University of South Carolina
Math 221: Math for Elementary Educators
Section 001
Fall 2011
Test 3
Please write only your name on the test sheet.
Place all work and answers on the blank sheets provided.
Computation
Evaluate each of the following expressions using the specificed algorithm.
(If you do not know the algorithm but can evaluate it another way, you may do so for
partial credit.)
1. 2.4 + 0.89, expanded form
Solution:
9
4
8
+
2.4 + 0.89 = 2 +
+
10
10 100
4
8
9
=2+
+
+
10 10 100
9
12
+
=2+
10 100
10
2
9
=2+
+
+
10 10 100
2
9
=2+1+
+
10 100
9
2
+
=3+
10 100
= 3.29
2. 3.2 − 1.45, standard algorithm
Solution:
3. 2.5 × 0.43, fraction form
Solution:
43
25
×
10 100
25 × 43
=
10 × 100
1075
=
1000
= 1.075
2.5 × 0.43 =
4. 4.5 ÷ 0.12, standard algorithm
Solution: We start by multiplying both numbers by 100 to avoid having a decimal in
the divisor.
Word Problems
Set up and solve an equation for each of the following scenarios.
1. Alice weighs 160 pounds on Earth and 416 pounds on Jupiter. Baalzech’k weighs 650
pounds on his home planet of Jupiter. How much does he weigh on Earth?
Solution: If we let x denote Baalzech’k’s (lol double apostrophe) weight on Earth,
then we have the following proportion:
160
x
=
416
650
650 × 160
=x
416
250 = x.
2. A shirt originally cost $40 but is now on sale for 30% off. How much does the shirt
cost now?
Solution: One way is to start with $40 and take away 30% of $40.
40 − (0.3 × 40) = 40 − 12
= 28.
Alternatively, you could view the problem as asking “What is 70% of the shirt’s cost?”
and evaluate.
0.7 × 40 = 28
3. Alice donated $250 to a charity event, which turned out to be 8% of the total funds
raised. How much money did the charity raise in total?
Solution:
0.08 × base = 250
base = 250 ÷ 0.08
base = 3125
Short Answer
Provide a brief discussion for each question.
1. In converting
4
11
to a decimal, one stage of the process might look like the following:
Explain why the division does not need to be carried out any further. What is the
4
decimal representation of 11
?
Solution: We have a remainder of 4, which has occurred already in the division (we
started with a 4). This means the subsequent divisions will give the same result as
before, namely another copy of “36”. This will repeat for all eternity, so we’ll get 0.36
4
as the decimal representation of 11
.
2. In converting 0.12 to a rational number, one stage of the process might look like the
following:
x = 0.12121212 · · ·
100x = 12.121212 · · ·
100x = 12 + x
Justify what is happenning on the third line. Finish the computation to determine the
representation of 0.12 as a rational number in reduced form.
Solution: We defined x to be 0.12. Since 12.12 is the same as 12 + 0.12, we are
justified in writing 12 + x instead. We can finish the computation by solving for x.
100x = 12 + x
99x = 12
12
x=
99
4
x=
33
√
3. In this problem, we will prove that 3 is irrational. Provide details for each bulleted
item.
√
We prove this indirectly by first believing 3 actually can be written as a rational
number. That is, we believe there are integers a and b such that
a √
= 3
b
and ab is in reduced form. Doing a little algebra, we can determine that
a2 = 3b2 .
• Based on the previous equation, how can you determine that there is exactly one
factor of 3 in the prime factorization of a2 ?
Solution: We know that a2 has at least one factor of 3 in it’s prime factorization from
the equation a2 = 3b2 (there is a 3 right there in the expression). There can’t be any
other 3’s inside of b2 , because then ab would not be reduced, but we specifically chose
a and b so that the fraction would be reduced.
Next comes the pivotal question: How many factors of 3 are in the prime factorization
of a?
• Explain why this question has no answer.
Solution: It cannot be that a has no factors of 3, since then a2 would also have no
factors of 3. It also cannot be that a has one or more factors of 3, since then a2 would
have two or more factors of 3.
Since the question above has no answer, no such a can possibly exist.
• Having arrived at an impossible state of affairs, what must we conclude?
√
Solution: The only statement we took on faith was that 3 could be written as a
rational number. Believing this forced us to believe in the existence of a number a
which both cannot have factors of 3 and also must have factors of 3. Since
√ such an
a cannot possibly exist, we have to throw away our original belief that 3 could be
written as a rational number.