Week 5: Miscellaneous Topics

SAT Prep

I. PROBABILITY and COUNTING:

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?

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.

 

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 $4

113 109 1 5

113 109 1 3

B.) Between Integers – 1 endpt. included subtractBoth endpts. included subtract + 1Neither 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? 

6 3 6 5 4 120P

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?

H

Odd Square

A D B

GFE

CPrime

6,8,10,12,14

Odd Square

15 1,94

GF3,5,7,

11,13

2Prime

G and F - A number

can't be a perfect square

and prime at the same time

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

Sp. Fr.

240-x 180-x

445 410x

25 240 180 410x x x

35x

A.) PROBABILITY =

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)?

Favorable Outcomes

Total Outcomes

2(5)

30

II. PROBABILITY

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

999 100 1 900 36P

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?

2 2

2

5 4 9

5 25P

A.) Alphanumeric Problems – Letters in place of numbers 

Ex. If AB + AB = BCC, what is the value of A?

AB

AB

BCC

III. LOGICAL REASONING:

has to be 1B1

1

1

A

A

CC

has to be 2C1

1

122

A

A

has to be 6A

Ex. If in the following problem, each letter represents a different number, what is the value of A + B + C + D?

X

5

ABA

A

CBD

has to be 2C

has to be 7D has to be 9B

has to be 5A

5 5

X 5

5

B

CBD

5 5

X 5

2 5

B

BD

5 9 2 7 =23 A B C D

B.) Sequences – Never answer without writing first five terms !!

1.) Arithmetic – Common difference

2.) Geometric – Common ratio

1 ( 1) na a d n

11

nna a r

3 4 1na n 3,7,11,15,19,...

100 3 4 100 1a

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?

100 399a

13 4

n

na

3,12,48,192,768,...

10 1

10 3 4a

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?

10 786,432a

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

6 5008

4820

3

182

2 digits in the pattern nd

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

IV. DATA INTERPRETATION

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.

1 2 3 4 5 60

10

20

30

40

50

Price per Share

Stock A Stock B

Ex. What is the difference, in dollars, between the highest and lowest values of a share of stock A?

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

$45 $25 $20

Ex. In what year was the percent increase in the value of stock B the greatest?

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

1993

B.) Circle Graphs – 360 degrees – Usually sectors are in percents Use the circle graph below to answer the next two examples:

Blue25%

Yellow20%

Green

Orange

Red30%

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) 2x x

300 3x

100x

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 20

300 (400 ) 25

x

x

178x

V. FUNCTIONS AND THEIR GRAPHS

A.) Function – a 1 to 1 mapping – No repeats in the domain – Vertical line test

B.) Function notation f(x) replaces y.

Ex. If , what is 2( ) 2f x x x (3) ( 3)? f f

2(3) 3 2 3 15 f

2( 3) 3 2 3 3 f (3) ( 3) 15 3 18 f f

Ex. If , what is 2( ) 2f x x x ( 2)?f x

2( 2) 2 2 2 f x x x

2( 2) 4 4 2 4 f x x x x

2( 2) 6 8 f x x x

Ex. -If , for what value of a is it true that

( ) 3 3 f x x

3 ( ) (2 )?f a f a

3 3 3 3(2 ) 3 a a

9 9 6 3 a a

3 6a

2a

C.) DOMAIN and RANGE

Domain – values which can be substituted for xRange – values which are returned by the

function for y

Ex. - What is the domain of ( ) 4 ? f x x

4 0 x

4 x

4x

Ex. What is the range of

Graph it and look at the -valuesy

2( ) 3? f x x

3y

Ex. Which of the following could be the equation of the graph shown below?A.) 2 4y x B.) 2 4y x 2C.) y x

2D.) 2 4y x 2E.) 4 4y x x 2

-2

-4

-2 2

D.) TRANSLATIONS –

( ) r vertically

( ) opposite r horizontally

( ) reflect over x axis

y f x r

y f x r

y f x

Ex. If the figure below is the graph of f (x), which of the following is the graph of f (x) + 2?

2

-2

-4

-2 2 4

2

-2

-4

-2 2 4

The End!!!