Part 3
Any two of the quantities x + y x^2 + y^2 xy x - y x^3 + y^3 x^3 - y^3 x^2 + xy + y^2 x^2 - xy + y^2 given.
x + y = 5, x^2 - xy + y^2 = 7.
METHOD: Solve for x + y and x - y; then add to get x, subtract to get y.
CASE IV.
Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree. x^5 + y^5 = 242, x + y = 2.
METHOD: Let x = u + v and y = u - v, and substitute in both equations.
~Special Devices~
I. Consider some compound quantity like xy, [x - y]^(1/2), [xy]^(1/2), x/y, etc., as the unknown, at first. Solve for the compound unknown, and combine the resulting equation with the simpler original equation.
x^2 y^2 + xy = 6, x + 2y = -5.
II. Divide the equations member by member. Then solve by Case I, II, or III.
x^3 - y^3 = 152, x - y = 2.
III. Eliminate the quadratic terms. Then solve by Case I, II, or III.
xy + x = 15, xy + y = 16.
SIMULTANEOUS QUADRATICS
Solve:
1. x + y = 7, x^2 + 4xy = 57.
2. 2x^2 = 46 + y^2, xy + y^2 = 14.
3. x^2 + y^2 = 25, x + y = 1.
4. x^4 + y^4 = 2, x - y = 2.
5. x^3 + y^3 = 28, x + y = 4.
6. x^2 y^2 + xy - 12 = 0, x + y = 4.
7. 2xy - x + 2y = 16, 3xy + 2x - 4y = 10.
8. (3x - 2y)(2x - 3y) = 26, x + 1 = 2y.
9. 4x^2 + 3xy + 2y^2 = 18, 3x^2 + 2xy - y^2 = 3.
10. x^5 + y^5 = 242, x + y = 2.
11. x - y + [x - y]^(1/2) = 6, xy = 5.
12. 4x^2 - x + y = 67, 3x^2 - 3y = 27.
13. x - y - [x - y]^(1/2) = 2, x^3 - y^3 = 2044. (_Yale._)
14. x^2 + xy + x = 14, y^2 + xy + y = 28. (_Princeton._)
15. x^2 + y^2 = 13, y^2 = 4(x - 2). Plot the graph of each equation. (_Cornell._)
16. x^2 + y^2 = xy + 37, x + y = xy - 17. (_Columbia._)
_In grouping the answers, be sure to associate each value of x with the corresponding value of y._
17. The course of a yacht is 30 miles in length and is in the shape of a right triangle one arm of which is 2 miles longer than the other. What is the distance along each side?
~Reference:~ The chapter on Simultaneous Quadratics in any algebra.
RATIO AND PROPORTION
1. Define ratio, proportion, mean proportional, third proportional, fourth proportional.
2. Find a mean proportional between 4 and 16; 18 and 50; 12m^2n and 3mn^2.
3. Find a third proportional to 4 and 7; 5 and 10; a^2 - 9 and a - 3.
4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14.
5. Write out the proofs for the following, stating the theorem in full in each case:
(_a_) The product of the extremes equals etc.
(_b_) If the product of two numbers equals the product of two other numbers, either pair etc.
(_c_) Alternation.
(_d_) Inversion.
(_e_) Composition.
(_f_) Division.
(_g_) Composition and division.
(_h_) In a series of equal ratios, the sum of the antecedents is to the sum of the consequents etc.
(_i_) Like powers or like roots of the terms of a proportion etc.
6. If x : m :: 13 : 7, write all the possible proportions that can be derived from it. [See (5) above.]
7. Given rs = 161m; write the eight proportions that may be derived from it, and quote your authority.
8. (_a_) What theorem allows you to change any proportion into an equation?
(_b_) What theorem allows you to change any equation into a proportion?
9. If xy = rg, what is the ratio of x to g? of y to r? of y to g?
10. Find two numbers such that their sum, difference, and the sum of their squares are in the ratio 5 : 3 : 51. (_Yale._)
~Reference:~ The chapter on Ratio and Proportion in any algebra.
