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Math 105
Final Exam
Name _____________________
December 15, 2004
There are 25 problems on this exam. Obviously, you can’t spend the same amount of time on
each problem as on the midterm exams. The instructions have been modified to reflect this, so
please follow them carefully. Your strategy is to do as many of the problems as you can, starting
with the easiest ones and skipping the harder, longer ones until the end. After all, in a given 15
minute period you can do either one hard problem and earn five points or five easier ones and
earn twenty-five points. You should first build up a solid foundation score, then finish it off by
doing the harder ones.
For #1-11, draw an appropriate diagram (or two or three if necessary to show a progression of
ideas on your overhead projector). Then give a brief explanation of what’s going on.
Example: Show 6 + 3 using the set model of addition.
Sample Answer:
Start with 6 marbles.
Add in 3 more marbles.
Now you’ve got nine
marbles. So 6 + 3 = 9
1. Show 437 + 685 using place-value cards. (Be sure to explain exchanging)
2. Show –23+7 using the number line/kinesthetic model of addition.
3. Show 3 7 using the Cartesian Product model of multiplication.
4. Show 28  4 using the repeated subtraction model of division.
5. Show 28  4 using the partition model of division.
6. Subtract 11 from –5 using +’s and –’s.
7. Show  5  (6) on a number line.
8. Show 6 5 on a number line.
9. Show why –13 is less than –8 even though 8 is less than 13.
3
1
from 2 using fraction strips. (Be sure to explain what the LCD represents and
5
3
why you need it.)
10. Subtract 1
11. Multiply 3 by 4.56 using place value cards. (Be sure to explain exchanging.)
For #12-14, demonstrate your ability to use the instructional algorithms for arithmetic. Note,
as in the example, what each line represents.
Example: Add 23, 37, and 52 using the instructional algorithm for addition.
23
37
Sample Answer:
 52
12
100
112
3 + 7 + 2 = 12
20 + 30 + 50 = 100
12 + 100 = 112
For #12-14, see instructions on previous page.
12. Add 5876 + 7218 + 493 using the instructional algorithm for addition.
13. Multiply 5.2 by 6.74 using the instructional algorithm for multiplication, appropriately
modified for decimals. Then explain in one sentence why it is “legal” to “scoot” the
decimal in three places in the answer, as you’ve done.
14. Divide 12731 by 43 using the Scaffold Algorithm. Express your answer according to the
notation of The Division Algorithm ( a  bq  r ).
15. (a) Complete this definition: Two sets are equivalent if ____________________________.
(b) Use your definition to prove that the sets F and R below are equivalent

2 3 1 5 6
7
2 1 10 
1 1 1 1 1 1 1 1 1 1 
F   , , , , , , , , , ,... and R  1,
,
, ,
,
,
,
, ,
, ...
1 2 3 4 5 6 7 8 9 10 
 2 3 2 5 6 77 4 3 10

(c) Use your definition to prove that the two line segments to the right
are equivalent.
16. Use the method of prime factorizations to find GCD(1980, 1050) and LCM(1980, 1050).
17. Use the Euclidean algorithm to find GCD(1980, 1050) and use that answer to find
LCM(1980, 1050).
18. (a) The irrational numbers are dense in the real numbers. Explain what this means.


(b) The principal square roots, 1, 2 , 3 , 2, 5 , 6 , 7 , 2 2 , 3,... , are not dense in the
set of real numbers. Explain what this means.
19. Find three fractions between the two below in three different ways then explain why there are
infinitely many fractions between these two.
4
5
and .
5
6
20. Prove that
2 is irrational.
21. (a) State the definition of an infinite set. (b) Then give an example of an infinite set and
verify that it really is an infinite set by showing how it satisfies the definition. (c) Also, give
an example of a finite set and show why it is not infinite based on this definition.
22. At a meeting of registered nurses, 86% were women. There were 35 male nurses in
attendance. How many female nurses were at the meeting?
For Problems 23-25, It’s time to make another trip to ToonTown. Recall that since cartoon
characters only have eight fingers, everything in ToonTown is done in base eight. That
includes decimals and fractions!
23. (a) Which is bigger,
(b) Which is bigger,
3
3
in Modesto or in ToonTown? Explain.
4
4
13
13
in Modesto or
in ToonTown? Explain.
10
10
24. How does a teacher in ToonTown explain to a student how to combine the following
fractions? (Be sure to discuss the LCD. Do all work in base eight.)
2 7 3
 
5 10 4
25. Suppose you go into a store in ToonTown with a 25% off coupon. Now in ToonTown, 25%
means, similar to us, 25 per 100, but recall that the 25 and 100 are in base eight. Sales tax is
charged in ToonTown at a rate of 7%. Also recall that their money isn’t dollars, it’s Disneys.
If you buy a new frying pan to hit Popeye on the head with, and if it’s regularly priced at 25
Disneys, how much to you pay total? Round your answer to two octal places. (octal – like
“decimal” but “deci-“ means ten!)