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Miscellaneous Questions, producing Simple Equations. 1. To find two numbers, such that their sum shall be 40, and their difference 16.
Ans. 12 = least number, and 28 = greater number. 2. A prize of 10001. is to be divided between two persons, whose shes therein are in the proportion of 7 to 9; required the share of each.
Ans. 4371. 10s. = first share, and 5621. 10s, = second share. 5. What number is that whose one-third part exceeds its one-fourth part by 16?
Ans. 192. 4. Divide 10001. between A, B, and C, so that A shall have 721. more than B, and C. 1001. more than A.
Ans. 2521. B.'s share ; 3241. A.'s share ; and 4241. C.'s share. 5. What fraction is that to the numerator, of which, if i be added, the value will be one-third ; but if i be added to the denominator, its value will be one-fourth ?
15 6. A labourer agreed to serve for 40 days, upon condition that for every day he worked he should receive 201. but for every day he played, or was absent, he should forfeit 8d. Now, at the end of his engagement, he had to receive 11. Ils. 8d. it is required to find how many days he worked, and how many he was idle.
Ans. He worked 25 days, and he was idle 15. 7. If the paving of a square, at 2s. a yard, costs as much as the inclosing of it at 5s. a yard ; required the side of the square.
Ans. 10 yards. 8. A market-woman bought a certain number of eggs, at 2 a penny, and as many at 3 a penny, and sold them all again at the rate of 5 for 2d. and by so doing lost 4d. what number of eggs had she?
Ans. 120. 9. If one agent A, alone, can produce an effect e, in the time a, and another agent B, alone, in the time b; in what time will they both produce the same effect?
bta 10. If A, alone, can do a piece of work in ten days, and B in thir. teen ; set them both about it, in what time will it be finished ?
15 Ans. In 5 days.
23 11. What two numbers are those whose difference is 7, and sum 33 ?
Ans. 13 and 20. 12. Two travellers set out at the same time from London and York, the distance being 150 miles ; one of them goes 8 miles a day, and the other 7; in what time will they meet ? Ans. in 10 days.
13. A post is in the niud, $ in the water, and 10 feet above the water ; what is its whole length ?
Ans. 24 feet. 14. In a mixture of cider and wine, one-half of the whole plus 25 gallons was wine, and one-third part minus 5 gallons was cider; how many gallons were there of each?
Ans. 85 of wine, and 35 of cider. 15. The tail of a fish weighs glb. its head weighs as much as its tail and half its body, and its body weighs as much as its head and its tail ; what is the whole weight of the fish)
Ans. 721b. 16. A bill of 1201. was paid in guineas and moidores, and the number of pieces of both denominations that were used amounted to 100; how many were there of each?
Ans. 50 of each. 17. After paying away one-fourth and one-fifth of my money, I had 66 guineas left; what had I at first? Ans. 120 guineas.
18. At an election, 375 persons voted, and the candidate chosen had a majority of 91 ; how many voted for each candidate ?
Ans. 233 for one, and 142 for the other. 19. A.'s age is double of B.'s, and B.'s is triple of C.'s, and the sum of all their ages is 140; what is the age of each?
Ans. A.'s=84, B.'s=42, and C.'s=14. 20. What number is that from which, if 5 be subtracted, two. thirds of the remainder will be 40 ?
Ans. 65, 21. Two persons, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A.'s money is now double of B.'s; what did each lay out?
Ans. 300l. 22. Out of a cask of wine, which bad leaked away one-third, 21 gallons were drawn ; and then, being gauged, it appeared to be half full ; how much did it hold ?
Ans. 126 gallons. 23. A gentleman bought a chaise, horse, and harness, for 60l. the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness; how much did he pay for each? Ans. 131. 6s. 8d. for the horse, 61. 13s. 4d, for the barness,
and 401. for the chaise. 24. A person was desirous of giving some beggars 3d. a-piece, but found that he had not money enough in his pocket by 8d. he therefore gave them 2d. each, and had then 3d remaining; how many beggars ·
Ans. 11. 25. Two persons, A and B, have both the same income; A saves one-fifth of his yearly, but B, by spending 50l. per annum more than A, at the end of four years finds himself 100l. in debt ; what is their income?
Ans. 1251. 26. A footman agreed to serve for 8l. a year and a livery, but was turned away at the end of seven months, receiving only 21. 135. 4d. and his livery; what was its value ?
Ans. yl. 16s.
27. A hare is fifty leaps before a greyhound, and takes four leaps to the greyhound's three; but two of the greyhound's leaps are as much as three of the hare's; how many leaps must the greyhound take to catch the hare ?
Ans. 300. 28. The hour and minute hand of a clock are exactly together at 12 o'clock; at what time are they next together ?
