Sidebilder
PDF
ePub

10. 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

[ocr errors]

11. Two persons, A and B, lay out equal sums of money in trade; A gains 1261. and в loses 877., and A's money is now double of B's; what did each lay out?

B

Ans. 3001.

12. A person bought a chaise, horse, and harness, for 601.; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness ; what did he give for each ?

Ans. 131. 6s. 8d for the horse, 6l. 13s. 4d.

for the harness, and 401. for the chaise

13. A person was desirous of giving 3d. apiece to some beggars, but found he had not money enough in his pocket by 8d., he therefore gave them each 2d.. and had then 3d. remaining; required the number of beggars?

Ans. 11

14. A servant agreed to live with his master for 31. a year, and a livery, but was turned away at the end of seven months, and received only 21. 13s. 4d. and his livery; what was its value ? Ans. 41. 16s.

15. A person left 5601. between his son and daughter, in such a manner, that for every half crown the son should have, the daughter was to have a shilling; what were their respective shares?

Ans. Son 400l., daughter 160l.

16. There is a certain number, consisting of two places of figures, which is equal to four times the sum of its digits; and if 18 be added to it the digits will be inverted ; what is the number? Ans. 24

17. Two persons, A and B, have both the same income; A saves a fifth of his yearly, but в, by spending 50 per annum more than A, at the end of four years, finds himself 1001. in debt; what was their income?

[graphic]

Ans. 125.

་་

18. When a company at a tavern came to pay their reckoning, they found, that if there had been three persons more, they would have had a shilling apiece less to pay, and if there had been two less, they would have had a shilling apiece more to pay; required the number of persons, and the quota of each ?

Ans. 12 persons, quota of each 5s.

19. A person at a tavern borrowed as much money as he had about him, and out of the whole spent 1s.; he then went to a second tavern, where he also borrowed as much as he had now about him, and out of the whole spent 1s.; and going on, in this manner, to a third and fourth tavern, he found, after spending his shilling at the latter, that he had nothing left; how much money had he at first? Ans. 111d.

20. It is required to divide the number 75 into two such parts, that three times the greater shall exceed seven times the less by 15. Ans. 54 and 21

[ocr errors]

21. In a mixture of British spirits and water, of the whole plus 25 gallons was spirits, and part minus 5 gallons was water; how many gallons were there of each? Ans. 85 of wine, and 35 of water

22. A bill of 120l. was paid in guineas and moidores, and the number of pieces of both sorts that were used were just 100; how many were there of each, reckoning the guinea at 21s., and the moidore at 27s.?

[graphic]

Ans. 50

23 Two travellers set out at the same time from London and York, whose distance is 197 miles; one of them goes 14 miles a day, and the other 16; in what time will they meet? Ans. 6 days 133 hours.

24. There is a fish whose tail weighs 91b., his head weighs as much as his tail and half his body, and his body weighs as much as his head and his tail; what is the whole weight of the fish? Ans. 72lb..

25. It is required to divide the number 10 into three

such parts, that, if the first be multiplied by 2, the second by 3, and the third by 4, the three products shall be all equal. Ans. 4, 3, and 21%

26. It is required to divide the number 36 into three such parts, that the first, of the second, and of the third, shall be all equal to each other.

Ans. The parts are 8, 12, and 16

27. A person has two horses, and a saddle, which, of itself, is worth 501.; now, if the saddle be put on the back of the first horse, it will make his value double that of the second, and if it be put on the back of the second, it will make his value triple that of the first: what is the value of each horse? Ans. One 301. and the other 401.

28. If A gives в 5s. of his money, в will have twice as much as the other has left; and if в gives A 5s. of his money, A will have three times as much as the other has left ; how much had each ? Ans. A 13s. and в 11s.

29. What two numbers are those whose difference, sum and product, are to each other as the numbers 2, 3, and 5, respectively? Ans. 10 and 2

30. A person in play lost a fourth of his money, and then won back 3s., after which he lost a third of what he now had, and then won back 2s. ; lastly, he lost a seventh of what he then had, and after this found he had but 12s. remaining; what had he at first?、 Ans. 20s.

31. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3, but 2 of the greyhound's leaps are as much as 3 of the hare's how many leaps must the greyhound take to catch the hare?

Ans. 300

32. It is required to divide the number 90 into four such parts, that if the first part 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?

Ans. The parts are 18, 22, 10, and 40

33. The quotient and remainder of a sum in division are, each, 21; and the divisor is 7 less than their sum: what is the number to be divided. Ans. 1050

34. A man and his wife usually drank out a cask of beer in 12 days, but when the man was from home it lasted the woman 30 days; how many days would the man alone be in drinking it? Ans. 20 days

35. A general, ranging his army in the form of a solid square, finds he has 284 men to spare, but increasing the side by one man, he wants 25 to fill up the square; how many soldiers had he? Ans. 24000

36. If A and B together can perform a piece of work in 8 days, a and c together in 9 days, and в and c in 10 days, how many days will it take each person to perform the the same work alone?

Ans. A 143 days, в 1743, and c 23,

QUADRATIC EQUATIONS.

A QUADRATIC EQUATION, as before observed, is that in which the unknown quantity is of two dimensions, or which rises to the second power; and is either simple or compound.

A simple quadratic equation, is that which contains only the square, or second power, of the unknown quantity, as

[blocks in formation]

A compound quadratic equation, is that which contains both the first and second power of the unknown quantity; as

[blocks in formation]

In which case, it is to be observed, that every equation of this kind, having any real positive root, will fall under one or other of the three following forms:

1. x2+ax=b.

a2

...

where x=

2. x2 ➡ax=b . . . where x=+&±√(~7+b).

3. x2 —ax=-b.. where x=+ ±√(———6).

2

4

Or, if the second and last terms be taken either positively or negatively, as they may happen to be, the general equation

b

C

ax2±bx=±c, or x2±-x=±

[blocks in formation]

which comprehends all the three cases above mentioned, may be resolved by means of the following rule:

RULE.

Transpose all the terms that involve the unknown quantity to one side of the equation, and the known terms to the other; observing to arrange them so, that the term which contains the square of the unknown quantity may be positive, and stand first in the equation.

Then, if this square has any coefficient prefixed to it, let all the rest of the terms be divided by it, and the equation will be brought to one of the three forms abovementioned.

In which case, the value of the unknown quantity x is always equal to half the coefficient, or multiplier of x, in the second term of the equation, taken with a contrary sign, together with the square root of the square of this number and the known quantity that forms the absolute, or third term, of the equation. (c)

(c) This rule, which is more commodious in its practical appli cation, than that usually given, is founded upon the same princi ple; being derived from the well known property. that in any quadratic

x2±ax+b, if the square of half the coefficient a
L

« ForrigeFortsett »