Review

1. ALGEBRA 2
Engr. Rennel Mallari

2. Progression/Sequence
 A sequence or progression is a set or
collection of numbers arranged in an orderly
manner such that the preceding and the
following numbers are completely specified.
 Arithmetic progression
 Geometric progression
 Harmonic progression

3. Arithmetic Progression
A sequence of numbers in which the difference of any two adjacent terms is
constant.
Ex. 4,7,10,13,16 … (common difference is 3)
Elements:
a1 = first term
an = nth term
am = any term before an
d = common difference
d= a2 –a1 = a4 –a3, etc
S= sum of all terms
nth term of AP
an = a1 + (n-1)d
or an = am + (n-m)d
Sum of n terms of AP
S = (n/2)(a1 + an)
or S = (n/2)[2a1 + (n –1)d]

4. Geometric Progression
 A sequence of numbers in which the ratio of any two adjacent
terms is constant.
 Ex. 2,6,18,54
 nth term of GP
an = a1*r^(n-1)
or an = am * r^(n-m)
 common ratio,
r = a2/a1 = a5/a4 = …
Sum of n terms of GP
S = a1(r^n - 1) when r>1
r - 1
S = a1(1 - r^n) when r<1
1 - r
Sum of Infinite GP
for a geometric progression where
–1< r <1 and n = ∞ “infinity”
Sum of IGP = a1 / (1– r)

5. Question 1
How many terms of the sequence –9, -6, -3, …
must be taken so that the sum is 66?
 A. 13
 B. 12
 C. 4
 D. 11

6. Question 2
The sum of all even numbers from 0 to 420 is:
 A. 43410
 B. 44300
 C. 44310
 D. 44130

7. Question 3
Find the 30th term of the AP 4,7,10 …
 A. 88
 B. 91
 C. 75
 D. 90

8. Question 4
Find the sum of the first 10 terms of the
geometric progression 2,4,8,16 …
 A. 1023
 B. 2046
 C. 1596
 D. 225

9. Question 5
The geometric mean of 4 and 64 is:
 A. 48
 B. 16
 C. 34
 D. 24

10. Question 6
Find the sum of the infinite geometric
progression 6,-2,2/3 …
 A. 5/2
 B. 9/2
 C. 7/2
 D. 11/2

11. Harmonic Progression
 A sequence of numbers in which their
reciprocals forms an arithmetic progression.
Example:
 Find the 12th term of the series 1/9,1/6,1/3 …
12th term = -1/24

12. Question 7
Find the fourth term of the progression 1/2,
0.2, 0.125 …
 A. 0.102
 B. 1/10
 C. 1/11
 D. 0.099

13. Question 8
A rubber ball is dropped from a height of 15 m.
On each rebound, it rises 2/3 of the height
from which it last fell. Find the distance
traveled by the ball before it comes to rest?
 A. 75 m
 B. 96 m
 C. 100 m
 D. 85 m

14. Question 9
In a benefit show, a number of wealthy men
agreed that the first one to arrive would pay 10
centavos to enter and each later arrival would pay
twice as much as the preceding man. The total
amount collected from all of them was
P104,857.50. How many wealthy men had paid?
 A. 18
 B. 19
 C. 20
 D. 21

15.  Worded Problem

16. How to solve worded problem?
 1. Read the entire problem.
 2. Look for words that signal specific
operation.
 3. Assign variables.
 4. Create equation.
 5. Convert units of measurement.
 6. Solve the equation

17. Work Problem
 Case 1: different rate of work
 Rate = 1 / time to finish the work
 Work done = Rate x time
 Suppose that a person can do a certain work in 5
days. This means that the said person can finish
1/5 of the work in one day. Thus, his rate is 1/5 of
the work per day.

18. Work Problem
 Case 2: same rate of work
 When there is a specific work and specific time
and manpower, the rate of doing the work may
be computed using the number of man-hour.
work done = no.of workers x time of doing the
job
no. of job done
 Example:
 If 20 bakers can bake 40 pizzas in 8 hours,
how many bakers can bake 10 pizzas in 2
hours?

