Whence x 5. The required fraction is, }. = EXERCISES II. = 1. What number added to the numerator and denominator of will give a fraction equal to ? 2. The sum of two numbers is 18, and the quotient of the less divided by the greater is equal to . What are the numbers? 3. The denominator of a fraction exceeds its numerator by 2, and if 1 be added to both numerator and denominator, the resulting fraction will be equal to . What is the fraction? 4. The sum of a number and seven times its reciprocal is 8. What is the number? 5. The value of a fraction, when reduced to its lowest terms, is §. If its numerator be increased by 7 and its denominator be decreased by 7, the resulting fraction will be equal to . What is the number? 6. What number must be added to the numerator and subtracted from the denominator of the fraction, to give its reciprocal ? 7. If be divided by a certain number increased by 4, and be subtracted from the quotient, the remainder will be . What is the number? 8. A train runs 200 miles in a certain time. If it were to run 5 miles an hour faster, it would run 40 miles further in the same time. What is the rate of the train ? 9. A number has three digits, which increase by 1 from left to right. The quotient of the number divided by the sum of the digits is 26. What is the number? 10. A number of men have $72 to divide. If $144 were divided among 3 more men, each one would receive $4 more. How many men are there? 11. It was intended to divide by a certain number, but by mistake was added to the number. The result was, nevertheless, the same. What is the number? 12. A steamer can run 20 miles an hour in still water. If it can run 72 miles with the current in the same time that it can run 48 miles against the current, what is the speed of the current? 13. A man buys two kinds of wine, 14 bottles in all, paying $9 for one kind and $12 for the other. If the price of each kind is the same, how many bottles of each does he buy? 14. A can do a piece of work in 10 days, B in 6 days; and A, B, and C together in 3 days. In how many days can C do the work? 15. A and B together can do a piece of work in 2 days, B and C together in 3 days, and A and C together in 21 days. In how many days can A, B, and C together do the work? 16. The circumference of the hind wheel of a carriage exceeds the circumference of the front wheel by 4 feet, and the front wheel makes the same number of revolutions in running 400 yards that the hind wheel makes in running 500 yards. What is the circumference of each wheel? 17. In a number of two digits, the digit in the tens' place exceeds the digit in the units' place by 2. If the digits be interchanged and the resulting number be divided by the original number, the quotient will be equal to . What is the number? 18. In a number of three digits, the digit in the hundreds' place is 2; if this digit be transferred to the units' place, and the resulting number be divided by the original number, the quotient will be equal to 17. What is the number? 19. In one hour a train runs 10 miles further than a man rides on a bicycle in the same time. If it takes the train 6 hours longer to run 255 miles than it takes the man to ride 63 miles, what is the rate of the train? 20. Two engines are used in different places in a mine to pump out water. The one pumps 11 gallons every 5 minutes from a depth of 155 yards; the other pumps 31 gallons every 10 minutes from a depth of 88 yards. The engines together represent the power of 54 horses. Each engine represents the power of how many horses? 21. A cistern has three pipes. To fill it, the first pipe takes one-half of the time required by the second, and the second takes two-thirds of the time required by the third. If the three pipes be open together, the cistern will be filled in 6 hours. In what time will each pipe fill the cistern? 22. A and B ride 100 miles from P to Q. They ride together at a uniform rate until they are within 30 miles of Q, when A increases his rate by of his previous rate. When B is within 20 miles of Q, he increases his rate by of his previous rate, and arrives at Q 10 minutes earlier than A. At what rate did A and B first ride? 23. A circular road has three stations, A, B, and C, so placed that A is 15 miles from B, B is 13 miles from C in the same direction, and C is 14 miles from A in the same direction. Two messengers leaving A at the same time, and traveling in opposite directions, meet at B. The faster messenger then reaches A, 7 hours before the slower one. What is the rate of each messenger? CHAPTER XI. LITERAL EQUATIONS IN ONE UNKNOWN NUMBER. 1. The unknown numbers of an equation are frequently to be determined in terms of general numbers, i.e., in terms of numbers represented by letters. The latter are commonly represented by the leading letters of the alphabet, a, b, c, etc. Such numbers as a, b, c, etc., are to be regarded as known. E.g., in the equation x + a = b, a and b are the known numbers, and x is the unknown number. From this equation we obtain x = b — a. 2. It is important to notice that the assumption that x, y, z, etc., are the unknown numbers of an equation, and that a, b, c, etc., are the known numbers, is arbitrary. In the equation x + a = b, either a or b could be taken as the unknown number. If a be taken as the unknown number, we have a = bx; if b be taken as the unknown number, we have b = x + a. 3. A Numerical Equation is one in which all the known numbers are numerals; as 2x+3=7; 4x-3y=7. A Literal Equation is one in which some or all of the known numbers are literal; as 2 ax+3b=5; ax + by = c. 4. Ex. 1. Solve the equation Clearing of fractions, 2 ax 2 a2+2 bx - 2 b2 a2 + 2 ab Transferring and uniting terms, 2(a + b) x = a2 + 2 ab + b2. Dividing by 2 (a + b), x= a + b (a - b)2 2 ab b2. Notice that the above equation, although algebraically fractional, is integral in the unknown number x. The equations which follow are fractional in the unknown number. a a2x = b2x-b+2 abx. Clearing of fractions, (a2+2ab+b2) x = a + b. + a b (x − b) (x − c) (x − b) (x − c) Uniting fractions with common denominator, Had we cleared of fractions at once, we should have introduced the root c of the factor x c equated to 0. 5. A linear equation in one unknown number has one, and only one, distinct root. Any linear equation in one unknown number can be reduced to the form ax = b. If both members of this equation be divided by a, when a 0, we obtain x = b a Since this value of x satisfies the equation, we conclude that every linear equation has at least one root. Let us assume that the equation ax = b has two distinct roots, and let us denote them by r1 and r2. Then, since they must both satisfy the given equation, we have ari = b (1), and ar2 = b (2). Subtracting (2) from (1), we obtain Therefore the assumption that the equation has two distinct roots is untenable. Hence the truth of the principle enunciated. 6. Observe that, in Art. 5, we have no authority for dividing both members of the equation ax = b by a, when a = 0. But if we assume that still gives the solution of a the equation when a = 0, the value of x will be indeterminate (8) or infinite (x), according as b = 0 or b‡0. Evidently, when a = 0 and b = 0, any finite value of x will satisfy the equation; while, when a = 0 and b0, no finite value of x will satisfy the equation. 8. 4 a22 abx + b2 + 3 a2x = 5 a2 — b2x + 2 a2x. 9. (2 a − b)x = 4 a2 - 3 a(b + x). |