An easy and powerful method of proving four expressions in proportion is illustrated by the following example:
Given a : b = c : d;
prove that 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.
Let a/b = r. Therefore a = br.
Also c/d = r. Therefore c = dr.
Substitute the value of a in the first ratio, and c in the second:
Then
(3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3b^3r^3 + 5b^3r)/(3b^3r^3 - 5b^3r) = [b^3r(3r^2 + 5)]/[b^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).
Also
(3c^3 + 5cd^2)/(3c^3 - 5cd^2) = (3d^3r^3 + 5d^3r)/(3d^3r^3 - 5d^3r) = [d^3r(3r^2 + 5)]/[d^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).
Therefore (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3c^3 + 5cd^2)/(3c^3 - 5cd^2).
Axiom 1.
Or, 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.
If a : b = c : d, prove:
1. a^2 + b^2 : a^2 = c^2 + d^2 : c^2.
2. a^2 + 3b^2 : a^2 - 3b^2 = c^2 + 3d^2 : c^2 - 3d^2.
3. a^2 + 2b^2 : 2b^2 = ac + 2bd : 2bd.
4. 2a + 3c : 2a - 3c = 8b + 12d : 8b - 12d.
5. a^2 - ab + b^2 : (a^3 - b^3)/a = c^2 - cd + d^2 : (c^3 - d^3)/c.
6. The second of three numbers is a mean proportional between the other two. The third number exceeds the sum of the other two by 20; and the sum of the first and third exceeds three times the second by 4. Find the numbers.
7. Three numbers are proportional to 5, 7, and 9; and their sum is 14. Find the numbers. (_College Entrance Board._)
8. A triangular field has the sides 15, 18, and 27 rods, respectively. Find the dimensions of a similar field having 4 times the area.
~ARITHMETICAL PROGRESSION~
1. Define an arithmetical progression.
Learn to derive the three formulas in arithmetical progression:
l = a + (n - 1)d, S = (n/2)(a + l), S = (n/2)[2a + (n - 1)d].
2. Find the sum of the first 50 odd numbers.
3. In the series 2, 5, 8, ···, which term is 92?
4. How many terms must be taken from the series 3, 5, 7, ···, to make a total of 255?
5. Insert 5 arithmetical means between 11 and 32.
6. Insert 9 arithmetical means between 7-1/2 and 30.
7. Find x, if 3 + 2x, 5 + 6x, 9 + 5x are in A. P.
8. The 7th term of an arithmetical progression is 17, and the 13th term is 59. Find the 4th term.
9. How can you turn an A. P. into an equation?
10. Given a = -5/3, n = 20, S = -5/3, find d and l.
11. Find the sum of the first n odd numbers.
12. An arithmetical progression consists of 21 terms. The sum of the three terms in the middle is 129; the sum of the last three terms is 237. Find the series. (Look up the short method for such problems.) (_Mass. Inst. of Technology._)
13. B travels 3 miles the first day, 7 miles the second day, 11 miles the third day, etc. In how many days will B overtake A who started from the same point 8 days in advance and who travels uniformly 15 miles a day?
~Reference:~ The chapter on Arithmetical Progression in any algebra.
~GEOMETRICAL PROGRESSION~
1. Define a geometrical progression.
Learn to derive the four formulas in geometrical progression:
{ I. l = ar^(n - 1). {II. S = (ar^n - a)/(r - 1).
{III. S = (rl - a)/(r - 1). { IV. S_{[infinity]} = (a)/(1 - r).
