General Exam Paper 1 Solutions 2002

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Scottish Standard Grade General Course General Exam Paper 1 Solutions 2002 Created by Graduate Bsc (Hons) MathsSci (Open) GIMA 1. (a) Given 9.2 – 3.71 + 6.47 8 11 1 9 . 2 0 - 3 . 7 1 5 . 4 9 5 . 49 + 6 . 47 11 . 96 1 (b). Given 7.29 x 8 7. 29 × 8 58.32 2 7 (c).

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Algebra I for Students Comfortable with Arithmetic
c 2001 Herman Chernoff
February 6, 20022Chapter 1
Introduction
Why should you learn Algebra I? I could elaborate on many reasons why
a knowledge of Mathematics beyond Arithmetic is of interest and of value
to people in the twenty-first century, but for the sake of brevity, I won’t.
It is enough to say that a high school curriculum demands it. What are
the basic concepts to master? They are the ability to translate quantitative
problemsoftherealworldintosymbolicstatementsandtomanipulatethose
statements to arrive at satisfactory answers.
As the title indicates, this book is written for students who are comfort-
able with Arithmetic. One problem with current books on Algebra I, is that
in order to be adopted by state agencies they must be suitable for students
of all levels of attainment, including some who are still unsure of themselves
with simple problems in Arithmetic. As a result, these books tend to be
full of detail and very large and heavy. To encourage the practice that is
required to fix the basic ideas, the students are asked to do many exercises,
which are essentially repetitions of the text, and not very challenging.
AssumingthatthereadersofthisbookarequitecomfortablewithArith-
metic, and willing to do some experimenting, permits bypassing some of the
tedious details in text and exercises. This does not mean that the student
can avoid the necessary practice. It is hoped that he or she will get that
practice in working on more demanding problems. The student should be
prepared to use lots of scratch paper and graph paper in reading this text
and in doing the problems. The custom will be to have problem sets with
threedifferenttypesofproblems. Therewillbeexercises. labeledEx, totest
whether the reader has understood the text well enough to solve a slightly
modified variation of the material in the text. There will be problems, la-
beled Pr, which demand a more comprehensive knowledge of the past text
34 CHAPTER 1. INTRODUCTION
and some imagination. A few of them are quite difficult. There are not
many problems, and the reader reader should try all of them. If one seems
too difficult or complicated, he or she should not get hung up on it, but go
on with the book after a valiant try. It might be rewarding to return to
it after completing the text. Then there will be some puzzles, labeled Pu,
which are almost impossible without an illuminating insight.
The object of the problems is to provide an opportunity to experiment
with alternative approaches. It is expected that this experimentation will
provide the necessary practice to fix the basic underlying techniques of the
subject without boring repetition. On the other hand one should be pre-
pared to deal with getting wrong answers. Perhaps more important than
doing the problems correctly in the first place, is the task of finding out
where one went wrong, how to correct for it and to avoid similar errors
later. The ability to recognize when an answer is wrong by testing it in
various ways is also a useful talent. Finally, even when an answer is correct,
a review of the work will often indicate that the reasoning went in unneces-
sary circles instead of a direct straight line. The task of refining a solution
to cut out unnecessary digressions often leads to a better understanding of
the essentials of the argument. Incidentally, mathematicians tend to refine
their work so much that the readers lose the excitement involved in tracking
the trail of the hunt, where strange ideas led to hints of how to proceed next
to get to the ultimate objective.
Therearenotmanyfiguresinthistext,butthereadershouldbeprepared
touseuplotsofgraphpaper,drawingmanyfigurestohelphisunderstanding
of the text, examples and problems.
In Appendix A, we shall present the answers to the exercises and prob-
lems. Some of them will be detailed solutions of the problems. Such so-
lutions are not necessarily unique, and if the reader has another approach,
that will not necessarily be better or worse. The solutions presented will
usually aim to be concise. That does not necessarily mean they will be as
insightful as they could be. Finally there will be another section with hints
for the puzzles. While it sometimes took me a couple of days to solve some
of these puzzles, they usually became rather easy once the hint occurred to
me. I have a Ph.D. in Applied Mathematics and have been a professor of
Mathematics and of Statistics, so don’t be surprised if some of them defy
your efforts without the hint.
