Transcript Week 5

SAT Prep
A.) Use arithmetic to answer the following. Be careful, not all
are the same answer.
Ex.- John bought some apples. If he entered the store with $113
and left with $109, how much did the apples cost?
$113  $109  $4
Ex. Kim was selling tickets for the school play. One day she sold
tickets numbered 109 through 113. How many tickets did she sell that
day.
113 109 1  5
Ex. John is the 109th person in a line, and Kim is the 113th person.
How many people are between John and Kim?
113 109 1  3
B.) Between Integers –
1 endpt. included  subtract
Both endpts. included  subtract + 1
Neither endpt. included  subtract – 1
Ex. - From 1:09 to 1:13, Elaine read pages 109 through
113 in her English book. What was her rate of reading in
pages per minute?
pages 113  109  1 5


minute 1:13  1: 09 4
C.) Making Lists – Use to look at all possibilities –
LAST RESORT
Ex. - Sally has 6 paintings from which to choose to
hang 1 different painting in each of 3 rooms. How
many possible ways can she choose which paintings go
in which room?
P  6  5  4  120
6 3
Use Counting Principle wherever possible.
Ex. How many integers are there between 100 and
1000 all of whose digits are odd?
5  5  5  125
D.) Venn Diagrams – Use to find intersections and
unions
Ex. The integers 1 through 15 are each placed in the
diagram below. Which of the following region(s) is
(are) empty?
Square
Odd
A
B
D
G
E
H
F
C
Prime
Square
Odd
15
G and F - A number
can't be a perfect square
and prime at the same time
6,8,10,12,
14
4
1,9
G
F
3,5,7,
11,13
2
Prime
Ex. - Of the 410 students at Kennedy High School, 240
study Spanish and 180 study French. If 25 students do
not take either of these languages, how many study both?
(Hint: Draw a Venn Diagram)
25  x  240  x  180  x  410
445  x  410
Fr.
Sp.
240-x
25
x  35
x
180-x
Favorable Outcomes
A.) PROBABILITY =
Total Outcomes
Ex. - In 2003, Thanksgiving was on Thursday,
November 27, and there are 30 days in November. If
one day in November 2003 was chosen at random for
a concert, what is the probability that the concert was
on a weekend (Saturday or Sunday)?
2(5)

30
1
3
Ex. An integer between 100 and 999, inclusive, is chosen
at random. What is the probability that all of the digits
are odd?
125
125 5
P


999  100  1 900 36
Ex. In the figure below, a white square whose sides are 4
has been pasted on a black square whose sides a 5. If a
point is chosen at random from the large square, what is
the probability that the point is in the black area?
5 4
9
P

2
5
25
2
2
A.) Alphanumeric Problems – Letters in place of numbers
Ex. If AB + AB = BCC, what is the value of A?
AB
 AB
BCC
B has to be 1
A1
 A1
1CC
C has to be 2
A1
 A1
122
A has to be 6
Ex. If in the following problem, each letter represents a
different number, what is the value of A + B + C + D?
A has to be 5
C has to be 2
ABA
5 B5
X A
X 5
5 B5
X 5
CBD5
CBD5
2 BD5
D has to be 7
B has to be 9
A  B  C  D  5  9  2  7 =23
B.) Sequences – Never answer without writing first five
terms !!
1.) Arithmetic – Common difference
an  a1  d (n  1)
2.) Geometric – Common ratio
an  a1r
n 1
Ex. - A sequence is formed as follows: the first term is 3,
and every other term is 4 more than the previous term.
What is the 100th term?
3, 7,11,15,19,...
an  3  4  n  1
a100  3  4 100 1
a100  399
Ex. - A sequence is formed as follows: the first term is 3,
and every other term is 4 times the previous term. What
is the 10th term?
3,12, 48,192, 768,...
an  3  4 
 n 1
a10  3  4 
101
a10  786, 432
C.) DIV/MOD to find terms in repeating sequence
Ex.- What is the 500th term of the sequence 1, 4, 2,
8, 5, 7, 1, 4, 2, 8, 5, 7,…? 6 digits until it repeats
83
6 500
48
20
18
2
2
nd
digits in the pattern
4
Ex. - What is the sum of the 800th through the 805th
term of the sequence above?
805  800 1  6
 the sum of all 6 digits
in the pattern
1  4  2  8  5  7  27
A.) Hint – Before reading the questions about a given graph,
take some time and analyze the graph.
The following 4 questions refer to the line graph below.
Price per Share
50
40
30
20
10
0
1
2
3
Stock A
4
Stock B
5
6
Ex. What is the difference, in dollars, between the highest
and lowest values of a share of stock A?
$45  $25  $20
Ex. On January 1 of what year was the ratio of the value of
a share of stock A to the value of a share of stock B the
greatest?
1992
Ex. In what year was the percent increase in the value of
stock B the greatest?
1993
Ex. If from 1995 to 2000 the value of each stock
increased at the same rate as it did from 1990 to 1995,
what would be the ratio of the value of stock B to the
value of a share of stock A?
40 8

45 9
B.) Circle Graphs – 360 degrees – Usually sectors are in
percents
Use the circle graph below to answer the next two
examples:
Blue
25%
Red
30%
Yellow
20%
Orange
Green
Ex. If the jar contains 1200 marbles and there are
twice as many orange marbles as there are green, how
many green marbles are there?
.25(1200)  x  2 x
300  3x
x  100
Ex. Assume that the jar contains 1200 marbles, and
that all of the red ones are removed and replaced by an
equal number of marbles, all of which are blue or
yellow. If the ratio of blue to yellow marbles remains
the same, how many additional yellow marbles are
there?
.3(1200)  400
240  x
20

300  (400  x) 25
x  178
A.) Function – a 1 to 1 mapping – No repeats in the domain
– Vertical line test
B.) Function notation f(x) replaces y.
Ex. If
f ( x )  x 2  2 x , what is f (3)  f (3) ?
f (3)   3  2  3  15
2
f (3)   3  2  3  3
2
f (3)  f (3)  15  3  18
Ex. If
f ( x )  x 2  2 x , what is
f ( x  2)   x  2   2  x  2 
2
f ( x  2)  x 2  4 x  4  2 x  4
f ( x  2)  x 2  6 x  8
f ( x  2) ?
Ex. -If f ( x )  3 x  3 , for what value of a is it true that
3 f (a )  f (2a ) ?
3  3a  3  3(2a)  3
9a  9  6a  3
3a  6
a  2
C.) DOMAIN and RANGE
Domain – values which can be substituted for x
Range – values which are returned by the function
for y
Ex. - What is the domain of f ( x ) 
4 x  0
 x  4
x4
4 x ?
Ex. What is the range of
f ( x ) x 2  3?
Graph it and look at the y-values
y  3
Ex. Which of the following could be the equation of the
graph shown below?
A.) y  2 x  4
B.) y  2 x  4
D.) y  2 x  4
2
2
-2
2
-2
-4
C.) y  x
2
E.) y  x  4 x  4
2
D.) TRANSLATIONS –
y  f ( x )  r  r vertically
y  f ( x  r ) opposite r horizontally
y   f ( x ) reflect over x axis
Ex. If the figure below is the graph of f (x), which
of the following is the graph of f (x) + 2?
2
-2
2
2
4
-2
2
-2
-2
-4
-4
4