1.2.3. Find four numbers in arithmetical progression, such that the sum of the first two is 1, and the sum of the last two is -19.4. What number added to 2, 20, 9, 34, will make the results proportional?5. Find the middle term of6. Solve(Princeton.)7. A strip of carpet one half inch thick andfeet long is rolled on a roller four inches in diameter. Find how many turns there will be, remembering that each turn increases the diameter by one inch, and that the circumference of a circle equals (approximately)times the diameter.(Harvard.)8. The sum of the first three terms of a geometrical progression is 21, and the sum of their squares is 189. What is the first term?(Yale.)9. Find the geometrical progression whose sum to infinity is 4, and whose second term is10. Solve11. Solve12. Two hundred stones are placed on the ground 3 feet apart, the first being 3 feet from a basket. If the basket and all the stones are in a straight line, how far does a person travel who starts from the basket and brings the stones to it one by one?
1.
2.
3. Find four numbers in arithmetical progression, such that the sum of the first two is 1, and the sum of the last two is -19.
4. What number added to 2, 20, 9, 34, will make the results proportional?
5. Find the middle term of
6. Solve(Princeton.)
7. A strip of carpet one half inch thick andfeet long is rolled on a roller four inches in diameter. Find how many turns there will be, remembering that each turn increases the diameter by one inch, and that the circumference of a circle equals (approximately)times the diameter.(Harvard.)
8. The sum of the first three terms of a geometrical progression is 21, and the sum of their squares is 189. What is the first term?(Yale.)
9. Find the geometrical progression whose sum to infinity is 4, and whose second term is
10. Solve
11. Solve
12. Two hundred stones are placed on the ground 3 feet apart, the first being 3 feet from a basket. If the basket and all the stones are in a straight line, how far does a person travel who starts from the basket and brings the stones to it one by one?
Solve graphically; and check by solving algebraically:
1.2.3.
1.
2.
3.
Determine the value ofmfor which the roots of the equation will be equal: (Hint:Seepage 40. To have the roots equal,must equal 0.)
4.5.6. Ifis a root offind the other root without solving the equation.(Univ. of Penn.)7. How many times does a common clock strike in 12 hours?8. Find the sum to infinity of···.9. Solve10. Find the value of the recurring decimal 2.214214....11. A man purchases a $500 piano by paying monthly installments of $10 and interest on the debt. If the yearly rate is 6%, what is the total amount of interest?12. The arithmetical mean between two numbers isand their geometrical mean is 42. Find the numbers.(College Entrance Exam. Board.)13. If the middle term ofis equal to the fourth term offind the value ofx.(M. I. T.)
4.
5.
6. Ifis a root offind the other root without solving the equation.(Univ. of Penn.)
7. How many times does a common clock strike in 12 hours?
8. Find the sum to infinity of···.
9. Solve
10. Find the value of the recurring decimal 2.214214....
11. A man purchases a $500 piano by paying monthly installments of $10 and interest on the debt. If the yearly rate is 6%, what is the total amount of interest?
12. The arithmetical mean between two numbers isand their geometrical mean is 42. Find the numbers.(College Entrance Exam. Board.)
13. If the middle term ofis equal to the fourth term offind the value ofx.(M. I. T.)
Linear Equations, One Unknown
1. A train running 30 miles an hour requires 21 minutes longer to go a certain distance than does a train running 36 miles an hour. How great is the distance?(Cornell.)2. A man can walkmiles an hour up hill andmiles an hour down hill. He walks 56 miles in 20 hours on a road no part of which is level. How much of it is up hill?(Yale.)3. A physician having 100 cubic centimeters of a 6% solution of a certain medicine wishes to dilute it to a% solution. How much water must he add? (A 6% solution contains 6% of medicine and 94% of water.)(Case.)4. A clerk earned $504 in a certain number of months. His salary was increased 25%, and he then earned $450 in two months less time than it had previously taken him to earn $504. What was his original salary per month?(College Entrance Board.)5. A person who possesses $15,000 employs a part of the money in building a house. He invests one third of the money which remains at 6%, and the other two thirds at 9%, and from these investments he obtains an annual income of $500. What was the cost of the house?(M. I. T.)6. Two travelers have together 400 pounds of baggage. One pays $1.20 and the other $1.80 for excess above the weight carried free. If all had belonged to one person, he would have had to pay $4.50. How much baggage is allowed to go free?(Yale.)7. A man who can rowmiles an hour in still water rows downstream and returns. The rate of the current ismiles per hour, and the time required for the trip is 13 hours. How many hours does he require to return?
1. A train running 30 miles an hour requires 21 minutes longer to go a certain distance than does a train running 36 miles an hour. How great is the distance?(Cornell.)
2. A man can walkmiles an hour up hill andmiles an hour down hill. He walks 56 miles in 20 hours on a road no part of which is level. How much of it is up hill?(Yale.)
3. A physician having 100 cubic centimeters of a 6% solution of a certain medicine wishes to dilute it to a% solution. How much water must he add? (A 6% solution contains 6% of medicine and 94% of water.)(Case.)
