6. Divide the number 75 into two such parts, that three times the greater may exceed seven times the less by 15.

Ans. 54 and 21.

7. In a mixture of wine and cider, of the whole plus 25 gallons was wine, and part minus 5 many gallons were there of each ?

gallons, was cider; how

Ans. 85 of wine, and 35 of cider.

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

Ans. 50.

9. Two travelers set out at the same time from London and York, whose distance apart is 150 miles; they travel toward each other; one of them goes 8 miles a day, and the other 7; in what time will they meet? Ans. In 10 days.

10. At a certain election, 375 persons voted for two candi dates, and the candidate chosen had a majority of 91; how many voted for each?

Ans. 233 for one, and 142 for the other.

11. A's age is double B's, and B's is triple C's, and the sum of all their ages is 140; what is the age of each?

12. A person bought a chaise, horse, and harness, for £60; 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 ? for the horse. for the harness.

£13 6s. 8d. Ans. £ 6 13s. 4d. £40

for the chaise.

13. A person has two horses, and a saddle worth £50; 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 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 £30, and the other £40.

14. Two persons, A and B, have each the same income. A saves of his yearly; but B, by spending £50 per annum mcre than A, at the end of 4 years finds himself £100 in debt; what is the income of each? Ans. £125.

15. To divide the number 36 into three such parts, that of the first,of the second, and of the third, may be all equal to each other. Ans. 8, 12, and 16.

16, A footman agreed to serve his master for £8 a year and a livery, but was turned away at the end of 7 months, and received only £2 13s. 4d. and his livery; what was its value?

Ans. £4 16s.

17. To divide the number 90 into four 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, so obtained, will be all equal to each other. Ans. The parts are 18, 22, 10, and 40.

18. The hour and minute hands of a clock are exactly together at 12 o'clock; when are they next together?

Ans. 1 h. 5 min.

19. 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 be in drinking it alone? Ans. 20 days.

20. If A and B together can 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 would it take each person to perform the same work alone? Ans. A 143 days, B 1723, and C 231

21. A laborer can do a certain work expressed by a, in a time expressed by b; a second laborer, the work c in a time d; a third, the work e in a time f. Required the time it would take the three laborers, working together, to perform the work g.

22. If 32 pounds of sea water contair 1 pound of salt, how much fresh water must be added to these 32 pounds, in order that the quantity of salt contained in 32 pounds of the new mixture shall be reduced to 2 ounces, or of a pound?

Ans. 224 lbs.

23. A number is expressed by three figures; the sum of these figures is 11; the figure in the place of units is double that in the place of hundreds; and when 297 is added to this number, the sum obtained is expressed by the figures of this number reversed. What is the number? Ans. 326.

24. A person who possessed 100000 dollars, placed the greater part of it out at 5 per cent. interest, and the other part at 4 per cent. The interest which he received for the whole amounted to 4640 dollars. Required the two parts.

25. A person possessed a certain capital, which he placed out at a certain interest. Another person possessed 10000 dollars more than the first, and putting out his capital 1 per cent. more advantageously, had an income greater by 800 dollars. A third, possessed 15000 dollars more than the first, and putting out his capital 2 per cent. more advantageously, had an income greater by 1500 dollars. Required the capitals, and the three rates of interest.

Sums at interest,

Rates of interest,

$45000.

$30000, $40000,
4
5

6 per cent.

26. A cistern may be filled by three pipes, A, B, C. By the two first it can be filled in 70 minutes; by the first and third it can be filled in 84 minutes; and by the second and third in 140 minutes. What time will each pipe take to do it in ? What time will be required, if the three pipes run together?

A in 105 minutes.

Ans.

B in 210 minutes.

C in 420 minutes.

All will fill it in one hour.

27. A, has 3 purses, each containing a certain sum of money. If $20 be taken out of the first and put into the second, it will contain four times as much as remains in the first. If $60 be taken from the second and put into the third, then this will contain 13 times as much as there remains in the second. Again, if $40 be taken from the third and put into the first, then the third will contain 2 times as much as the first. What were the contents of each purse? 1st. $120. Ans. 2d. $380. 3d. $500.

28. A banker has two kinds of money; it takes a pieces of the first to make a crown, and b of the second to make the same sum. Some one offers him a crown for pieces. How many of each kind must the banker give him?

persons, A, B, C, is worth, added to 7 times what B

29. Find what each of three knowing, 1st, that what A is worth and C are worth, is equal to p; 2d, that what B is worth added to m times what A and C are worth, is equal to q; 3d, that what C is worth added to n times what A and B are worth, is equal to r.

If we denote by s what A, B, and C, are worth, we introduce an auxiliary quantity, and resolve the question in a very simple manner.

30. Find the values of the estates of six persons, A, B, C, D, E, F, from the following conditions: 1st. The sum of the estates of A and B is equal to a; that of C and D is equal to b; and that of E and F is equal to c. 2d: The estate of A is worth m times that of C; the estate of D is worth n times that of E, and the estate of F is worth p times that of B.

This problem may be solved by means of a single equation, involving but one unknown quantity.

Of Indeterminate Equations and Indeterminate Problems.

88. An equation is said to be indeterminate when it may be satisfied for an infinite number of sets of values of the unknown quantities which enter it.

Every single equation containing two unknown quantities is inde terminate.

For example, let us take the equation

and any two corresponding values of x, y, being substituted in the given equation,

will satisfy it: hence, there are an infinite number of values for and y which will satisfy the equation, and consequently it is indeterminate; that is, it admits of an infinite number of solutions.

If an equation contains more than two unknown quantities, we may find an expression for one of them in terms of the others.

If, then, we assume values at pleasure for these others, we can find from this equation the corresponding values of the first; and the assumed and deduced values, taken together, will satisfy the given equation. Hence,

Every equation involving more than one unknown quantitu iɛ indeterminate.

In general, if we have n equations involving more than n unknown quantities, these equations are indeterminate; for we may, by combination and elimination, reduce them to a single equation containing more than one unknown quantity, which we have already seen is indeterminate.

If, on the contrary, we have a greater number of equations than we have unknown quantities, they cannot all be satisfied