PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 118. 1. A merchant has three kinds of flour. He can sell 1 bbl. of the first, 2 of the second, and 3 of the third for $85; 2 of the first, 1 of the second, and bbl. of the third for $45.50; and 1 of each kind for $41. What is the price per bbl. of each? Ans. 1st, $12; 2d, $14; 3d, $15. 2. Three boys, A, B, and C, divided a sum of money among themselves in such a manner that A and B received 18 cents, B and C 14 cents, and A and C 16. How much did each receive? Ans. A, 10; B, 8; C, 6 cents. 3. As three persons, A, B, and C, were talking of their ages, it was found that the sum of one half of A's age, one third of B's, and one fourth of C's was 33; that the sum of A's and B's was 13 more than C's age; while the sum of B's and C's was 3 less than twice A's age. What was the age of each? Ans. A's, 32; B's, 21; C's, 40. 4. As three drovers were talking of their sheep, says A to B, "If you will give me 10 of yours, and C one fourth of his, I shall have 6 more than C now has." Says B to C, "If you will give me 25 of yours, and A one fifth of his, I shall have 8 more than both of you will have left." Says C to A and B, "If one of you will give me 10, and the other 9, I shall have just as many as both of you will have left." How many did 5. Divide 32 into four such parts that if the first part is increased by 3, the second diminished by 3, the third multiplied by 3, and the fourth divided by 3, the sum, difference, product, and quotient shall all be equal. Ans. 3, 9, 2, and 18. 6. If A and B can perform a piece of work together days, B and C in 9 days, and A and C in 82 days, in how many days can each do it alone? in 8 Ans. A in 15, B in 18, and C in 21 days. 7. Find three numbers such that one half of the first, one third of the second, and one fourth of the third shall together be 56; one third of the first, one fourth of the second, and one fifth of the third, 43; one fourth of the first, one fifth of the second, and one sixth of the third, 35. 8. The sum of the three figures of a certain number is 12; the sum of the last two figures is double the first; and if 297 is added to the number, the order of its figures will be inverted. What is the number? Ans. 417. 9. A man sold his horse, carriage, and harness for $450. For the horse he received $25 less than five times what he received for the harness; and one third of what he received for the horse was equal to what he received for the harness plus one seventh of what he received for the carriage. What did he receive for each? Ans. Horse, $225; carriage, $175; harness, $50. 10. A man owned three horses, and a saddle which was worth $45. If the saddle is put on the first horse, the value of both will be $30 less than the value of the second; if the saddle is put on the second horse, the value of both will be $55 less than the value of the third; and if the saddle is put on the third horse, the value of both will be equal to twice the value of the second minus $10 more than one fifth of the value of the first. What is the value of each horse? Ans. 1st, $100; 2d, $175; 3d, $275. 11. The sum of the numerators of two fractions is 7, and the sum of their denominators 16; moreover the sum of the numerator and denominator of the first is equal to the denominator of the second; and the denominator of the second, minus twice the numerator of the first, is equal to the numerator of the second. What are the fractions? Ans. and §. 12. A man bought a horse, a wagon, and a harness, for $180. The horse and harness cost three times as much as the wagon, and the wagon and harness one half as much as the horse. What was the cost of each? 13. A gentleman gives $600 to be divided among three classes in such a way that each one of the best class is to receive $10, and the remainder to be divided equally among those of the other two classes. If the first class proves to be the best, each one of the other two classes will receive $5; if the second class proves to be the best each one of the other two classes will receive $44; but if the third class proves to be the best, each one of the other two classes will receive $2. What is the number in each class? 14. A cistern has 3 pipes opening into it. If the first should be closed, the cistern would be filled in 20 minutes; if the second, in 25 minutes; and if the third, in 30 minutes. How long would it take each pipe alone to fill the cistern, and how long would it take the three together? Ans. 1st, 855 minutes; 2d, 46 minutes; 3d, 355 minutes. The three together, 16 minutes. 15. Three men, A, B, and C, had together $24. Now if A gives to B and C as much as they already have and then B gives to A and C as much as they have after the first distribution, and again C gives to A and B as much as they have after the second distribution, they will all have the same sum. How much did each have at first? Ans. A, $13; B, $7, and C, $4. SECTION XVI. POWERS AND ROOTS. 119. A POWER of any quantity is the product obtained by taking that quantity any number of times as a factor; and the exponent shows how many times the quantity is taken (Art. 24). Thus, a = a1 is the first power of a; 120. In order to explain the use of negative indices, we form, by the rules of division, the following series: We form the first series as follows: a divided by a gives a*; a1 by a, gives a3; a3 by a, gives a2; a2 by a, gives a; a by a, gives 1 1 a2 1; 1 by a, gives; by a, gives a; by a, gives, and so on. 1 a The second series is formed in the same way from a5 to a; but if we follow the same rule of division from a toward the right as from a3 to a, viz. subtracting the index of the divisor from that of the dividend, a divided by a, gives a°; a° by a, gives a-1; read a, with the negative index one; a-1 by a, gives a-2; a-2 by a, gives a3; and so on. From this we learn, 1st. That the 0 power of every quantity is 1; 2d. That a1, a-2, a-3, &c., are only different ways of |