4. A clerk earned $504 in a certain number of months. His salary was increased 25%, and he then earned $450 in two months less time than it had previously taken him to earn $504. What was his original salary per month?(College Entrance Board.)
5. A person who possesses $15,000 employs a part of the money in building a house. He invests one third of the money which remains at 6%, and the other two thirds at 9%, and from these investments he obtains an annual income of $500. What was the cost of the house?(M. I. T.)
6. Two travelers have together 400 pounds of baggage. One pays $1.20 and the other $1.80 for excess above the weight carried free. If all had belonged to one person, he would have had to pay $4.50. How much baggage is allowed to go free?(Yale.)
7. A man who can rowmiles an hour in still water rows downstream and returns. The rate of the current ismiles per hour, and the time required for the trip is 13 hours. How many hours does he require to return?
Simultaneous Equations, Two and Three Unknowns
1. A manual training student in making a bookcase finds that the distance from the top of the lowest shelf to the under side of the top shelf is 4 ft. 6 in. He desires to put between these four other shelves of inch boards in such a way that the book space will diminish one inch for each shelf from the bottom to the top. What will be the several spaces between the shelves?2. A quantity of water, sufficient to fill three jars of different sizes, will fill the smallest jar 4 times, or the largest jar twice with 4 gallons to spare, or the second jar three times with 2 gallons to spare. What is the capacity of each jar?(Case.)3. A policeman is chasing a pickpocket. When the policeman is 80 yards behind him, the pickpocket turns up an alley; but coming to the end, he finds there is no outlet, turns back, and is caught just as he comes out of the alley. If he had discovered that the alley had no outlet when he had run halfway up and had then turned back, the policeman would have had to pursue the thief 120 yards beyond the alley before catching him. How long is the alley?(Harvard.)4. A and B together can do a piece of work in 14 days. After they have worked 6 days on it, they are joined by C who works twice as fast as A. The three finish the work in 4 days. How long would it take each man alone to do it?(Columbia.)5. In a certain mill some of the workmen receive $1.50 a day, others more. The total paid in wages each day is $350. An assessment made by a labor union to raise $200 requires $1.00 from each man receiving $1.50 a day, and half of one day's pay from every man receiving more. How many men receive $1.50 a day?(Harvard.)
1. A manual training student in making a bookcase finds that the distance from the top of the lowest shelf to the under side of the top shelf is 4 ft. 6 in. He desires to put between these four other shelves of inch boards in such a way that the book space will diminish one inch for each shelf from the bottom to the top. What will be the several spaces between the shelves?
2. A quantity of water, sufficient to fill three jars of different sizes, will fill the smallest jar 4 times, or the largest jar twice with 4 gallons to spare, or the second jar three times with 2 gallons to spare. What is the capacity of each jar?(Case.)
3. A policeman is chasing a pickpocket. When the policeman is 80 yards behind him, the pickpocket turns up an alley; but coming to the end, he finds there is no outlet, turns back, and is caught just as he comes out of the alley. If he had discovered that the alley had no outlet when he had run halfway up and had then turned back, the policeman would have had to pursue the thief 120 yards beyond the alley before catching him. How long is the alley?(Harvard.)
4. A and B together can do a piece of work in 14 days. After they have worked 6 days on it, they are joined by C who works twice as fast as A. The three finish the work in 4 days. How long would it take each man alone to do it?(Columbia.)
5. In a certain mill some of the workmen receive $1.50 a day, others more. The total paid in wages each day is $350. An assessment made by a labor union to raise $200 requires $1.00 from each man receiving $1.50 a day, and half of one day's pay from every man receiving more. How many men receive $1.50 a day?(Harvard.)
6. There are two alloys of silver and copper, of which one contains twice as much copper as silver, and the other three times as much silver as copper. How much must be taken from each to obtain a kilogram of an alloy to contain equal quantities of silver and copper?(M. I. T.)7. Two automobiles travel toward each other over a distance of 120 miles. A leaves at 9a.m., 1 hour before B starts to meet him, and they meet at 12:00m. If each had started at 9:15a.m., they would have met at 12:00m. also. Find the rate at which each traveled.(M. I. T.)
6. There are two alloys of silver and copper, of which one contains twice as much copper as silver, and the other three times as much silver as copper. How much must be taken from each to obtain a kilogram of an alloy to contain equal quantities of silver and copper?(M. I. T.)
7. Two automobiles travel toward each other over a distance of 120 miles. A leaves at 9a.m., 1 hour before B starts to meet him, and they meet at 12:00m. If each had started at 9:15a.m., they would have met at 12:00m. also. Find the rate at which each traveled.(M. I. T.)
Quadratic Equations
1. Telegraph poles are set at equal distances apart. In order to have two less to the mile, it will be necessary to set them 20 feet farther apart. Find how far apart they are now.(Yale.)2. The distancethat a body falls from rest intseconds is given by the formulaA man drops a stone into a well and hears the splash after 3 seconds. If the velocity of sound in air is 1086 feet a second, what is the depth of the well?(Yale.)3. It requires 2000 square tiles of a certain size to pave a hall, or 3125 square tiles whose dimensions are one inch less. Find the area of the hall. How many solutions has the equation of this problem? How many has the problem itself? Explain the apparent discrepancy.(Cornell.)4. A rectangular tract of land, 800 feet long by 600 feet broad, is divided into four rectangular blocks by two streets of equal width running through it at right angles. Find the width of the streets, if together they cover an area of 77,500 square feet.(M. I. T.)
