B and C in 70 minutes. How long will it take each separately

to fill it?

Let x, y, z= the number of minutes for A, B, and C respectively to fill it separately.

the parts A, B, C can fill in 1 minute respectively.

the part A and B can fill in 1 minute,

the part A and C can fill in 1 minute,

the part B and C can fill in 1 minute.

But A and B can fill of it in 1 minute, A and C of it, and B and C of it.

70

Therefore, A can fill it in 52 minutes, B in 105 minutes, and C in 210 minutes.

2. A certain number represented by three figures is such that the sum of the ones' and hundreds' digits equals the tens'

х

=

105

.. x =

digit; the ones' digit exceeds the hundreds' digit by 4, and the sum of the tens' and hundreds' digits is 2 more than the ones' digit. What is the number?

3. There are three numbers such that the sum of of the first, of the second, and of the third is 23; the sum of of the first, of the second, and of the third is 17, and the sum of the numbers is 68. Find the numbers.

4. Three cities, A, B, C, situated at the vertices of a triangle, are connected by straight roads. The distance from A to C by way of B is 118 kilometers, from B to A by way of C is 74 kilometers, and from C to B by way of A is 92 kilometers. Find the lengths of the three straight roads.

5. A certain number is expressed by three digits whose sum is 10. The tens' digit is equal to the sum of the ones' and hundreds' digits, and if 99 is added to the number, its digits will be reversed. Find the number.

6. A person finds he can buy for $31.10 either 10 bushels of wheat, 12 bushels of rye, and 9 bushels of oats; or 12 bushels of wheat, 6 bushels of rye, and 13 bushels of oats; or 16 bushels of wheat, 10 bushels of rye, and 2 bushels of oats. What is the cost of a bushel of each?

7. A loaned his money at 4%, B at 5%, and C at 6%. If A and B together received $1592 interest, B and C together $1766 interest, and A and C together $1638 interest, what sum did each loan?

8. A cistern has three pipes, A, B, and C. If A and B run in while C runs out, it will be filled in 284 minutes; if B and C run in while A runs out, it will be filled in 40 minutes; and if A and C run in while B runs out, it will be filled in 66% minutes. In what time would each alone fill it?

9. A number expressed by three digits, whose sum is 18, has its digits reversed if we add to it 22 times the sum of its digits; and the number so formed is 72 less than double the original number. What is the number?

10. A and B can do a piece of work in 12 days, B and C can do it in 18 days, and A and C can do half the work in 12 days. In how many days can each alone do the work?

11. A certain number consists of three digits, whose sum is 15. If the first two digits are reversed, the number becomes 180 larger; but if the last two digits are reversed, the number becomes but 18 larger. Find the number.

12. A, B, and C each had a pocket full of marbles. After each had given each of the others of the marbles, in his pocket, they found that A had 28, B 30, and C 32. How many had each at first?

13. A gave to B and C as much as each of them already had; B then gave to A and C as much as each of them then had; and C then gave to A and B as much as each of them had, after which each had $16. How much had each at first?

14. A man invests $7620 in bonds; he buys 4% bonds at 90, 5% bonds at 105, and 6% bonds at 120. How much is invested in each kind if he obtains the same income from each investment?

15. Two fractions having the same denominator are such that their sum is 3. If 1 is added to each numerator, the sum of the fractions is 14; but if 1 is subtracted from each numerator, their difference is 4. Find the fractions.

16. A, B, C, on a hunting trip killed 72 birds. In order to share them equally A gives to B and C as many as they already have; next, B gives to A and C as many as they each had after the first division; and then C gives to A and B as many as they each had after the second division, when each had the same number. How many had each at first?

17. Of three bars of metal, the first contains 2 kg. of silver, 3 kg. of copper, and 4 kg. of tin; the second contains 3 kg. of silver, 4 kg. of copper, and 5 kg. of tin; the third contains 4 kg. of silver, 3 kg. of copper, and 5 kg. of tin. How many kilograms is it necessary to take from each of these bars to form a bar which shall contain 9 kg. of silver, 10 kg. of copper, and 14 kg. of tin?

INDETERMINATE EQUATIONS

235. We have seen (Art. 218) that when a single linear equation contains two unknown numbers, an unlimited number of values of the unknown numbers may be found which satisfy the equation.

Such an equation is called an Indeterminate Equation.

236. Any required number of the roots of an indeterminate equation can usually be found by direct inspection. For example, consider the equation

Then for a=0, y = 5; for x=1, y=4; for x = 2, y = 3; for x=3, y=2; for x=5, y=0; for x=6, y=-1; and so on. It is thus seen that the value of either unknown number depends upon that of the other.

237. In general, if the number of independent equations is less than that of the unknown numbers, the solution is indeterminate. For instance, in the case of two equations, each containing three unknown numbers, we have, after eliminating either of the unknown numbers, a single equation containing only the other two unknown numbers.

238. Although the number of solutions of an indeterminate equation is in general unlimited, restrictions may be imposed upon the nature of the solution, as by requiring that the values of the unknown numbers be all integral, or integral and positive, which will render the solution indeterminate in a less general sense.

For practical purposes the values of the unknown numbers are usually limited to positive integers, and in this connection we shall consider only indeterminate linear equations in which such values are required.

239. The usual methods of finding the positive integral roots of indeterminate equations are illustrated in the following solutions:

1. Find the positive integral roots of 3x+5y=20.

Since the equation may be written

5 y 20 - 3x,

it is seen that x cannot be greater than 6, and integral, for if x>6, y is negative.

If x = 6, 5, 4, 3, 2, 1, then y = a fraction in each case (which is readily seen by inspection), except for the value x = 5, when y = 1.

.. x = 5, y 1 are the only positive integral roots.

2. Find the positive integral roots of 3 a + 4 y = 34.

Solving for x, because it has the smaller coefficient, we have

1

3

Since x and y are to be integers, x + y

11 will be integral, and hence

will be integral, which is the case when y has such a value that 1 — y