73. A fox is 70 leaps ahead of a hound and takes 5 leaps while the hound takes 3; but 3 of the hound's leaps equal 7 of the fox's. How many leaps must the hound take to catch the fox? 74. A rabbit makes 5 leaps while a dog makes 4; but 3 of the dog's leaps are equal to 4 of the rabbit's. If the rabbit has a start of 20 leaps, how many leaps will each take before the rabbit is caught? 75. A hound is 39 of his leaps behind a rabbit that takes 7 leaps while the hound takes 8. If 6 leaps of the rabbit are equal to 5 leaps of the hound, how many leaps must the hound take to catch the rabbit? 76. A wheelman and a pedestrian leave the same place at the same time to go to a point 54 miles distant, the former traveling 3 times as fast as the latter. The wheelman makes the trip and returning meets the pedestrian in 63 hours from the time they started. What is the rate of each? 77. If 1 pound of lead loses of a pound, and 1 pound of iron loses of a pound when weighed in water, how many pounds of lead and of iron are there in a mass of lead and iron that weighs 159 pounds in air and 143 pounds in water? 78. If 97 ounces of gold weighs 92 ounces when it is weighed in water, and 21 ounces of silver weighs 19 ounces when it is weighed in water, how many ounces of gold and of silver are there in a mass of gold and silver that weighs 320 ounces in air and 298 ounces in water? 79. A merchant increases his capital annually by † of it, and at the end of each year takes out $800 for expenses. At the end of three years, after taking out his expenses, he finds that his capital is $6325. What was his original capital? 80. A merchant added annually to his capital of it, and at the end of each year took out $1000 for expenses. If at the end of the third year, after taking out the last $1000, he had g of his original capital, what was his original capital? 81. A cistern can be filled by one pipe in 20 minutes, by another in 15 minutes, and it can be emptied by a third in 10 minutes. If the three pipes are running at the same time, how long will it take to fill the cistern? 82. A man walked from A to B at the rate of 2 miles an hour, and rode back at the rate of 3 miles an hour, being gone 13 hours. How far is it from A to B? 83. An express train whose rate is 40 miles an hour starts 1 hour and 4 minutes after a freight train and overtakes it in 1 hour and 36 minutes. How many miles does the freight train run per hour? 84. The distance from Albany to Syracuse is 148 miles. A canal boat leaves Albany for Syracuse, moving at the rate of 3 miles in 2 hours; another leaves Syracuse for Albany, moving at the rate of 5 miles in 4 hours. How far from Albany do they meet? 85. A steamer goes 5 miles downstream in the same time that it goes 3 miles upstream; but if its rate each way is diminished 4 miles an hour, its rate downstream will be twice its rate upstream. How fast does it go in each direction? 86. A, B, and C together can do a piece of work in 3 days; B can do as much as A, and C can do as much as B in a day. In how many days can each do the work alone? 87. A can do a piece of work in 6 days that B can do in 14 days. A began the work, and after a certain number of days B finished the work in 10 days from the time How many days did B work? took his place and it was begun by A. 88. In a certain weight of gunpowder the niter composed 10 pounds more than of the weight, the sulphur 3 pounds more than, and the charcoal 3 pounds less than of the niter. What was the weight of the gunpowder? 89. A library containing 16,000 volumes was divided into five departments. In the department of history there were twice as many volumes as in the department of science, and 500 more than as many volumes as in the juvenile department. Of fiction there were 7 times as many volumes as of science, and 500 less than 8 times as many as in the reference department. How many volumes were there in each department? 90. An estate was divided among four heirs, A, B, C, and D. If the value of the estate had been $1000 less, what A received would have been of it, and what B received of it; if the value of the estate had been greater, what C received would have been of it, and what D received of it. What sum did each receive? 91. A father takes 3 steps while his son takes 5; but 2 of the father's steps are equal to 3 of the son's. How many steps will the son require to overtake his father, who is 36 steps ahead? 92. A purse contained some money and a ring worth $10 more than the money. If the purse was worth as much as the money it contained, and the purse and the money together were worth as much as the ring, what was the value of each? 93. Brass is 83 times as heavy as water, and iron 71⁄2 times as heavy as water. A mixed mass weighs 57 pounds, and when immersed displaces 7 pounds of water. How many pounds of each metal does the mass contain? 94. A man began business with $4725, and annually added to his capital of it. At the end of each year he put aside a certain sum for expenses. If at the end of the third year, after taking out the sum for expenses, his capital was $3800, what were his annual expenses? 95. The sum of two numbers is s, and their difference d. What are the numbers? If the sum of two numbers is 30, and their difference is 6, or 12. A problem in which the numbers assumed to be known are represented by letters to which any values may be assigned is called a General Problem. Problem 95 is a general problem. The results obtained in solving a general problem may be considered formula for solving similar problems. 96. Divide c cents between two boys so that one shall have d cents more than the other. If c = 50 and d = 10, how much will each receive? 97. A horse and a saddle are together worth a dollars, and the horse is worth m times as much as the saddle. What is the value of each? What, if a = 160 and m = = 3? 98. Divide b into two parts one of which represents m times the other. What will they be, if b represents 100, and m, 4? 99. An estate of a dollars is divided between two heirs in the proportion of m to n. What is the share of each? What is the share of each, if a = 40,000, m = = 5, and n = 3? 100. If A can do a piece of work in a days, and B in b days, in what time can both do it working together? Give the result, if a 10 and b = 15. = 101. An alloy of two metals is composed of m parts of one to n parts of the other. How many pounds of each are required in the composition of a pounds of the alloy ? Bell metal is an alloy of 5 parts of tin and 16 parts of copper. How many pounds of tin and of copper are there in a bell which weighs 4200 pounds? 102. A wheelman set out from B at the rate of r miles an hour. a hours later another started in pursuit at the rate of p miles an hour. How far from B will the second wheelman overtake the first? What will be the distance, if r = 10, p 12, and a = = =8? 103. A man traveled from home at the rate of a miles an hour and returned at the rate of b miles an hour. If he made the entire journey in h hours, how far from home did he go? How 4, b=3, and h = 15? far, if a = SIMULTANEOUS SIMPLE EQUATIONS TWO UNKNOWN NUMBERS 201. 1. If x+y=12, what is the value of a? of y? How many values may have? How many may y have? = 2. In the expression x + y = 12, x and y each may have an indefinite number of values, but if, at the same time, x what is the value of x? of y? - y = 4, 3. Although one equation containing two unknown numbers has an indefinite number of values for each unknown number, or is indeterminate, what can be said about the values of the unknown numbers, when two equations are given involving the same values of the unknown numbers, but in different relations, that is, when two independent equations are given? 202. Two or more equations in which the unknown numbers have the same values are called Simultaneous Equations. If x and y represent the same numbers in 2x + 3y = 19 as they represent in 5 x − y = 22, 2 x + 3y = 19 and 5 x − y = 22 are simultaneous equations. 203. Equations that represent different relations between the unknown numbers, and so cannot be reduced to the same form, are called Independent Equations. 3x + 3y = 18 and 2x + 2y = 12 really express but one relation between x and y; viz., that their sum is 6. Hence, both equations may be reduced to the same form, as x + y = 6. But 3x + 3y 18 and x + 3y = 14 express different relations between x and y, and cannot be reduced to the same form. Hence, they are independent equations. = 204. An equation whose unknown numbers may have an infinite number of values is called an Indeterminate Equation. x + y = 6 is an indeterminate equation, because, if x = 2, y = 4; if x = 3, y = 3; if x = 21, y = 31; etc. |