19. Question 10
Mr. Ostan can wash his car in 15 minutes,
while his son John takes twice as long to do
the same job. If they work together, how many
minutes can they do the washing?
 A. 6
 B. 8
 C. 10
 D. 12

20. Question 11
A pump can pump out a tank in 11 hours.
Another pump can pump out the same tank in
20 hours. How long will it take both pumps
together to pump out the tank?
 A. 1/2 hours
 B. 1/4 hours
 C. 6 hours
 D. 7 hours

21. Question 12
One pipe can fill a tank in 5 hr and another
pipe can fill the same tank in 4 hrs. A drain
pipe can empty the full content of the tank in
20 hrs. With all the three pipes open, how long
will it take to fill the tank?
 A. 2 hrs
 B. 2.5 hrs
 C. 1.92 hrs
 D. 1.8 hrs

22. Age Problems
This type of problem must be solve meticulously by
giving more emphasis to the tenses (i.e. past, present
or future) of the statements.
 Example:
 The ages of a certain person in the past present and
future in terms of x are as follows:
 6 years ago = x –6
 present = x
 5 years hence = x +5

23. Question 13
 A father is three times as old as his son. Four
years ago, he was four times as old as his son.
How old is his son?
 A. 36 y/o
 B. 24 y/o
 C. 32 y/o
 D. 12 y/o

24. Question 14
A man is 41 years old and in seven years, he
will be four times as old as his son at that time.
How old is his son now?
 A. 9
 B. 4
 C. 5
 D. 8

25. Mixture Problems
 The easiest way to solve a mixture
problem is to draw a rectangle or square which
will illustrate the content of the mixture.

26. Question 15
A 40-gram alloy containing 35% gold is to be
melted with a 20-gram alloy containing 50%
gold. How much percentage of gold is the
resulting alloy?
 A. 40%
 B. 30%
 C. 45%
 D. 35%

27. Digit Problem
 Let h, t and u be the hundreds’, tens’ and
units’ digit, respectively. A three-digit number
must be represented in the following manner:
 Number = h(100) + t(10) + u
 A two-digit number is represented by:
 Number = t(10) + u

28. Digit Problem
 The number: 100h + 10t + u
 The number with reversed digit: 100u +
10t + h
 Sum of digits: u + t + h
 Product of digits: u*t*h

29. Question 16
In a two-digit number, the unit’s digit is 3
greater than the ten’s digit. Find the number if
it is 4 times as large as the sum of its digits.
 A. 47
 B. 58
 C. 36
 D. 25

30. Motion Problems
 This problems pertains to the motion with a
uniform velocity, i.e. no acceleration nor
deceleration in the process.

31. Motion Problems
 x=velocity of boat/airplane in still water/air
 y=velocity of the water/air
 x+y = velocity when going downstream / with
the wind (tailwind)
 x-y = velocity when going upstream / against
the wind (headwind)

32. Question 17
Corpuz left Pikit to drive to Davao at 6:15 PM
and arrived at 11:45 PM. If the averaged
30mph and stopped 1 hour for dinner, how far
is Davao from Pikit?
 A. 128
 B. 135
 C. 160
 D. 256

33. Question 18
Juan can walk from his home to his office at
the rate of 5mph and back at the rate of 2mph.
What is the average speed in mph?
 A. 2.86
 B. 3.56
 C. 4.12
 D. 5.89

34. Question 19
A boat travels downstream in 2/3 of the time as
it goes going upstream. If the velocity of the
river’s current is 8 kph, determine the velocity
of the boat in still water.
 A. 40 kph
 B. 50 kph
 C. 30 kph
 D 60 kph

35. Question 20
 A man rows downstream at the rate of 5 mph
and upstream at the rate of 2 mph. How far
downstream should he go if he is to return in
7/4 hours after leaving.
 A. 2.5 miles
 B. 2 miles
 C. 1.5 miles
 D. 3 miles

36. Coin Problems
Under American denominations, US dollar, the
following are the coins and their corresponding
equivalent:
 Penny = 1 cent
 Nickel = 5 cents
 Dime = 10 cents
 Quarter = 25 cents
 Half = 50 cents

37. Variation
 A mathematical function that relates the
values of one variable to those of other
variables.
Direct Variation:
 if x is directly proportional to y, then,
 x α y or x = ky
 k = proportionality constant

38. Variation
Inversely Variation
 if x is inversely proportional to y, then,
 x α 1/y or x = k/y
Joint Variation
 if x is directly proportional to y and inversely
proportional to the square of z, then,
 x α y/z^2 or x = ky/ z^2

39. Question 21
Given that w varies directly as the product of x
and y and inversely as the square of z and that
w = 4 when x = 2, y=6, and z =3. Find w when
x=1, y=4 and z=2.
 A. 4
 B. 2
 C. 1
 D. 3

