1. Colegio San Patricio
2nd Period Math Review´s ANSWERS
School Year 2009 – 2010
Counting Techniques
5) The school’s cafeteria has new options. Now they have: 3 types of salads, 3 types of
hamburgers, 5 types of fruit beverages, and 5 dessert options. How many different
combinations you to choose from?
MULTIPLICATION Technique (you have 4 groups)
Salads = 3
Hamburgers = 3
Fruit Beverages = 5
Desserts = 5
3*3*5*5 = 225 different combinations
6) How many ways are there for choosing any 4 books from the library’s desk from the 12
books they are on the top of the desk?
COMBINATION Technique (you have 1 group = books & order in which you choose the books
does NOT matters.)
n Cr = n! (until r) = 12 C4 = 12*11*10*9 = 495 combinations
r! 4*3*2*1
7) In how many ways can a jury choose the best 3 essays from the top 20 essays they
received? (The first chosen person will have the 3rd. place; the second one will have the 2nd.
place, and the last person will be the 1st. place.)
PERMUTATION Technique (you have 1 group = essays & order in which the jury selects the
essays DOES matters.)
n Pr = n! (until r) = 20 C3 = 20*19*18 = 6840 combinations
Grouping and Analyzing Data
8) The following data shows the grades of different students.
74 95 96 85 77 65
83 88 93 12 81 92
100 20 40 35 98 87
a) Create a class interval table (interval, tally, frequency). The first interval is 0-20
b) Graph the information using a frequency polygon graph.

2. a) Interval TableTable:
Intervals Tallies Frequency
0 - 20 II 2
21 - 40 II 2
41 - 60 0 0
61 - 80 III 3
81 - 100 IIIII IIIII II 11
b)Graph
F r e q u e n c y P o l y g o n Gr a p h
12
10
8
6 Fr ecuency
4
2
0
0-20 21-40 41-60 61-80 81-100
Int e r v a l s
Multiplying Polynomials
9) (x – 5) (3x + 7) 10) (3a - 2)(a2 -5 a + 5)
3x2 + 7x – 15x – 35 3a3 – 15a2 + 15a – 2a2 + 10a - 10
3x2 – 8x – 35 3a3 – 17a2 + 25a - 10
11) (3x - 4)2 12) (5x + 6y) (5x – 6y)
9x2 – 24x + 16 25x2 – 36y2
13) Find an expression for the area of a trapezoid with short base length x + 3, long base 2x +
9, and height is 10.
* Remember: Area = (B1 + b2)H = (x + 3 + 2x + 9) 10 = (3x + 12)10 = (3x + 12)5 = 15x + 60
of trapezoid 2 2 2
14) What is the area of the rectangle? ___b___
a) 7x² + 5xz + 2xy +yz
5x
b) 10x2 + 5xz + 2xy + yz
c) 7x² + 10xz + xy + 2yz
d) 10x² + 5xz + 2xy + 2yz² y
2x z
(Remember: You need to multiply each side, because you have 2 binomials. (5x + y)(2x + z). Or you can
“reflect” each side, because a property of parallelograms is that opposite sides are the same.)

3. Solid Geometry
15) Draw and answer the following
a) Pentagonal Prism Faces: 5 + 2 = 7
Edges: 5 * 3 = 15
Vertices: 5 * 2 = 10
b) Nonagon Pyramid Faces: 9 + 1 = 10
Edges: 9 * 2 = 10
Vertices: 9 + 1 = 10
16) Draw the different views for the following structure:
Front Left Side Right Side Top
17. Volume
Figure Formula Procedure
a) a = 2cm V = Ab*H V = Ab*8cm = 30cm2(8cm) = 240cm3
Each side = 3cm A = (10*3cm)2cm = 30cm2
A = Pa 2
H = 8cm 2
b) a = 4m V = Ab*H V = Ab*12cm = 72cm2(12cm) = 864cm3
Each side = 6m
A = (6*6cm)4cm = 72cm2
H = 12 m A = Pa 2
2
c) a = 3cm V = Ab*H V = Ab*10cm = 37.5cm2(10cm)= 375cm3
3 3 3
Each side = 5cm A = (5*5cm)3cm = 37.5cm2
H = 10cm A = Pa 2
2

4. 18) What is the volume of a cube if the length is 8cm, width is 5cm, and it’s height is 9cm?
V = l*w*h = 8cm(5cm)(9cm) = 360cm3
19) Find the volume of the following figure.
V= (4cm*5cm*6cm) – (2cm*2cm*6cm)
= (120cm3) – (24cm3) = 96cm3
Remember: You need to “take away” the volume of the
empty space (the hole, which has the same deepness of the
complete figure). That’s why you subtract the volume of the
hole FROM the volume of the COMPLETE figure.
20) A cereal company decided to increase the height of its boxes by 30% and reduce the
width in order to maintain the same volume. What will be the new height and width if the
length stays the same?
Initially: Length = 20 cm
Width = 30cm
Height = 40cm
Answer:
Actual cereal’s box volume= (20cm)(30cm)(40cm) = 24,000cm3.
NEW MEASURES =
L= 20cm (stays the same)
H= 52cm (it increased a 30% = 40 +30% = 40 + 12 = 52cm)
W= ?
* They what the SAME volume, so:
24,000cm3 =(20cm)(52cm)W =
24,000cm3 = 1040cm2 * W =
24,000cm3 / 1040cm2 = W
W = 23.08cm.
So this is the new size of the width in order to keep the same volume of the boxes.