1. Telegraph poles are set at equal distances apart. In order to have two less to the mile, it will be necessary to set them 20 feet farther apart. Find how far apart they are now.(Yale.)
2. The distancethat a body falls from rest intseconds is given by the formulaA man drops a stone into a well and hears the splash after 3 seconds. If the velocity of sound in air is 1086 feet a second, what is the depth of the well?(Yale.)
3. It requires 2000 square tiles of a certain size to pave a hall, or 3125 square tiles whose dimensions are one inch less. Find the area of the hall. How many solutions has the equation of this problem? How many has the problem itself? Explain the apparent discrepancy.(Cornell.)
4. A rectangular tract of land, 800 feet long by 600 feet broad, is divided into four rectangular blocks by two streets of equal width running through it at right angles. Find the width of the streets, if together they cover an area of 77,500 square feet.(M. I. T.)
5. (a) The heightyto which a ball thrown vertically upward with a velocity of 100 feet per second rises inxseconds is given by the formula,In how many seconds will the ball rise to a height of 144 feet?5.(b) Draw the graph of the equation(College Entrance Board.)6. Two launches race over a course of 12 miles. The first steamsmiles an hour. The other has a start of 10 minutes, runs over the first half of the course with a certain speed, but increases its speed over the second half of the course by 2 miles per hour, winning the race by a minute. What is the speed of the second launch? Explain the meaning of the negative answer.(Sheffield Scientific School.)7. The circumference of a rear wheel of a certain wagon is 3 feet more than the circumference of a front wheel. The rear wheel performs 100 fewer revolutions than the front wheel in traveling a distance of 6000 feet. How large are the wheels?(Harvard.)8. A man starts from home to catch a train, walking at the rate of 1 yard in 1 second, and arrives 2 minutes late. If he had walked at the rate of 4 yards in 3 seconds, he would have arrivedminutes early. Find the distance from his home to the station.(College Entrance Board.)
5. (a) The heightyto which a ball thrown vertically upward with a velocity of 100 feet per second rises inxseconds is given by the formula,In how many seconds will the ball rise to a height of 144 feet?
5.(b) Draw the graph of the equation(College Entrance Board.)
6. Two launches race over a course of 12 miles. The first steamsmiles an hour. The other has a start of 10 minutes, runs over the first half of the course with a certain speed, but increases its speed over the second half of the course by 2 miles per hour, winning the race by a minute. What is the speed of the second launch? Explain the meaning of the negative answer.(Sheffield Scientific School.)
7. The circumference of a rear wheel of a certain wagon is 3 feet more than the circumference of a front wheel. The rear wheel performs 100 fewer revolutions than the front wheel in traveling a distance of 6000 feet. How large are the wheels?(Harvard.)
8. A man starts from home to catch a train, walking at the rate of 1 yard in 1 second, and arrives 2 minutes late. If he had walked at the rate of 4 yards in 3 seconds, he would have arrivedminutes early. Find the distance from his home to the station.(College Entrance Board.)
Simultaneous Quadratics
1. Two cubical coal bins together hold 280 cubic feet of coal, and the sum of their lengths is 10 feet. Find the length of each bin.2. The sum of the radii of two circles is 25 inches, and the difference of their areas issquare inches. Find the radii.
1. Two cubical coal bins together hold 280 cubic feet of coal, and the sum of their lengths is 10 feet. Find the length of each bin.
2. The sum of the radii of two circles is 25 inches, and the difference of their areas issquare inches. Find the radii.
3. The area of a right triangle is 150 square feet, and its hypotenuse is 25 feet. Find the arms of the triangle.4. The combined capacity of two cubical tanks is 637 cubic feet, and the sum of an edge of one and an edge of the other is 13 feet. (a) Find the length of a diagonal of any face of each cube. (b) Find the distance from upper left-hand corner to lower right-hand corner in either cube.5. A and B run a mile. In the first heat A gives B a start of 20 yards and beats him by 30 seconds. In the second heat A gives B a start of 32 seconds and beats him byyards. Find the rate at which each runs.(Sheffield.)6. After street improvement it is found that a certain corner rectangular lot has lostof its length andof its width. Its perimeter has been decreased by 28 feet, and the new area is 3024 square feet. Find the reduced dimensions of the lot.(College Entrance Board.)7. A man spends $539 for sheep. He keeps 14 of the flock that he buys, and sells the remainder at an advance of $2 per head, gaining $28 by the transaction. How many sheep did he buy, and what was the cost of each?(Yale.)8. A boat's crew, rowing at half their usual speed, row 3 miles downstream and back again in 2 hours and 40 minutes. At full speed they can go over the same course in 1 hour and 4 minutes. Find the rate of the crew, and the rate of the current in miles per hour.(College Entrance Board.)9. Find the sides of a rectangle whose area is unchanged if its length is increased by 4 feet and its breadth decreased by 3 feet, but which loses one third of its area if the length is increased by 16 feet and the breadth decreased by 10 feet.(M. I. T.)
