, the last term 23, and the number of terms 14. Ascertain the sum of the series. Ans. 210. 2. If the first term is 2, the last term 51, and the number of terms 18, what is the sum of the series ? Ans. 477. 3. One hundred apples are a yard apart in a straight line; the first is a yard from the basket, into which they are to be put one at a time. How far must you travel to pick them up and carry them one by one to the basket ? Ans. 5 mi. 5 fur. 36 r. 2 yds. 4. How many strokes will a clock strike in 24 hours, if it is made so as to strike the number of hours from 1 to 24? Ans. 300. 5. The first term is 3, the last term 58, and the number of terms 12. What is the common difference? What is the sum of the series? 1. Ans. 5. 2. Ans. 366. 6. If the extremes are 2 and 53, and the common difference 3, what is the number of terms ? Ans. 18. GEOMETRICAL PROGRESSION. 1. If the first term of a series in geometrical progression is 1, the last term 2187, and the ratio 3, what is the sum of the series? Ans. 3280. 2. 1 and 65536 are extremes; the ratio is 4; what is the sum of the series? Ans. 87381. 3. If I buy a horse and pay 1 cent for the first nail in his shoes, 2 for the second, 4 for the third, &c., and there are 32 nails in his shoes, how much will he cost? Ans. $9265100944259.20. 4. What is the 12th term of a series, whose first term is 3, and the ratio the same? Ans. 531441. 5. A agreed to work for B 40 years, on condition of receiving one kernel of corn for the first year, ten for the second, one hundred for the third, and so on. If one thousand kernels make a pint, and the corn was sold for half a dollar a bushel, how much did his wages amount to? Ans. $8,680,555,555,555,555,555,555,555,555,555,555,555,555. 6. A man rode 252 miles. The first day he went 4, the next 8, and so on. How far did he ride the last day? Ans. 128 miles. 7. A owes B 252 dollars. The first payment is to be 4 dollars, the last 128. In what ratio will the payments exceed each other? Ans. 2 (i. e. double.) (For other sums in Progression, see Miscellaneous Examples.) SECTION X X VIII. SINGLE POSITION. 1. A and B found a bag of money ; A said the half, third and fourth of it made 130 £., and if B could tell how much there was, he should have all, otherwise none; how much was in the bag? Ans. 120 €. 2. A lent B a sum of money, to be paid at four payments; when three were made, and the fourth demanded, B would pay no more, except A would tell what was already paid; A said the first payment was a fourth, the second a fifth, and the third a sixth of the sum first lent, and all made 74 £.; what was the sum lent ? Ans. 120 €. 3. A person after spending } and 4 of his money, has yet remaining 60 dollars ; what had he at first ? Ans. $144. 4. A gentleman had a certain number of dollars in his purse; the sum of the third, fourth, and sixth part of them made 54; how many were in the purse ? Ans. 72. 5. Seven eighths of a certain number exceeds four fifths by 6; what is that number? Ans. 80. 6. A gentleman bought a chaise, horse and harness, for 500 dollars; the horse cost 7 more than the harness, and the chaise $ more than the horse ; what was the price of each? (Harness $127.66937. Ans. Horse 159.57434. Chaise 212.7651 DOUBLE POSITION. 1. A and B laid out equal sums of money in trade; A gained a sum equal to 1 of his stock, and B lost 225 dollars, when A's money was double that of B; what did each expend ? Ans. $600. 2. What number is that, which, being increased by its half, its fourth and 5 more, will be doubled ? Ans. 20. 3. A man dying, bequeathed 100 dollars to his three sons, A, B and C, in this manner : A was to have a certain part, B twice as much as A, wanting 8 dollars, and C thrice as much as A, wanting 15 dollars ; how must it be divided ? Ans. A $20.50; B=33; C= 46.50. 4. Three persons talking of their ages--said A, I am 20 years old; said B, I am as old as A and half as old as C; said C, I am as old as you both; I demand their several ages. Ans. A= 20 years; B, 60; C, 80. 5. A and B have the same income; A saves f of his; but B, by spending 30 €. per annum more than A, at the end of 8 years finds himself 40 £. in debt; what is their income, and what does each spend per annum? Their income is £ ann. per ann. 6. Two men, A and B, lay out equal sums of money in trade ; A gains 126 £. and B loses 87 £., after which A's money is double to B's; what did each lay out? Ans. 300 £. Ans. {Topenas 175 F., and B 205 Ł. SECTION XXIX. ALLIGATION MEDIAL. 1. A farmer mixed 15 bushels of rye, at 64 cts. a bushel, 18 bushels of corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; what is the worth of a bushel of this mixture ? Ans. 47 cts. 2. A grocer mixed 2 cwt. of sugar at 56 s. per cwt., 1 cwt. at 43 s. per cwt., and 2 cwt. at 50 s. per cwt. together ; I demand the price of 3 cwt. of this mixture ? Ans. 7 £. 13 s. 3. Suppose 5 lbs. of gold of 22 carats fine-2 lbs. of 21 carats fine—and 1 lb. of alloy, be melted together ; what is the fineness of the compound ! Ans. 19 carats fine, |