Challenging Problems in Algebra

Alfred S. Posamentier, Charles T. Salkind, Charles T. Salkind

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486691480
• eBook ISBN: 9780486131542
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Designed for high-school students and teachers with an interest in mathematical problem-solving, this stimulating collection includes more than 300 problems that are "off the beaten path" — i.e., problems that give a new twist to familiar topics that introduce unfamiliar topics. With few exceptions, their solution requires little more than some knowledge of elementary algebra, though a dash of ingenuity may help.
Readers will find here thought-provoking posers involving equations and inequalities, diophantine equations, number theory, quadratic equations, logarithms, combinations and probability, and much more. The problems range from fairly easy to difficult, and many have extensions or variations the author calls "challenges."
By studying these nonroutine problems, students will not only stimulate and develop problem-solving skills, they will acquire valuable underpinnings for more advanced work in mathematics.

SECTION II
Second Year Algebra

9 Diophantine Equations:
The Whole Answer

An equation with two or more variables whose values are restricted to integers is known as a Diophantine equation after Diophantus of Alexandria, who studied them about 1800 years ago. They may arise in describing situations involving objects that occur only in integral quantities. Some of the problems here, for example, concern people, coins, or pieces of merchandise. A solution of a Diophantine equation is an ordered pair (or triple, or quadruple, etc.) of integers. When solutions exist, there are generally an infinite number of them. If further restrictions are imposed on the values of the variables, such as that they must be positive or less than a certain integer, there may be a finite number of solutions or even just one.

9-1 A shopkeeper orders 19 large and 3 small packets of marbles, all alike. When they arrive at the shop, he finds the packets broken open with all the marbles loose in the container. Can you help the shopkeeper make new packets with the proper number of marbles in each, if the total number of marbles is 224?

Challenge Redo the problem with 19 small packets and 3 large packets.

9-2 Find the integral solutions of 6x + 15y = 23.

Challenge 1 Solve in positive integers 13x + 21y = 261.

Challenge 2 Show that there are no positive integral solutions of 17x + 15y = 5 but that 17x – 15y = 5 has infinitely many positive integral solutions.

9-3 A picnic group transported in n buses (where n > 1 and not prime) to a railroad station, together with 7 persons already waiting at the station, distribute themselves equally in 14 railroad cars. Each bus, nearly filled to its capacity of 52 persons, carried the same number of persons. Assuming that the number of picnickers is the smallest possible for the given conditions, find the number of persons in each railroad car.

Challenge Solve the problem with the following changes:

(a) 11 persons are waiting at the station instead of 7.

(b) There are 22 railroad cars and each of 21 cars has the same number of persons, but in the 22nd car there are 10 vacant seats.

9-4 Find the number of ways that change can be made of $1.00 with 50 coins (U.S.).

Challenge Solve the problem restricting the change to dimes, nickels, and cents.

9-5 Let x be a member of the set {1, 2, 3, 4, 5, 6, 7}, y a member of the set {8, 9, 10, 11, 12, 13, 14}, and z, a member of the set {15, 16, 17, 18, 19, 20, 21}. If a solution of x + y + z = 33 is defined a...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Introduction
• Preparing to Solve a Problem
• Section I First Year Algebra
• Section II Second Year Algebra
• Answers
• Appendices