3. The area of a right triangle is 150 square feet, and its hypotenuse is 25 feet. Find the arms of the triangle.
4. The combined capacity of two cubical tanks is 637 cubic feet, and the sum of an edge of one and an edge of the other is 13 feet. (a) Find the length of a diagonal of any face of each cube. (b) Find the distance from upper left-hand corner to lower right-hand corner in either cube.
5. A and B run a mile. In the first heat A gives B a start of 20 yards and beats him by 30 seconds. In the second heat A gives B a start of 32 seconds and beats him byyards. Find the rate at which each runs.(Sheffield.)
6. After street improvement it is found that a certain corner rectangular lot has lostof its length andof its width. Its perimeter has been decreased by 28 feet, and the new area is 3024 square feet. Find the reduced dimensions of the lot.(College Entrance Board.)
7. A man spends $539 for sheep. He keeps 14 of the flock that he buys, and sells the remainder at an advance of $2 per head, gaining $28 by the transaction. How many sheep did he buy, and what was the cost of each?(Yale.)
8. A boat's crew, rowing at half their usual speed, row 3 miles downstream and back again in 2 hours and 40 minutes. At full speed they can go over the same course in 1 hour and 4 minutes. Find the rate of the crew, and the rate of the current in miles per hour.(College Entrance Board.)
9. Find the sides of a rectangle whose area is unchanged if its length is increased by 4 feet and its breadth decreased by 3 feet, but which loses one third of its area if the length is increased by 16 feet and the breadth decreased by 10 feet.(M. I. T.)
1. Ifandfind the value of:(a)(b)2. Reduce to a mixed number:
1. Ifandfind the value of:
(a)
(b)
2. Reduce to a mixed number:
Simplify:
3.4.5. A's age 10 years hence will be 4 times what B's age was 11 years ago, and the amount that A's age exceeds B's age is one third of the sum of their ages 8 years ago. Find their present ages.6. Draw the lines represented by the equationsand6.and find by algebra the coördinates of the point where they intersect.7. Solve the equations8. Solve
3.
4.
5. A's age 10 years hence will be 4 times what B's age was 11 years ago, and the amount that A's age exceeds B's age is one third of the sum of their ages 8 years ago. Find their present ages.
6. Draw the lines represented by the equations
and
6.and find by algebra the coördinates of the point where they intersect.
7. Solve the equations
8. Solve
1. Solve by factoring:2. Show that3. How many pairs of numbers will satisfy simultaneously the two equations3.Show by means of a graph that your answer is correct.3.What is meant by eliminatingxin the above equations by substitution? by comparison? by subtraction?4. Find the square root of 223,728.5. Simplify: (a)5. Simplify:(b)6. Solve the equation7. How far must a boy run in a potato race if there arenpotatoes in a straight line at a distancedfeet apart, the first being at a distanceafeet from the basket?
1. Solve by factoring:
2. Show that
3. How many pairs of numbers will satisfy simultaneously the two equations
3.Show by means of a graph that your answer is correct.
3.What is meant by eliminatingxin the above equations by substitution? by comparison? by subtraction?
4. Find the square root of 223,728.
5. Simplify: (a)
5. Simplify:(b)
6. Solve the equation
7. How far must a boy run in a potato race if there arenpotatoes in a straight line at a distancedfeet apart, the first being at a distanceafeet from the basket?
Time: Three Hours
Six questions are required; two from GroupA, two from GroupB, and both questions of GroupC. No extra credit will be given for more than six questions.
Group A
1. (a) Resolve the following into their prime factors:(1)(2)1.(b) Find the H. C. F. and the L. C. M. of2. (a) Simplify2.(b) Ifprove thatzis a mean proportional betweenxandy.3. A crew can row 10 miles in 50 minutes downstream, and 12 miles in an hour and a half upstream. Find the rate of the current and of the crew in still water.
1. (a) Resolve the following into their prime factors:
(1)
(2)
1.(b) Find the H. C. F. and the L. C. M. of
2. (a) Simplify
2.(b) Ifprove thatzis a mean proportional betweenxandy.
3. A crew can row 10 miles in 50 minutes downstream, and 12 miles in an hour and a half upstream. Find the rate of the current and of the crew in still water.
Group B
4. (a) Determine the values ofkso that the equation4. (a)shall have equal roots.4.(b) Solve the equations4.(c) Plot the following two equations, and find from the graphs the approximate values of their common solutions:5. Two integers are in the ratioIncrease each by 15, and the difference of their squares is 999. What are the integers?6. A man has $539 to spend for sheep. He wishes to keep 14 of the flock that he buys, but to sell the remainder at a gain of $2 per head. This he does and gains $28. How many sheep did he buy, and at what price each?
4. (a) Determine the values ofkso that the equation
4. (a)shall have equal roots.