40. Clock Problems
 By principle, the minute hand (MH) always moves faster
than the (HH). The relation between minute hand and the
hour hand is
 HH = MH/12
 where: MH is in number of minutes
 Also, the hour hand in terms of second hand is expressed as
 HH = SH/720
 where: SH is in number of seconds

41. Question 22
How many minutes after 10:00 o’clock will the
hands of the clock be opposite each other for
the first time?
 A. 21.41
 B. 22.31
 C. 21.81
 D. 22.61

42.  end

43. Question (quiz)
 A jogger starts a course at a steady rate of 8
kph. Five minutes later, a second jogger starts
the same course at 10 kph. How long will it
take the second jogger to catch the first?
 A. 20 min
 B. 21 min
 C. 22 min
 D. 18 min

44. Question (quiz)
 Mary is 24 years old. Mary is twice as old as
Ann was when Mary was as old as Ann is now.
How old is Ann?
 a. 16
 b. 18
 c. 20
 d. 22

45. Question (quiz)
 Gravity causes a body to fall 16.1 ft in the first
second, 48.3 in the 2nd second, 80.5 in the 3rd
second. How far did the body fall during the
10th second?
 a. 248.7 ft
 b. 308.1 ft
 c. 241.5 ft
 d. 305.9 ft

46. Question (quiz)
 If the sum is 220 and the first term is 10, find
the common difference if the last term is 30.
 a. 2
 b. 5
 c. 3
 d. 2/3

47. Question (quiz)
 Three cats can kill 3 rats in 3 minutes. How
long will it take 100 cats to kill 100 rats?
 a. 100 minutes
 b. 3 minutes
 c. 300 minutes
 d. 30 minutes

48. Question (quiz)
 In a box there are 25 coins consisting of
quarters, nickels and dimes with a total
amount of $ 2.75. If the nickels were dimes,
the dimes were quarters, and the quarters
were nickels, the total amount would be $
3.75. How many quarters are there?
 a. 4
 b. 5
 c. 6
 d. 7

49. Question (quiz)
 At what time after 3 and 5 o’clock will the
hands be perpendicular for the second time?
 a. 3:32:58
 b. 3:32:12
 c. 3:32:44
 d. 3:33:58

50. Question (quiz)
 The sum of the two digits in a two-digit number
is 12. If the digits are reversed, the new
number is 18 more than the original number.
Find the original number.
 a. 66
 b. 39
 c. 57
 d. 48

51.  Seatwork

52. Question 1
 If the first term of an arithmetic progression is
25 and the fourth term is 13, what is the third
term?
 a. 17
 b. 18
 c. 19
 d. 20

53. Question 2
 How many terms of the progression 3,5,7…
must be taken in order that their sum will be
2600?
 a. 48
 b. 49
 c. 50
 d. 51

54. Question 3
 In a pile of logs, each layer contains one more
log than the layer above and the top contains
just one log. If there are 105 logs in the pile,
how many layers are there?
 a. 11
 b. 12
 c. 13
 d. 14

55. Question 4
 What is the sum of the progression
4,9,14,19… up to the 20th term?
 a. 1030
 b. 1035
 c. 1040
 d. 1045

56. Question 5
 When all odd numbers from 1 to 101 are
added, the result is
 a. 2500
 b. 2601
 c. 2501
 d. 3500

57. Question 6
 The 3rd term of a harmonic progression is 15
and the 9th term is 6. Find the 11th term.
 a. 4
 b. 5
 c. 6
 d. 7

58. Question 7
 An airplane flying with the wind, took 2 hours
to travel 1000 km and 2.5 hours in flying back.
What was the wind velocity in kph?
 a. 50
 b. 60
 c. 70
 d. 40

59. Question 8
 Find the 9th term of the harmonic progression
3,2,3/2 …
 a. 3/5
 b. 3/8
 c. 4/5
 d. 4/9

60. Question 9
 For a particular experiment you need 5 liters of
a 10% solution. You find 7% and 12% solution
on your shelves. How much of the 7% solution
should you mix with the appropriate amount of
the 12% solution to get 5 liters of a 10%
solution?
 a. 1 liter
 b. 2 liters
 c. 3 liters
 d. 4 liters

61. Question 10
 The sum of the digits of a two-digit number is
11. If the digits are reversed, the resulting
number is seven more than twice the original
number. What is the original number?
 a. 32
 b. 34
 c. 36
 d. 38

62.  Pass your papers.