22. If 32 pounds of sea water contain 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.

Ans. $64000 and $36000.

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.

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?

All will fill it in one hour.

A in 105 minutes. Ans. B in 210 minutes.

C in 420 minutes.

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 I 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

29. Find what each of three persons, A, B, C, is worth, knowing, 1st, that what A is worth added to 7 times what B 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,

5x 3y = 12,

will satisfy it: hence, there are an infinite number of values for x 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 quantity is indeterminate.

In general, if we have n, equations involving more than » 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

unless some of them are dependent upon the others. If we combine them, we may eliminate all the unknown quantities, and the resulting equations, which will then contain only known quantities, will be so many equations of condition, which must be satisfied in order that the given equations may admit of solution. For example, if we have

and by substituting these in the third, we shall find

which expresses the relation between a, c and d, that must exist, in order that the three equations may be simultaneous.

88*. A Problem is indeterminate when it admits of an infinite number of solutions. This will always be the case when its enunciation involves more unknown quantities than there are given conditions; since, in that case, the statement of the problem will give rise to a less number of equations than there are unknown quantities.

1st. Let it be required to find two numbers such that 5 times the first diminished by 3 times the second shall be equal to 12.

If we denote the numbers by x and y, the conditions of the problem will give the equation

which we have seen is indeterminate :-Hence, the problem admits of an infinite number of solutions, or is indeterminate.

2. Find a quantity such that if it be multiplied by a and the product increased by b, the result will be equal to c times the quantity increased by d.

Let x denote the required quantity. Then from the condition,

If now we make the suppositions that db and a = c, the

If we make these substitutions in the first equation, it be

comes

ax + b = ax + b,

an identical equation (Art. 75), which must be satisfied for all values of x. These suppositions also render the conditions of the problem so dependent upon each other, that any quantity whatever will fulfil them all.

Hence, the result oe indicates that the problem admits of an

infinite number of solutions.

3. Find two quantities such that a times the first increased by b times the second shall be equal to c, and that d times the first increased by f times the second shall be equal to g. If we denote the quantities by x and y, we shall have from the conditions of the problem,

we shall find by multiplying these equations together, member by member,

cf=bg.

These suppositions, reduce the values of both x and y to