of the work would have been done; how long will it take the man and the boy to do the work, supposing the man to work 5 hours longer than the boy? 17. A person rows 20 miles down a river and back again in 10 hours, and he finds that he can row two miles against the stream in the same time that he can row three miles with it; required the rate of the stream, and the times of his going and returning. Ans. of a mile per hour, and 4 and 6 hours. 18. A body of men are just sufficient to form a hollow equilateral wedge, three deep, and if 597 be taken away, the remainder will form a hollow square four deep, the front of which contains one man more than the square root of the number contained in a front of the wedge; what is the number of men? Ans. 693. 19. Two merchants enter into partnership with 100l.; one has his money in business for three months, and the other for two months; and each receives 991. for his capital and profit; find the contribution of each. Ans. 45l. and 55l. 20. Two detachments of infantry are ordered to a station distant 39 miles; they begin their march at the same time, but one party by travelling of a mile an hour more than the other, arrives one hour sooner; required the rates of marching. Ans. 3 and 3 miles an hour. 21. A vintner sold 7 dozen of sherry and 12 dozen of claret for 50%.; he sold 3 dozen more of sherry for 101. than he did of claret for 6l.; required the price of each. Ans. Sherry, 21. per dozen; claret, 31. per dozen. 22. The number of men in both fronts of two columns of troops, A and B, when each consisted of as many ranks as it had men in front, was 84; but when the columns changed ground, and A was drawn up with the front B had, and B with the front A had, the number of ranks in both columns was 91; required the number of men in each column. Ans. 2304 and 1296 23. A and B lay out some money in speculation; A disposes of his bargain for 11 and gains as much per cent. as B lays out; B's gain is 367., and it appears that A gains four times as much per cent. as B; required the capital of each. A's capital, 5l., and B's, 120. 24. A detachment from an army was marching in regular column, with 5 men more in depth than in front; but upon the enemy coming in sight, the front was increased by 845 men, and by this movement the detachment was drawn up in five lines; required the number of men. Ans. 4550 men. 25. In a mixture of wine and cider, one-half, together with 25 gallons, was wine, and the cider was less than one-third part of the mixture by 5 gallons; how many gallons of each did it contain? Ans. 85 of wine, and 35 of cider. = 26. The difference between the hypothenuse and base of a right-angled triangle is 6, and the difference between the hypothenuse and the perpendicular is 3; what are the sides? Ans. 15, 12, and 9. = 27. There is a number, consisting of two digits; and being multiplied by the digit on the left hand, the product is 46; but if the sum of the digits be multiplied by the same digit, the product is only 10; required the number. Ans. 23. 28. A and B gained by trading 1007.; half of A's stock was less than B's by 1007. and A's gain was three-twentieths of B's stock; what did each put into stock, and what are the respective shares of the gain? Ans. A's stock was 600l., and B's 400l.; A's gain was 607., B's 401. 29. From two places, at the distance of 320 miles, two persons, A and B, set out at the same time to meet each other; A travelled 8 miles a day more than B, and the number of days until they met was equal to half the number of miles B went in a day; how many miles did each travel per day, and how far did each travel? Ans. A went 24, B 16 miles per day; A went 192, B 128 miles. 30. There are two rectangular vats, the greater of which contains 20 solid feet more than the other. Their capacities are in the ratio of 4 to 5, and their bases are squares, a side of each of which is equal to the depth of the other; what are the depths? Ans. 5 feet, and 4 feet. 31. Three persons divide a certain sum of money amongst them in the following manner: A takes the nth part of the whole, together with -£, B takes the nth part of the remain a der, together with - £, C takes the nth part of that which now n a remains, together with -£, after which nothing remains; find n 32. The distance between two places is a, and on the first 1 1 dayth of the journey is performed, on the second day - th m 1 1 n of the remainder, then th and th of the remainders alter nately on succeeding days; find the distance gone over in 2p days. Ans. a { 1 − (1 − 1 ) . ( 1 − 1)}. m -- CHAPTER VII. INEQUALITIES, RATIO, PROPORTION, AND XVIII. Inequalities are indicated by the sign > greater than, or less than; thus 5 > 3, 4 < 6, a > b, are inequalities. They may be treated in the same manner as equations, excepting that when all the terms have their signs changed, the sign must be changed to <, and the sign < to >; for whenever a is > b, a is necessarily <b; for instance, 5 > 3. but - 5 < 3. Examples. 1. Let m and n be any two unequal quantities, then m2 + n22 > 2mn. For no square quantity can be negative, .. (mn)2 or m2 + n2. 2mn is positive; .. the positive part of this expression is the negative part; that is, m2 + n2 > 2 m n 2. Show that√11 + √7 is greater than √19 + √2 ✓11 + 7> or <√19 + √2 according as (✓11 + √7)3 > or < (√19 + √2)2; or, or, 18+ 2 √77> or < 21 + 2 √38, tracting 18 from each, 2 √77 > or < 3 + 2 √38, by sub or, 308> or < 161 +638; by squaring each, or, 147 > or < 638, by subtracting 161 from each. Now it is evident that 147 is greater than 6 √38, ../11 + 7 is greater than√19+ √2. 3. Show that every fraction + its reciprocal is > 2. m n Let be the fraction, then is its reciprocal. n m Ans. x must = 2, and then y will = 5; for the value of x above found, namely, 25-y, becomes impossible when y> 5; and when y becomes 5, the expression ± √5-y vanishes, and 2. [For additional Examples, see Appendix.] x = = 5. Which is greater, 7+10, or 3+ Ans. 6. If 4x 7 < 2x + 3. and 3x + 1> 13 integral value of x. 7. Show that 8. Show that n3 n2 + n + 1 for all real values of n. RATIO. [Although Ratio and Proportion have been occasionally used already, yet the general theory remains to be established.] XIX. Ratio is the relation which one quantity bears to another with respect to magnitude; thus 12 is 3 times as great as 4, and the ratio of 12 to 4 is 3; this is expressed 12: 4, the former 12 being called the antecedent, and the latter 4 the consequent of the ratio. It is obvious that this ratio might be expressed by the frac value of the ratio of a: b may be expressed by The value of a ratio is not altered by multiplying or dividing both its terms by the same quantity. |