4.(b) Solve the equations
4.(c) Plot the following two equations, and find from the graphs the approximate values of their common solutions:
5. Two integers are in the ratioIncrease each by 15, and the difference of their squares is 999. What are the integers?
6. A man has $539 to spend for sheep. He wishes to keep 14 of the flock that he buys, but to sell the remainder at a gain of $2 per head. This he does and gains $28. How many sheep did he buy, and at what price each?
Group C
7. (a) Find the seventh term of7.(b) Derive the formula for the sum ofnterms of an arithmetic progression.8. A ball falling from a height of 60 feet rebounds after each fall one third of its last descent. What distance has it passed over when it strikes the ground for the eighth time?
7. (a) Find the seventh term of
7.(b) Derive the formula for the sum ofnterms of an arithmetic progression.
8. A ball falling from a height of 60 feet rebounds after each fall one third of its last descent. What distance has it passed over when it strikes the ground for the eighth time?
1. Find the H. C. F.:2. Solve the following set of equations:3. Expand and simplify:4. An automobile goes 80 miles and back in 9 hours. The rate of speed returning was 4 miles per hour faster than the rate going. Find the rate each way.5. Simplify:6. Solve forx:7. A, B, and C, all working together, can do a piece of work indays. A works twice as fast as C, and A and C together could do the work in 4 days. How long would it take each one of the three to do the work alone?
1. Find the H. C. F.:
2. Solve the following set of equations:
3. Expand and simplify:
4. An automobile goes 80 miles and back in 9 hours. The rate of speed returning was 4 miles per hour faster than the rate going. Find the rate each way.
5. Simplify:
6. Solve forx:
7. A, B, and C, all working together, can do a piece of work indays. A works twice as fast as C, and A and C together could do the work in 4 days. How long would it take each one of the three to do the work alone?
1. Solve the following set of equations:2. Simplify: (a)(b)3. Find, and simplify, the 23d term in the expansion of4. The weight of an object varies directly as its distance from the center of the earth when it is below the earth's surface, and inversely as the square of its distance from the center when it is above the surface. If an object weighs 10 pounds at the surface, how far above, and how far below the surface will it weigh 9 pounds? (The radius of the earth may be taken as 4000 miles.)5. Solve the following pair of equations forxandy:6. Find the value ofwhen7. From a square of pasteboard, 12 inches on a side, square corners are cut, and the sides are turned up to form a rectangular box. If the squares cut out from the corners had been 1 inch larger on a side, the volume of the box would have been increased 28 cubic inches. What is the size of the square corners cut out? (See the figure on the blackboard.)
1. Solve the following set of equations:
2. Simplify: (a)(b)
3. Find, and simplify, the 23d term in the expansion of
4. The weight of an object varies directly as its distance from the center of the earth when it is below the earth's surface, and inversely as the square of its distance from the center when it is above the surface. If an object weighs 10 pounds at the surface, how far above, and how far below the surface will it weigh 9 pounds? (The radius of the earth may be taken as 4000 miles.)
5. Solve the following pair of equations forxandy:
6. Find the value ofwhen
7. From a square of pasteboard, 12 inches on a side, square corners are cut, and the sides are turned up to form a rectangular box. If the squares cut out from the corners had been 1 inch larger on a side, the volume of the box would have been increased 28 cubic inches. What is the size of the square corners cut out? (See the figure on the blackboard.)
Time: One Hour and a Half
Arrange your work neatly and clearly, beginning each question on a separate page.
1. Simplify the following expression:2. (a) Write the middle term of the expansion ofby the binomial theorem.2.(b) Find the value ofifand2. (b)and reduce the result to a form having only positive exponents.3. Find correct to three significant figures the negative root of the equation4. Prove the rule for finding the sum ofnterms of a geometrical progression of which the first term isaand the constant ratio isr.4.Find the sum of 8 terms of the progression5. A goldsmith has two alloys of gold, the first beingpure gold, the secondpure gold. How much of each must he take to produce 100 ounces of an alloy which shall bepure gold?
1. Simplify the following expression:
2. (a) Write the middle term of the expansion ofby the binomial theorem.
2.(b) Find the value ofif
and
2. (b)and reduce the result to a form having only positive exponents.
3. Find correct to three significant figures the negative root of the equation
4. Prove the rule for finding the sum ofnterms of a geometrical progression of which the first term isaand the constant ratio isr.
4.Find the sum of 8 terms of the progression
5. A goldsmith has two alloys of gold, the first beingpure gold, the secondpure gold. How much of each must he take to produce 100 ounces of an alloy which shall bepure gold?
Time: One Hour and a Half
1. Solve the simultaneous equations1.and verify your results.2. Solve the equationobtaining the values of the roots correct to three significant figures.3. Write out the first four terms ofFind the fourth term of this expansion when3.expressing the result in terms of a single radical, and without fractional or negative exponents.4. Reduce the following expression to a polynomial inaandb:5. The cost of publishing a book consists of two main items: first, the fixed expense of setting up the type; and, second, the running expenses of presswork, binding, etc., which may be assumed to be proportional to the number of copies. A certain book costs 35 cents a copy if 1000 copies are published at one time, but only 19 cents a copy if 5000 copies are published at one time. Find (a) the cost of setting up the type for the book, and (b) the cost of presswork, binding, etc., per thousand copies.