Ans. 1 ho. 5 min. 29. A gamester lost one-fourth of his money at play, and then won three shillings; after which he lost one-third of what he then had, and then won two shillings ; lastly, he lost one-seventh of what he then bad; and, this done, found he had only 12s. remaining ; what bad he at first ?
Ans. 20s. 30. A man ana his wife usually drank a cask of beer in twelve days; but when the man was from home, it lasted his wife thirty days; how many days would the man alone be in drinking it?
Ans. 20 days. 31. A merchant has two kinds of tea, one worth gs. 6d. per pound, the other 13s. 6d. How many pounds of each must he take to form a chest of 104lbs. which shall be worth 561. ?
Ans. 33 at 13s. 6d. and 71 at gs. 6d. 32. Divide the number 90 into 4 such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2 ; the sum, difference, product, and quotient, shall be all equal to each other.
Ans. The parts are 18, 22, 10, and 40, respectively. 33. If A and B together perform a piece of work in 8 days; A and C together in 9 days; and B and C in 10 days; how many days will it take each person to perform the same work ?
Ans. A 14 days, B 1744, and C 231. 34. How much foreign brandy at 8s. per gallon, and British spirits at 3s. per gallon, must be mixed, that in selling a compound of 65 gallons, at gs. per gallon, the distiller may clear 30 per cent. ?
Ans. 51 gallons of brandy, and 14 of spirits. 35. If three agents, A, B, and C, produce the effects a, b, c, m the times e, f, g, respectively; in what time would they jointly produce the effect d?
f 8 36. A gentleman has wo horses, and a saddle worth 50l. ; now if the saddle be put on the back of the first horse, it will make his value double that of the second ; but if it be pa.. on the back of the second, i will make his value triple that of the first ; what is the value of each aorse?
Ans. One 301. and the other 401.
37. There is a certain number, to the sum of whose digits if you add 7, the result will be three times the left-hand digit; and if from the number itself you subtract 18, the digits will be inverted; what is the number?
38. A vessel, containing 120 gallons, is filled in 10 minutes by two spouts running successively; the one runs 14 gallons in a minute, the other 9 gallons in a minute ; for what time has each spout to run ?
Ans. 14 gallon spout runs 6 minutes.
9 gallon spout runs 4 minutes. 39. In the composition of a certain quantity of gunpowder, twothirds of the whole plus 10 was nitre ; one-sixth of the whole minus four and a half was sulphur; and the charcoal was one-seventh of the nitre minus two; how many pounds of gunpowder were there?
OP QUADRATIC EQUATIONS,
CONTAINING ONLY ONE UNKNOWN QUANTITY.
(68.) There are two kinds of Quadratic Equations with one unknown quantity.
1. When the equation contains only the square of the unknown quantity, it is called a pure quadratic equation.
Thus, ro=36; x2 +5=54; 3.r?–4=71; aro-l=c; are pure quadratic equations.
2. When the equation, together with the square of the unknown quantity, contains also the unknown quantity itself, it is then called an adfected quadratic equation.
Thus, x2 + 4x=45 ; 38-— 2x=21 ; xo + ax=b; axo +21x=c+d; are adfected quadratic equations.
(69.) The solution of pure quadratic equations is effected by the following
Rule Transpose the terms of the equation in such a manner, that the term containing ro may stand on one side of the equation, and the known quantities on the other ; divide both sides of the equation by the co-efficient of x', (if it has any co-efficient) and then extract the square root of each side of the equation.
Examples. 1. Sappose +5=54, to find x.
Toen, by transposition, x=54-5=49 Extract now the square root of both sides of the equation, and
then rt=749=7 Ana 2. Let 6r2-8=142, to find x. Then, by transposition, 6.xo=142+8=150.
150 Dividing by 6, x= = 25.
4. Let ar?– b=c; to find x.
By transposition, are=c+b; and zo=C+%;
5. Let ar?-5c=\x2—3c+d; to find x.
By transposition, ax-bx'=5c-3c+d;
; and x=v2c+d
Examples for Practice. 6. Given 5x-1=244, to find x......
Ans. x=7. 7. Given 9x® +9=3x2 +63, to find x........ Ans. x=3.
4x2 + 5 8. Let =45, to find x...
Ans. X=10. 9
C+2 9. Let bx? +c+3=26x2 +1, to find x...... Ans. x=
6 10. Given 2ax2 +1-4 = cx*-5+d-ars, to find x.
d-1-1 Ans. r=r
3a-C (70.) The solution of adfected equations is performed thus :
Rule.- 1. Bring the proposed equation to the form x' + ax=++, hy the rules of simple equations.
2. Add to each side of this equation the square of half th3