1. Divide the number n into two such parts that the greater increased by a shall be equal to the less increased by b. n+ b α 2. In the last example, what will be the two parts if n = 84, a = 16, and b = 58? Ans. 63 and 21. 3. The sum of three numbers is s; the second exceeds the first by a, and the third exceeds the second by b. Required the numbers. S - 2a bs + a -b s + a +26 Ans. 3 3 4. My indebtedness to three persons, A, B, and C, amounts to a dollars; and I owe B, n times the sum which I owe A, and C, m times the sum which I owé A. What is my indebtedness to A? 5. In the last example, what is the sum due to A when $786, 2, and m = 3 ? Ans. $131. α= 6. A person engaged to work a days on these conditions: For each day he worked he was to receive b cents, and for each day he was idle he was to forfeit c cents; at the end of a days he received d cents. How many days was he idle? ab d Ans. days. b + c 7. My horse and saddle are together worth horse is worth n times the price of my saddle. of each ? a a dollars, and my Ans. Saddle, $ ; horse, $ na n+1 8. The rent of an estate is n per cent. greater this year than it was last. This year it is a dollars; what was it last year? 9. A person after spending a dollars more than of his income, had remaining b dollars more than of it. Ans. Required his income. 21(a + b) dollars. 11 10. A person after spending a dollars more than 1 n come, had remaining b dollars more than th of it. income. Required his mn(a + b) Ans. dollars. 11. If A can perform a certain piece of work in a days, and B. can do the same in b days, and C the same in c days, in how many days can all together perform the work? Ans. 2 days. 12. In the last example, what will be the time required, when α = 6, b = 8, and c = 12? 13. If from a times a certain number, c be subtracted, the remainder will be equal to b times the number increased by d. Required the number. Ans. c + d a b 14. A farmer would mix oats worth a cents a bushel with peas worth b cents a bushel, to form a mixture of c bushels worth d cents a bushel. How many bushels of each kind must he take? 15. There were a boys in one party, and b boys in another, and each party had the same number of nuts. Each boy in the first party snatched m nuts from the second party, and ate them; then each boy in the second party snatched m nuts from the first party, and ate them. Each party then divided the nuts remaining to it equally among its members, when the boys in the two parties found that they had the same number of nuts apiece; how many nuts had each party at first? Ans. m(a+b). 16. Find four numbers, such that if a times the first be added to the second, b times the second be added to the third, c times the third be added to the fourth, and d times the fourth be added to the first, each sum shall be m. 17. A sent n pupils regularly to a certain school during a term of a days, and B sent m pupils regularly to another school for a term of b days. The two schools had the same number of pupils in attendance, and raised the same amount of money by rate-bill. There were c days' absence allowed for at the school to which A sent, and d days' absence at the school to which B sent; and A and B found that they had equal sums to pay. What was the number of pupils attending each school? bcm - adn Ans. ab(m — n) 18. Divide the number m into four parts, such that the second shall be a times the first, the third a times the second, and the fourth a times the third. Ans. 1st part, as + a2 + a + 1 m 19. The sum of two numbers is s, and their difference is d. Required the numbers. Ans. Greater, s + d S d ; less, 2 2 20. There are three numbers, such that the sum of the first and second is a, the sum of the first and third is b, and the sum of the second and third is c. What are the numbers? 21. There is a number consisting of two digits; the number is equal to a times the sum of its digits; and if c be number, the order of the digits will be reversed. two digits. Digit in units' place, Ans. Digit in tens' place, added to the Required the c(10 − a) 9(11-2a) c(a1) 9(11-2a) 22. Find what each of three persons, A, B, and 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. We give here a solution of this example, partly to illustrate the method of simplifying algebraic formulas by the use of auxiliary quantities. Let x = A's money, y = B's money, z= C's money. Then, by the conditions, y + mx + mz = q........ (2), z + nx + ny = r (3). m 1 ns r %= (7). n y ms ls ns r 8 = + + (8), or 7 1 m 1 n 1 Now the parenthetical expressions in equation (9) are known quantities. Hence, to simplify the results, DISCUSSION OF PROBLEMS INVOLVING SIMPLE EQUATIONS. 180. The Discussion of a problem consists in attributing certain values and relations to the arbitrary quantities which enter the equation, and in interpreting the results. 181. When a problem has been solved in a general manner, we may proceed to make an unlimited number of suppositions upon the arbitrary quantities involved in the formulas, and thus obtain a variety of results. But our experience with algebraic equations would lead us to expect that the problem might not be rational, or possible, under every hypothesis. Now the principal object in the discussion of a problem is to examine the peculiar or anomalous forms which present themselves, and ascertain whether the problem is rational or absurd, or how it is to be understood, under the suppositions which lead to these peculiarities. We shall commence with the INTERPRETATION OF NEGATIVE RESULTS. 1. What number must be added to a that the sum may be b? Let x represent the required number. Then, by the conditions of the question, a result which satisfies the conditions; for, we perceive that 8 is the number which must be added to 20, or a, to make 28, or b. Second, suppose a = 20 and b = 12; then by the formula, a negative result. x12-20= 89 In order to ascertain the meaning of the minus sign in this case, let us enunciate the question according to the supposition that gave this result; thus, What number must be added to 20, that the sum may be 12? |