In summary, this should be a short book, freely available on the world
wide web, and its mastery should not take more than a couple of months by
the student willing to spend about 4 hours a week. It would be appreciated
if you spread the news about the availability of the book to friends near and5
far. It is a good idea to collaborate with one or two others of comparable
ability. You should not be in the position where your collaborator does
all the heavy lifting or where you do all of it. That would put one of
you into too passive a mode to get what you should out of the work. An
occasional hint from a better trained friend is ok, but major dependence
will be counter productive. In the near future you may write to me with
complaints, suggestions for improvement (which will become my property),
and questions. I plan to spend a couple of hours a week on dealing with
such comments. My e-mail address is chernoff@stat.harvard.edu and I hope
to use the comments to improve the book. In the event that this book
becomes very popular, I might plan to publish it as a small soft cover text
to be sold at a modest price for students who don’t want to download it
from the computer.
Finally, a warning. This book is not appropriate for students who are
notcomfortablewithArithmetic. Ontheotherhand, thefactthatastudent
has the ability to master Algebra I does not automatically imply that this
book will be right for him or her. The approach that brings out the best in
a student differs very much from one student to another. It is possible that
a reasonably gifted student will not find this presentation suitable. If I knew
howtodescribethecharacteristicsthatdeterminewhetherastudentwillfind
this text useful, I would do so, but I don’t. I do feel that the willingness to
make some effort and to learn from one’s mistakes are important ingredients
for a successful use of this presentation.6 CHAPTER 1. INTRODUCTIONChapter 2
Arithmetic
2.1 Introduction
How well do we understand Arithmetic? With the availability of electronic
calculators, we no longer need to be able to add, subtract, multiply, or divide.
On the other hand, the ability to perform these operations is closely related
to a basic understanding. Moreover the ability to perform these operations
accurately and quickly requires a knowledge of the 10 10 addition and
multiplication tables.
I don’t recommend feats of memory in this text. Memorizing rules for
solving problems is usually a way of avoiding understanding. Without un-
derstanding, great feats of memory are required to handle a limited class of
problems, and there is no ability to handle new types of problems. However,
memorizing the multiplication and addition tables should not be a major
task, and studying them can even be informative. I strongly recommend it,
although I must admit that a friend who had a Ph.D. in Physics had to use
his ngers to add 8 and 5.
In this chapter we review Arithmetic and emphasize a couple of points
that are so well accepted that they are possibly not as well appreciated as
they should be. For example the fact that 35 = 53 is seldom questioned,
but perhaps it should be, because it means that 5+5+5 = 3+3+3+3+3 and
why is that obviously the case? The usual demonstration is to construct a
rectangle of 3 lines of ve points each to represent 3 5. Turning this around, we have 5 lines of 3 points each without changing the
total number of points. See Figure 2.1.1.
Readers who nd this chapter di cult are probably not ready for the
rest of this text. Incidentally, the example discussed above is an example
78 CHAPTER 2. ARITHMETIC
of a general rule that goes under the name of the commutative property of
multiplication. To forget this name is no disaster as long as you can apply
the principle. Chapter 10 contains a list of such principles, major results,
and notation for reference when the text uses them. I doubt that you will
have many occasions to use this list.
Figure 2.1.1
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• • •
• • • • • • • •
• • • • • • • •
• • • • • • • •
2.2 Integers
The basic rules for Arithmetic with integers (whole numbers) are taught
in the early grades. The reader should test his ability to add a list of 4
three digit numbers with many large digits. Then subtraction of several
examples of 6 digit numbers, also with some large digits, should be tried.
Then multiplication of a couple of 3 digit numbers should be tried. Finally
we are ready for long division. Multiply 439 by 6347 to get a product. Now
see if dividing the product by 439 retrieves 6347. You can do several other
examples, using a calculator to save time in checking your reults.
If you have trouble at this stage, it is likely that this text is not appro-
priate for you.
2.3 Fractions
If a lawyer wants to divide