Motion Problems 1

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Motion Problems 1
One hour after Yolanda started walking from X to
Y, a distance of 45 miles, Bob started walking
along the same road from Y to X. If Yolandas
walking rate was 3 miles per hour and Bobs was 4
miles per hour, how many miles had Bob walked
when they met? (A) 24 (B) 23 (C) 22 (D) 21 (E)
19.5
2
Motion Problems 2
How many miles long is the route from Houghton
to Callahan? (1) It will take 1 hour less time
to travel the entire route at an average rate of
55 miles per hour than at an average rate of 50
miles per hour. (2) If will take 11 hours to
travel the first half of the route at an average
rate of 25 miles per hour.
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Variation 1

• On a certain airline, the price of a ticket is
  directly proportional to the number of miles to
  be traveled. If the ticket for a 900-mile trip on
  this airline costs 120, which of the following
  gives the number of dollars changed for a k-mile
  trip on this airline?
• (2k)/15
• 2/(15k)
• 15/(2k)
• (15k)/2
• (40k)/3

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Variation 2

• In a certain formula, p is directly
  proportional to s and inversely proportional to
  r. If p 1 when r 0.5 and s 2, what is the
  value of p in terms of r and s ?
• s/r
• r/(4s)
• s/(4r)
• r/s
• (4r)/s

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Set/Overlapping 1

• In company X, 30 percent of the employees live
  over ten miles from work and 60 percent of the
  employees who live over 10 miles from work are in
  car pools. If 40 percent of the employees of
  company X are in car pools, what percent of the
  employees of company X live ten miles or less
  from work and are in car pools?
• 12 (D) 28
• 20 (E) 32
• 22

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Set/Overlapping 2
In each production lot for a certain toy, 25
percent of the toys are red and 75 percent of the
toys are blue. Half the toys are size A and half
are size B. If 10 out of a lot of 100 toys are
red and size A, how many of the toys are blue and
size B? (A) 15 (B) 25 (C) 30 (D)
35 (E) 40
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Set/Overlapping 3
Out of a total of 1,000 employees at a certain
corporation, 52 percent are female and 40 percent
of these females work in research. If 60 percent
of the total number of employees work in
research, how many male employees do NOT work in
research? (A) 520 (B) 480 (C) 392 (D) 208 (E)
88
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Venn Diagrams
Out of 40 students, 14 are taking English and 29
are taking Chemistry. If 5 students are in both
classes, how many students are in neither classes?
(A) 1 (B) 2 (C) 4 (D) 8 (E) 10
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Counting Techniques

• Worst case.
• Ordered places or for each.
• 3. Permutation (Can be done by number 2).
• 4. Combination.

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Worst Case

• Of the science books in a certain supply room,
  50 are on botany, 65 are on zoology, 90 are on
  physics. 50 are on geology, and 110 are on
  chemistry. If science books are removed randomly
  from the supply room, how many must be removed to
  ensure that 80 of the books removed are on the
  same science?
•  
•   (A) 81  (B) 159  (C) 166  (D) 285 
  (E) 324

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Ordered Places 1

• Katie must place five stuffed animals--a
  duck, a goose, a panda, a turtle and a swan in a
  row in the display window of a toy store. How
  many different displays can she make if the duck
  and the goose must be either first or last?
•  
• (A) 120 (B) 60 (C) 24 (D) 12 (E)
   6

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Ordered Places 2

• The president of a country and 4 other
  dignitaries are scheduled to sit in a row on the
  5 chairs represented above. If the president must
  sit in the center chair, how many different
  seating arrangements are possible for the 5
  people?
•  
•  

 (A) 4  (B) 5  (C) 20  (D) 24  (E) 120
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Ordered Places 3

• In how many arrangements can a teacher seat 3
  girls and 3 boys in a row of 6 if the boys are to
  have the first, third, and fifth seats?
•  
• (A)      6
• (B)     9
• (C)     12
• (D)     36
• (E)    720

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FOR EACH

• If a customer makes exactly 1 selection from
  each of the 5 categories listed below, what is
  the greatest number of different ice cream
  sundaes that a customer can create? 
•  
• 12 ice cream flavours
• 10 kinds of candy
•   8 liquid toppings
•   5 kinds of nuts
•   With or without whip cream.
•  
• (A) 9600 (B) 4800 (C) 2400 (D) 800 (E) 400

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Ordered Places or Permutation

• Given a selected committee of 8, in how many
  ways, can the members of the committee divide the
  responsibilities of a president, vice president,
  and secretary?
• (A) 120 (B) 336 (C) 56 (D) 1500 (E) 100

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Ordered Places or Permutation

•  How many four-digit numbers can you form using
  ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if
  the numbers can be used only once?
• (A) 72 (B) 5000 (C) 4536 (D) 10 000 (E) 210

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Combination 1

• From a group of 8 secretaries, select 3
  persons for a promotion. How many distinct
  selections are there?
• (A) 56 (B) 336 (C) 512 (D) 9 (E) 200

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Combination 2
A person has the following bills 1, 5, 10,
20, 50. How many unique sums can one form using
any number of these bills only once? (A) 10
(B) 15 (C) 31 (D) 35 (E) 40
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Permutation/Combination 1
In a certain contest, Fred must select any 3 of
5 different gifts offered by the sponsor. From
how many different combinations of 3 gifts can
Fred make his selection? (A) 10 (B) 15 (C) 20
(E) 30 (F) 60
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Permutation/Combination 2
Ben and Ann are among 7 contestants from which 4
semifinalist are to be selected. Of the different
possible selections, how many contain neither Ben
nor Ann? (A) 5 (B) 6 (C) 7 (D) 14 (E) 21
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Probability (Simple)
P(A) ( favorable outcomes of A)/( total
possible outcomes of A)
For example, consider a deck of 52 cards P (A
ace) 4/52 1/13 P(A
spade) 13/52 1/4
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Example 1
Two fair six-sided dice are rolled what is the
probability of having 5 as the sum of the number?
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Example 2
Two six-sided dice are rolled what is the
probability of having 12 as the sum of the
numbers?
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Probability (Compound)

• P(A and B) P(A) P(B)
• P(A or B) P(A) P(B) P(A and B)
• P( not A) 1 P(A)
• NOTE if A and B have nothing in common
• then P(A and B) 0

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Example 1
If a fair coin is tossed twice, what is the
probability that on the first toss the coin lands
heads and on the second toss the coin lands
tails? (A)    1/6 (B)      1/3 (C)     1/4
(D)    1/2 (E)     1
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Example 2

• If a fair coin is tossed twice what is the
  probability that it will land either heads both
  times or tails both times?
• (A)      1/8
• (B)      1/6
• (C)     1/4
• (D)       1/2
• (E)     1

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Example 3
A bowman hits his target in 1/2 of his shots.
What is the probability of him missing the target
at least once in three shots?
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Example 4
What is the probability that a card selected from
a deck will be either an ace or a spade?
(A)      2/52 (B)      2/13 (C)       7/26
(D)      4/13 (E)      17/52
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Example 5

• If someone draws a card at random from a deck and
  then, without replacing the first card, draws a
  second card, what is the probability that both
  cards will be aces?

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Example 6
If there are 30 red and blue marbles in a jar,
and the ratio of red to blue marbles is 23, what
is the probability that, drawing twice, you will
select two red marbles if you return the marbles
after each draw?
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Example 7

• Now consider the same question as example 6 with
  the condition that you do not return the marbles
  after each draw.