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 Yolanda’s walking rate was 3 miles per hour and Bob’s 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
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.
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?
(A)
(2k)/15
(B)
2/(15k)
(C)
15/(2k)
(D)
(15k)/2
(E)
(40k)/3
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 ?
(A)
s/r
(B)
r/(4s)
(C)
s/(4r)
(D)
r/s
(E)
(4r)/s
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?
(A)
12%
(D)
28%
(B)
20%
(E)
32%
(C)
22%
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
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
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
Counting Techniques
1. Worst case.
2. Ordered places or “for each”.
3. Permutation (Can be done by number 2).
4. Combination.
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
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
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
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
“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
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
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
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
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
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
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
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
Example 1
Two fair six-sided dice are rolled; what is the probability of
having 5 as the sum of the number?
Example 2
Two six-sided dice are rolled; what is the probability of
having 12 as the sum of the numbers?
Probability (Compound)
1. P(A and B) = P(A) • P(B)
2. P(A or B) = P(A) + P(B) – P(A and B)
3. P( not A) = 1 – P(A)
NOTE: if A and B have nothing in common
then P(A and B) = 0
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
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
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?
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
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?
Example 6
If there are 30 red and blue marbles in a jar, and the
ratio of red to blue marbles is 2:3, what is the
probability that, drawing twice, you will select two red
marbles if you return the marbles after each draw?
Example 7
Now consider the same question as example 6 with
the condition that you do not return the marbles after
each draw.