2. How many terms must be taken from the series 9, 18, 36, ··· to make a total of 567?
3. In the G. P. 2, 6, 18, ···, which term is 486?
4. Find x, if 2x - 4, 5x - 7, 10x + 4 are in geometrical progression.
5. How can you turn a G. P. into an equation?
6. Insert 4 geometrical means between 4 and 972.
7. Insert 6 geometrical means between 5/16 and 5120.
8. Given a = -2, n = 5, l = -32; find r and S.
9. If the first term of a geometrical progression is 12 and the sum to infinity is 36, find the 4th term.
10. If the series 3-1/3, 2-1/2, ··· be an A. P., find the 97th term. If a G. P., find the sum to infinity.
11. The third term of a geometrical progression is 36; the 6th term is 972. Find the first and second terms.
12. Insert between 6 and 16 two numbers, such that the first three of the four shall be in arithmetical progression, and the last three in geometrical progression.
13. A rubber ball falls from a height of 40 inches and on each rebound rises 40% of the previous height. Find by formula how far it falls on its eighth descent. (_Yale._)
~Reference:~ The chapter on Geometrical Progression in any algebra.
~THE BINOMIAL THEOREM~
1. Review the Binomial Theorem laws. (See Involution.)
Expand:
2. (b - n)^7.
3. (x + x^(-1))^5.
4. [a/x - x/a]^6.
5. [x/2y - [xy]^(1/2)]^5.
6. (x^2 - x + 2)^3.
7. [(2[b^2]^(1/3))/(y) + (3[y^(1/2)])/(b^3)]^4.
8. (a + b)^n = a^n + na^(n - 1)b + [n(n - 1)]/(1·2) a^(n - 2)b^2 + [n(n - 1)(n - 2)]/(1·2·3) a^(n - 3)b^3 + [n(n - 1)(n - 2)(n - 3)]/(1·2·3·4) a^(n - 4) b^4 + ···.
Show by observation that the formula for the
(r + 1)th term = [n(n - 1)(n - 2)···(n - r + 1)]/[1·2·3·4 ··· r] a^(n - r)b^r.
9. Indicate what the 97th term of (a + b)^n would be.
10. Using the expansion of (a + b)^n in (8), derive a formula for the rth term by observing how each term is made up, then generalizing.
Using either the formula in (8) or (10), whichever you are familiar with, find:
11. The 4th term of [a + 1/a]^(30).
12. The 8th term of (1 + x[y^(1/2)])^(13).
13. The middle term of (2a^(3/4) - y[a^(1/3)])^(10).
14. The term not containing x in [x^3 - 2/x]^(12).
15. The term containing x^(18) in [x^2 - a/x]^(15).
~Reference:~ The chapter on The Binomial Theorem in any algebra.
~MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND~
1. Solve the equation x^2 - 1.6x - .23 = 0, obtaining the values of the roots correct to three significant figures. (_Harvard._)
2. Write the roots of (x^2 + 2x)(x^2 - 2x - 3)(x^2 - x + 1) = 0. (_Sheffield Scientific School._)
3. Solve 2[2x + 2]^(1/2) + [2x + 1]^(1/2) = (12x + 4)/([8x + 8]^{1/2}). (_Yale._)
4. Solve the equation V = (H/3)(B + x + [Bx]^(1/2)) for x, taking H = 6, B = 8, and V = 28; and verify your result. (_Harvard._)
5. Solve { x : y = 2 : 3, { x^2 + y^2 = 5(x + y) + 2.
6. Solve 2x^2 - 4x + 3[x^2 - 2x + 6]^(1/2) = 15. (_Coll. Ent. Board._)
7. Find all values of x and y which satisfy the equations: { x^(1/2) + y^(1/2) = 4, { 1/[[x + 1]^(1/2) - x^(1/2)] - 1/[[x + 1]^(1/2) + x^(1/2)] = y. (_Mass. Inst. of Technology._)
8. If [alpha] and [beta] represent the roots of px^2 + qx + r = 0, find [alpha] + [beta], [alpha] - [beta], and [alpha][beta] in terms of p, q, and r. (_Princeton._)
9. Form the equation whose roots are 2 + [3]^(1/2) and 2 - [-3]^(1/2).
10. Determine, without solving, the character of the roots of 9x^2 - 24x + 16 = 0. (_College Entrance Board._)
11. If a : b = c : d, prove that a + b : c + d = [a^2 + b^2]^(1/2) : [c^2 + d^2]^(1/2). (_College Entrance Board._)
12. Given a : b = c : d. Prove that a^2 + b^2 : (a^3)/(a + b) = c^2 + d^2 : (c^3)/(c + d). (_Sheffield._)
13. The 9th term of an arithmetical progression is 1/6; the 16th term is 5/2. Find the first term. (_Regents._)
Solve graphically:
1. x^2 - x - 6 = 0.
2. x^2 + 3x - 10 = 0.
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 [3a^5 + (b^(3/4))/(2)]^8.
6. Solve (x + 1)/(3x + 2) = (2x - 3)/(3x - 2) - 1 - 36/(4 - 9x^2). (_Princeton._)
7. A strip of carpet one half inch thick and 29-6/7 feet 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) 22/7 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 3/4.
10. Solve 4x + 4[3x^2 - 7x + 3]^(1/2) = 3x^2 - 3x + 6.
11. Solve { 2x^2 + 3xy - 5y^2 = 4, { 2xy + 3y^2 = -3.
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. { x^2 + y^2 = 25, { x + y = 1.
2. x^2 - 3x - 18 = 0.
3. x^2 + 3x - 10 = 0.
Determine the value of m for which the roots of the equation will be equal: (HINT: See page 40. To have the roots equal, b^2 - 4ac must equal 0.)
4. 2x^2 - mx + 12-1/2 = 0.
5. (m - 1)x^2 + mx + 2m - 3 = 0.
6. If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find 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 2/(2^(1/2)), 1/(2^(1/2)), 1/(2[2]^(1/2)), ···.
9. Solve [x/2 + 6/x]^2 - 6[x/2 + 6/x] + 8 = 0.
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 is 42-1/2, and their geometrical mean is 42. Find the numbers. (_College Entrance Exam. Board._)
13. If the middle term of [3x - (1)/(2[x^(1/2)])]^4 is equal to the fourth term of [2[x^(1/2)] + 1/2x]^7, find the value of x. (_M. I. T._)
~PROBLEMS~
~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 walk 2-1/2 miles an hour up hill and 3-1/2 miles 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 3-1/2% 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 row 4-1/3 miles an hour in still water rows downstream and returns. The rate of the current is 2-1/4 miles 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._)
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 9 A.M., 1 hour before B starts to meet him, and they meet at 12:00 M. If each had started at 9:15 A.M., they would have met at 12:00 M. 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 distance S that a body falls from rest in t seconds is given by the formula S = 16t^2. A 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 height y to which a ball thrown vertically upward with a velocity of 100 feet per second rises in x seconds is given by the formula, y = 100x - 16x^2. In how many seconds will the ball rise to a height of 144 feet?
(_b_) Draw the graph of the equation y = 100x - 16x^2. (_College Entrance Board._)
6. Two launches race over a course of 12 miles. The first steams 7-1/2 miles 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 arrived 2-1/2 minutes 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 is 125[pi] square 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 by 9-5/11 yards. Find the rate at which each runs. (_Sheffield._)
6. After street improvement it is found that a certain corner rectangular lot has lost 1/10 of its length and 1/15 of 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._)
COLLEGE ENTRANCE EXAMINATIONS
~UNIVERSITY OF CALIFORNIA~
ELEMENTARY ALGEBRA
1. If a = 4, b = -3, c = 2, and d = -4, find the value of: (_a_) ab^3 - 3cd^2 + 2(3a - b)(c - 2d). (_b_) 2a^3 - 3b^4 + (4c^3 + d^3)(4c^2 + d^2).
2. Reduce to a mixed number: (3a^4 - 4a^3 - 10a^2 + 41a - 28)/(a^2 - 3a + 4).
Simplify: