5. What number is that whose eighth part multiplied by its fifth part, and the product divided by 4, will give a quotient equal to 40? Let By the conditions of the question, x= the number. Ans. 80. 6. Find a number such that one third of it multiplied by one fourth shall be equal to 108. Ans. 36. 7. What number is that whose sixth part multiplied by its fifth part, and the product divided by 10, will give a quotient equal to 3? Ans. 30. 8. What number is that whose square plus 18 will be equal to half the square plus 301? Ans. 5. 9. What numbers are those which are to each other as 1 to 2, and the difference of whose squares is equal to 75? 10. What two numbers are those which are to each other as 5 to 6, and the difference of whose squares is 44? 11. What two numbers are those which are to each other as 3 to 4, and the difference of whose squares is 28? Ans. 6 and 8. 12. What two numbers are those which are to each other as 5 to 11, and the sum of whose squares is 584? Ans. 10 and 22. 13. A says to B, "My son's age is one quarter of yours, and the difference between the squares of the numbers representing their ages is 240." What were their ages ? Ans. Elder, 16; younger, 4. EQUATIONS CONTAINING Two Unknown QUANTITIES. 157. When there are two or more unknown quantities, Eliminate one of the unknown quantities by § 113. Then extract the square root of both members of the equation. PROBLEMS FOR SOLUTION. 1. There is a room of such dimensions that the difference of the sides multiplied by the less is equal to 36, and the product of the sides is equal to 360. What are the sides? 2. A merchant sells two pieces of muslin, which together measure 12 yards. He received for each piece just as many dollars per yard as the piece contained yards. Now, he gets four times as much for one piece as for the other. How many yards in each piece? Ans. 8 and 4. X= 2y, by extracting the square root. Substituting this value of x in the first equation, we have 3. What two numbers are those whose product is 30, and the quotient of the greater by the less, 3? Ans. 10 and 3. 4. The product of two numbers is a, and their quotient b. What are the numbers? Ans. Vab and 5. The sum of the squares of two numbers is 117, and the difference of their squares 45. What are the numbers? 7. What two numbers are those which are to each other as 3 to 4, and the sum of whose squares is 225? Ans. 9 and 12. 8. What two numbers are those which are to each other as m to n, and the sum of whose squares is equal to a2? 9. What two numbers are those which are to each other as 1 to 2, and the difference of whose squares is 75? Ans. 5 and 10. 10. What two numbers are those which are to each other as m to n, and the difference of whose squares is equal to b2? 11. A certain sum of money is placed at interest for six months, at 8 per cent per annum. Now, if the sum put at interest be multiplied by the number expressing the interest, the product will be $562500. What is the principal at interest? Ans. $3750. 12. A person distributes a sum of money between a number of women and boys. The number of women is to the number of boys as 3 to 4. Now, the boys receive one half as many dollars as there are persons; and the women, twice as many dollars as there are boys; and together they receive $188. How many women were there, and how many boys? Ans. 36 women, 48 boys. COMPLETE EQUATIONS. 158. The reduced form of the complete equation (§ 153) is x2+2px = q. Comparing the first member of this equation with the square of a binomial (§ 54), we see that it needs but the square of half the coefficient of x to make it a perfect square. Adding p2 to both members (§ 102, Axiom 1), we have x2+2px+p2=q+p2. Then, extracting the square root of both members (Axiom 5), we have x + p = ±√q+p2. Transposing p to the second member, we have x=- · p±√q+p2. Hence there are two roots, one corresponding to the plus sign of the radical, and the other to the minus sign. Denoting these roots by x' and x", we have x' = − p +√q+p2, and x"=-p-√q+p2. The root denoted by x' is called the first root; that denoted by x" is called the second root. |