Romeyn Henry Rivenburg

“ 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 of the ratio of the fourth term to the fifth is Find x.
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?
ALGEBRA TO QUADRATICS
1. Simplify
2. Simplify
3. Factor (a)
3. Factor (b)
3. Factor (c)
4. Find H. C. F. of and
5. Solve
6. The sum of three numbers is 51 ”

“ 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. ”

“ 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. ”

“ Find correct to three significant figures the negative root of the equation
4. Prove the rule for finding the sum of n terms of a geometrical progression of which the first term is a and the constant ratio is r.
4. Find the sum of 8 terms of the progression
5. A goldsmith has two alloys of gold, the first being pure gold, the second pure gold. How much of each must he take to produce 100 ounces of an alloy which shall be pure gold?
[pg 65] HARVARD UNIVERSITY ELEMENTARY ALGEBRA Time: One Hour and a Half
1. ”

“ 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? ”

“ I. To find the sum and the product of the roots.
The general quadratic equation is
(1)
Or, (2)
To derive the formula, we have by transposing
Completing the square,
Extracting square root,
Transposing,
Hence,
These two values of x we call roots. For convenience represent them by and
[pg 39]
Hence we have shown that
and
Or, referring to equation (2) above, we have the following rule:
When the coefficient of is unity, the sum of the roots is the coefficient of x with the sign changed; the product of the roots is the independent term.
Examples:
1. ”

“ In going 7500 yd. a front wheel of a wagon makes 1000 more revolutions than a rear one. If the wheels were each 1 yd. greater in circumference, a front wheel would make 625 more revolutions than a rear one. ”

“ 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.
[pg 68] MASSACHUSETTS INSTITUTE OF TECHNOLOGY ALGEBRA B Time: One Hour and Three Quarters
1. ”

“ Cube root of algebraic expressions. Cube root of arithmetical numbers.
Theory of Exponents
Proofs:
Meaning of
fractional exponent. zero exponent. negative exponent.
Four rules
To multiply quantities having the same base, add exponents. To divide quantities having the same base, subtract exponents. To raise to a power, multiply exponents. To extract a root, divide the exponent of the power by the index of the root.
Radicals
Radical in its simplest form. ”

“ A man who can row miles an hour in still water rows downstream and returns. The rate of the current is miles per hour, and the time required for the trip is 13 hours. How many hours does he require to return? ”

“ The distance that a body falls from rest in t seconds is given by the formula 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? ”

“ It takes of a second for a ball to go from the pitcher to the catcher, and of a second for the catcher to handle it and get off a throw to second base. It is 90 feet from first base to second, and 130 feet from the catcher's position to second. ”

“ Arithmetic
Geometry
Physics
[pg 36] QUADRATIC EQUATIONS
1. Define a quadratic equation; a pure quadratic; an affected (or complete) quadratic; an equation in the quadratic form.
2. Solve the pure quadratic
Review the first (or usual) method of completing the square. Solve by it the following:
3.
4.
5.
6.
Review the solution by factoring. Solve by it the following:
7.
8.
9.
10.
Solve, by factoring, these equations, which are not quadratics:
11.
12.
13.
Review the solution by formula. Solve by it the following:
14. ”

“ Problems—Quadratic Equations, Simultaneous Quadratics 53-57 College Entrance Examinations 58-80
[pg 7] OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA
Important Definitions
Factors; coefficient; exponent; power; base; term; algebraic sum; similar terms; degree; homogeneous expression; linear equation; root of an equation; root of an expression; identity; conditional equation; prime quantity; highest common factor (H. C. F.) ; lowest common multiple (L. C. M.) ; involution; evolution; imaginary number; real number; rational; similar radicals; binomial surd; pure quadratic equation ”

“ Determine the value of m for which the roots of the equation will be equal: (Hint: See page 40. To have the roots equal, must equal 0.)
4.
5.
6. If is a root of 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 ···.
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? ”

“ Sum of two cubes.
8. Difference of two cubes.
9. Sum or difference of two like powers.
Cases in Factoring
1. Common monomial factor.
2. Trinomial that is a perfect square.
3. The difference of two squares.
(a) Two terms.
(b) Four terms.
(c) Six terms.
(d) Incomplete square.
4. Trinomial of the form
5. Trinomial of the form
[pg 9]
6. Sum or difference of
two cubes. See "Special Rules," 7 and 8. two like powers. See "Special Rules," 9.
7. Common polynomial factor. Grouping.
8. Factor Theorem.
H. C. F. and L. C. M.
H. C. F.
L. C. M.
Fractions
Reduction to lowest terms. ”

“ The H. C. F. of two expressions is and their L. C. M. is If one expression is what is the other?
5. Solve (short method) :
6. Solve
7. Simplify
8. Does ? Does ?
9. Write the fraction with rational denominator, and find its value correct to two decimal places.
10. Simplify (Princeton.)
[pg 33]
1. ”