1. Solve the simultaneous equations
1.and verify your results.
2. Solve the equationobtaining the values of the roots correct to three significant figures.
3. Write out the first four terms ofFind the fourth term of this expansion when
3.expressing the result in terms of a single radical, and without fractional or negative exponents.
4. Reduce the following expression to a polynomial inaandb:
5. The cost of publishing a book consists of two main items: first, the fixed expense of setting up the type; and, second, the running expenses of presswork, binding, etc., which may be assumed to be proportional to the number of copies. A certain book costs 35 cents a copy if 1000 copies are published at one time, but only 19 cents a copy if 5000 copies are published at one time. Find (a) the cost of setting up the type for the book, and (b) the cost of presswork, binding, etc., per thousand copies.
Time: One Hour and a Half
1. Find the highest common factor and the lowest common multiple of the three expressions2. Solve the quadratic equation2.computing the value of the larger root correct to three significant figures.3. In the expression3.substitute forxandythe values3.and reduce the resulting expression to its simplest form.4. State and prove the formula for the sum of the firstnterms of a geometric progression in whichais the first term andrthe constant ratio.5. A state legislature is to elect a United States senator, a majority of all the votes cast being necessary for a choice. There are three candidates, A, B, and C, and 100 members vote. On the first ballot A has the largest number of votes, receiving 9 more votes than his nearest competitor, B; but he fails of the necessary majority. On the second ballot C's name is withdrawn, and all the members who voted for C now vote for B, whereupon B is elected by a majority of 2. How many votes were cast for each candidate on the first ballot?
1. Find the highest common factor and the lowest common multiple of the three expressions
2. Solve the quadratic equation
2.computing the value of the larger root correct to three significant figures.
3. In the expression
3.substitute forxandythe values
3.and reduce the resulting expression to its simplest form.
4. State and prove the formula for the sum of the firstnterms of a geometric progression in whichais the first term andrthe constant ratio.
5. A state legislature is to elect a United States senator, a majority of all the votes cast being necessary for a choice. There are three candidates, A, B, and C, and 100 members vote. On the first ballot A has the largest number of votes, receiving 9 more votes than his nearest competitor, B; but he fails of the necessary majority. On the second ballot C's name is withdrawn, and all the members who voted for C now vote for B, whereupon B is elected by a majority of 2. How many votes were cast for each candidate on the first ballot?
Time: One Hour and Three Quarters
1. Factor the expressions:2. Simplify the expression:3. Find the value ofwhen4. Solve the equations:5. Solve the equations:6. Two squares are formed with a combined perimeter of 16 inches. One square contains 4 square inches more than the other. Find the area of each.7. A man walked to a railway station at the rate of 4 miles an hour and traveled by train at the rate of 30 miles an hour, reaching his destination in 20 hours. If he had walked 3 miles an hour and ridden 35 miles an hour, he would have made the journey in 18 hours. Required the total distance traveled.
1. Factor the expressions:
2. Simplify the expression:
3. Find the value ofwhen
4. Solve the equations:
5. Solve the equations:
6. Two squares are formed with a combined perimeter of 16 inches. One square contains 4 square inches more than the other. Find the area of each.
7. A man walked to a railway station at the rate of 4 miles an hour and traveled by train at the rate of 30 miles an hour, reaching his destination in 20 hours. If he had walked 3 miles an hour and ridden 35 miles an hour, he would have made the journey in 18 hours. Required the total distance traveled.
Time: One Hour and Three Quarters
1. How many terms must be taken in the series 2, 5, 8, 11, ··· so that the sum shall be 345?2. Prove the formulafor solving the quadratic equation3. Find all values ofafor whichis a root ofand check your results.4. Solveand sketch the graphs.5. The sum of two numbersxandyis 5, and the sum of the two middle terms in the expansion ofis equal to the sum of the first and last terms. Find the numbers.6. Solve6.(Hint:Divide byand substitute)7. In anticipation of a holiday a merchant makes an outlay of $50, which will be a total loss in case of rain, but which will bring him a clear profit of $150 above the outlay if the day is pleasant. To insure against loss he takes out an insurance policy against rain for a certain sum of money for which he has to pay a certain percentage. He then finds that whether the day be rainy or pleasant he will make $80 clear. What is the amount of the policy, and what rate did the company charge him?
1. How many terms must be taken in the series 2, 5, 8, 11, ··· so that the sum shall be 345?
2. Prove the formulafor solving the quadratic equation
3. Find all values ofafor whichis a root ofand check your results.
4. Solveand sketch the graphs.
5. The sum of two numbersxandyis 5, and the sum of the two middle terms in the expansion ofis equal to the sum of the first and last terms. Find the numbers.
6. Solve
6.(Hint:Divide byand substitute)
7. In anticipation of a holiday a merchant makes an outlay of $50, which will be a total loss in case of rain, but which will bring him a clear profit of $150 above the outlay if the day is pleasant. To insure against loss he takes out an insurance policy against rain for a certain sum of money for which he has to pay a certain percentage. He then finds that whether the day be rainy or pleasant he will make $80 clear. What is the amount of the policy, and what rate did the company charge him?
Time: Two Hours
1. Simplify2. Find the prime factors of(a)(b)3. (a) Simplify3.(b) Show that4. Definehomogeneous terms.4.For what value ofnisa homogeneous binomial?5. Extract the square root of6. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water asand in the second asWhat quantity must be taken from each, so as to form a third mixture which shall contain 5 gallons of wine and 9 gallons of water?7. Find a quantity such that by adding it to each of the quantitiesa,b,c,d, we obtain four quantities in proportion.8. What values must be given toaandb, so thatmay be equal?
1. Simplify
2. Find the prime factors of
(a)
(b)
3. (a) Simplify
3.(b) Show that
4. Definehomogeneous terms.
4.For what value ofnisa homogeneous binomial?
5. Extract the square root of
6. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water asand in the second asWhat quantity must be taken from each, so as to form a third mixture which shall contain 5 gallons of wine and 9 gallons of water?
7. Find a quantity such that by adding it to each of the quantitiesa,b,c,d, we obtain four quantities in proportion.
8. What values must be given toaandb, so thatmay be equal?
Time: Two Hours
1. Factor the following expressions:1.(a)1.(b)1.(c)2. (a) Simplify2.(b) Extract the square root of3. Solve the following equations:3.(a)3.(b)3.(c)4. Simplify:4.(a)4.(b)4.(c) Find
1. Factor the following expressions:
1.(a)
1.(b)
1.(c)
2. (a) Simplify
2.(b) Extract the square root of
3. Solve the following equations:
3.(a)
3.(b)
3.(c)
4. Simplify:
4.(a)
4.(b)
4.(c) Find
5. Plot the graphs of the following system, and determine the solution from the point of intersection:6. (a) Derive the formula for the solution of6.(b) Determine the value ofmfor which the roots ofare (i) equal, (ii) real, (iii) imaginary.6.(c) Form the quadratic equation whose roots areand7. A page is to have a margin of 1 inch, and is to contain 35 square inches of printing. How large must the page be, if the length is to exceed the width by 2 inches?8. (a) In an arithmetical progression the sum of the first six terms is 261, and the sum of the first nine terms is 297. Find the common difference.8.(b) Three numbers whose sum is 27 are in arithmetical progression. If 1 is added to the first, 3 to the second, and 11 to the third, the sums will be in geometrical progression. Find the numbers.8.(c) Derive the formula for the sum ofnterms of a geometrical progression.9. (a) Expand and simplify9.(b) For what value ofxwill the ratiobe equal to the ratio?
5. Plot the graphs of the following system, and determine the solution from the point of intersection:
6. (a) Derive the formula for the solution of
6.(b) Determine the value ofmfor which the roots ofare (i) equal, (ii) real, (iii) imaginary.
6.(c) Form the quadratic equation whose roots are
and
7. A page is to have a margin of 1 inch, and is to contain 35 square inches of printing. How large must the page be, if the length is to exceed the width by 2 inches?
8. (a) In an arithmetical progression the sum of the first six terms is 261, and the sum of the first nine terms is 297. Find the common difference.
8.(b) Three numbers whose sum is 27 are in arithmetical progression. If 1 is added to the first, 3 to the second, and 11 to the third, the sums will be in geometrical progression. Find the numbers.
8.(c) Derive the formula for the sum ofnterms of a geometrical progression.
9. (a) Expand and simplify
9.(b) For what value ofxwill the ratiobe equal to the ratio?
Time: Three Hours
1. Simplify:2. Find the H. C. F. and L. C. M. of3. A grocer buys eggs at 4 for 7¢. He sellsof them at 5 for 12¢, and the rest at 6 for 11¢, making 27¢ by the transaction. How many eggs does he buy?4. Solve fort:5. Find the square root of6. (a) For what values ofmwill the roots ofbe equal?6.(b) Ifis a root offind the other root without solving the equation.7. (a) Solve forx:7.(b) Solve form:8. Solve the system:9. Two boats leave simultaneously opposite shores of a rivermi. wide and pass each other in 15 min. The faster boat completes the tripmin. before the other reaches the opposite shore. Find the rates of the boats in miles per hour.10. Write the sixth term ofwithout writing the preceding terms.11. The sum of the 2d and 20th terms of an A. P. is 10, and their product isWhat is the sum of sixteen terms?
1. Simplify:
2. Find the H. C. F. and L. C. M. of
3. A grocer buys eggs at 4 for 7¢. He sellsof them at 5 for 12¢, and the rest at 6 for 11¢, making 27¢ by the transaction. How many eggs does he buy?
4. Solve fort:
5. Find the square root of
6. (a) For what values ofmwill the roots ofbe equal?
6.(b) Ifis a root offind the other root without solving the equation.
7. (a) Solve forx:
7.(b) Solve form:
8. Solve the system:
9. Two boats leave simultaneously opposite shores of a rivermi. wide and pass each other in 15 min. The faster boat completes the tripmin. before the other reaches the opposite shore. Find the rates of the boats in miles per hour.
10. Write the sixth term ofwithout writing the preceding terms.
11. The sum of the 2d and 20th terms of an A. P. is 10, and their product isWhat is the sum of sixteen terms?
Time: Two Hours
Candidates who are at this time takingbothAlgebra A and Algebra B may omit from Algebra A questions 4, 5, and 6, and from Algebra B questions 1 (a), 3, and 4.
1. Simplify2. (a) Divideby2.(b) Simplify3. Factor: (a)3. Factor:(b)4. Solve5. Solve forxandy:6. The road from A to B is uphill for 5 mi., level for 4 mi., and then downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks halfway from A to B and back again to A in 3 hr. and 55 min.; and later he walks from A to B in 3 hr. and 52 min. What are his rates of walking uphill, downhill, and on the level, if these do not vary?
1. Simplify
2. (a) Divideby
2.(b) Simplify
3. Factor: (a)
3. Factor:(b)
4. Solve
5. Solve forxandy:
6. The road from A to B is uphill for 5 mi., level for 4 mi., and then downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks halfway from A to B and back again to A in 3 hr. and 55 min.; and later he walks from A to B in 3 hr. and 52 min. What are his rates of walking uphill, downhill, and on the level, if these do not vary?
1. Solve (a)1. Solve(b)1. Solve(c)
1. Solve (a)
1. Solve(b)
1. Solve(c)
2. Solve forxandy, checking one solution in each problem:2.(a)2.(b)3. A man arranges to pay a debt of $3600 in 40 monthly payments which form an A. P. After paying 30 of them he still owesof his debt. What was his first payment?4. If 4 quantities are in proportion and the second is a mean proportional between the third and fourth, prove that the third will be a mean prop. between the first and second.5. In the expansion ofthe ratio of the fourth term to the fifth isFindx.6. Two men A and B can together do a piece of work in 12 days; B would need 10 days more than A to do the whole work. How many days would it take A alone to do the work?
2. Solve forxandy, checking one solution in each problem:
2.(a)
2.(b)
3. A man arranges to pay a debt of $3600 in 40 monthly payments which form an A. P. After paying 30 of them he still owesof his debt. What was his first payment?
4. If 4 quantities are in proportion and the second is a mean proportional between the third and fourth, prove that the third will be a mean prop. between the first and second.
5. In the expansion ofthe ratio of the fourth term to the fifth isFindx.
6. Two men A and B can together do a piece of work in 12 days; B would need 10 days more than A to do the whole work. How many days would it take A alone to do the work?
1. Simplify2. Simplify3. Factor (a)3. Factor(b)3. Factor(c)4. Find H. C. F. ofand5. Solve6. The sum of three numbers is 51; if the first number be divided by the second, the quotient is 2 and the remainder 5; if the second number be divided by the third, the quotient is 3 and the remainder 2. What are the numbers?
1. Simplify
2. Simplify
3. Factor (a)
3. Factor(b)
3. Factor(c)
4. Find H. C. F. ofand
5. Solve
6. The sum of three numbers is 51; if the first number be divided by the second, the quotient is 2 and the remainder 5; if the second number be divided by the third, the quotient is 3 and the remainder 2. What are the numbers?
1. Factor2. Solve3. The second term of a geometrical progression isand the fifth term isFind the first term and the ratio.4. Solve the following equations and check your results by plotting:5. Solve6. In an arithmetical progressionFindaandl.7. Expand by the binomial theorem and simplify:8. The diagonal of a rectangle is 13 ft. long. If each side were longer by 2 ft., the area would be increased by 38 sq. ft. Find the lengths of the sides.
1. Factor
2. Solve
3. The second term of a geometrical progression isand the fifth term isFind the first term and the ratio.
4. Solve the following equations and check your results by plotting:
5. Solve
6. In an arithmetical progressionFindaandl.
7. Expand by the binomial theorem and simplify:
8. The diagonal of a rectangle is 13 ft. long. If each side were longer by 2 ft., the area would be increased by 38 sq. ft. Find the lengths of the sides.
1. Find the H. C. F. ofand2. Solve:2.(a)2.(b)3. A farmer sold a horse at $75 for which he had paidxdollars. He realizedxper cent profit by his sale. Findx.4. Find the 13th term and the sum of 13 terms of the arithmetical progression···.5. The difference between two numbers is 48. Their arithmetical mean exceeds their geometrical mean by 18. Find the numbers.6. Expand by the binomial theorem and simplify7. Solve:8. Solve the following equations and check the results by finding the intersections of the graphs of the two equations:
1. Find the H. C. F. ofand
2. Solve:
2.(a)
2.(b)
3. A farmer sold a horse at $75 for which he had paidxdollars. He realizedxper cent profit by his sale. Findx.
4. Find the 13th term and the sum of 13 terms of the arithmetical progression
···.
5. The difference between two numbers is 48. Their arithmetical mean exceeds their geometrical mean by 18. Find the numbers.
6. Expand by the binomial theorem and simplify
7. Solve:
8. Solve the following equations and check the results by finding the intersections of the graphs of the two equations: