MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
I = {... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...}
ALGEBRA
3.
DEFINITION
Algebra - A branch of mathematics that substitutes letters for numbers.
-can include real numbers, complex numbers, matrices, vectors etc.
Rational Numbers (Z) – are numbers which can be
expressed as a quotient (ratio) of two integers. The term
“rational” comes from the word “ratio”.
Examples: 0.7, 4/7, -5, 0.555…
- allows the operations of arithmetic, such as addition, subtraction,
multiplication, to be performed without using specific numbers.
4.
Irrational Numbers – (Z’) – are numbers which cannot be
expressed as a quotient of two integers.
THE REAL NUMBER SYSTEM
The Number and the Set of Real Numbers
Real Numbers
Rational
Numbers
Irrational
Numbers
Examples:
, e, π
Note: The numbers in the examples above can never be
expressed exactly as a quotient of two integers. They are in
fact, a terminating and non – terminating decimal.

Positive integers except 1 can be classified as
either prime or composite.

The number 1 is neither prime nor composite.
Prime numbers are those integers greater than 1
whose only factors are 1 and itself.
Composite numbers are expressed as a product
of two or more factors.
Integers
Negative
Numbers
Zero
Natural
Number
s
The number system is divided into two categories namely, real numbers
and imaginary numbers.
Real Numbers are classified as follows:
1.
2.
Natural Numbers (N) -also called positive integers.
- considered as the counting numbers and used when you are
counting one to one objects.
N = {1, 2, 3,…}
Integers (I) - include whole numbers and their opposites. The
opposite of a whole number is the negative of that number.
The number 0 is also considered an integer, but 0 is the
opposite of itself.
Examples:
9+(6+3) = (9+6)+3
( 6.3 ). 4 = 6. ( 3.4 )
Identity Property:
For any real number a, there exist two real numbers 0 and 1, called
the additive identity and multiplicative identity, respectively, such that
a+0 = a
and
a-1=a
Examples:
2+0=2
2.1=2
Inverse Property:
For any real number a, there exist two distinct real numbers
–a and 1/a, called the opposite od a (or additive inverse of a) and
reciprocal ( or multiplicative inverse of a), respectively, such that
a+(-a) =0
and
a(1/a)=1
Note: a = 0 for multiplicative inverse property.
Examples:
Closure Property:
Given
6
2/3
-3/7
-8
If a and b are real numbers, then a + b (their sum) and ab (their
product) are unique real numbers.
Distributive Property:
Properties of Real Numbers
Whole
Numbers
SEPTEMBER 2022
Example: Given two real number, 3 and 5; their sum 8, and their
product 15, are also real numbers.
Additive Inverse
-6
-2/3
3/7
8
Multiplicative Inverse
1/6
3/2
-7/3
-1/8
If a, b, and c are real numbers,
a(b+c) =ab+ac
Commutative Property:
If a and b are real numbers,
a + b = b + a and a b = b a
Examples:
5.3 = 3.5
3+9 = 9+3
Associative Property:
If a, b, c are real numbers,
a + ( b + c ) = ( a + b ) +c and a ( b c ) = ( a b ) c
franciscan2009@gmail.com
Example:
a.
b.
2 (3 + 5) = 2 (3) + 2 (5)
2 (x + 6) = 2 (x) + 2 (6)
LAWS OF EXPONENTS AND RADICALS
am
1. am an  amn
2.
 am n
an
3.
a 
5.
am
a
 b   bm
 
m
n
 amn
4.
 ab
m
m
6. a m 
 ambm
1
am
MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
7.
9.
a0  1
n
m
8.
a  a
1
mn
10.
n
n
a a
1
n
n
a

b
n
n
a
b
12.
ab  n a n b
n
am  a
Sample Problem
1
1
1
1
11.
SEPTEMBER 2022
1
m
n
1
1
2
3
4
5
1
3
6
10
1. Solve for x in the given sets of equation
4 x + 2 y = 5 , 13 x – 3 y = 2
a) ½
b) 3/2
c) ¼
d) 3/4
1
4
10
1
5
1
Properties :
SPECIAL PRODUCTS AND FACTORING
1) a ( x + y ) = ax + ay
1.
2.
3.
2) ( x + y )( x – y ) = x2 – y2
3) ( x +/- y ) 2 = x2 +/- 2xy + y2
4.
The first term is x n
The last term is y n
The succeeding term contain xy in which the exponent of x
decreases by 1 and the exponent of y increases by 1 and the
sum of the exponents in each term is always equal to n
The number of terms in the expansion is always equal to n+1
2. Solve for x in the given sets of equation.
Solve for x in the expression 8x  2y  2 and 163x  y  4y
a) 1
b) 3
c) 2
d) 4
QUADRATIC EQUATION
- A quadratic equation is of the form a x2 + b x + c = 0
4) ( x + a)( x + b) =
x2
+ ( ax + bx ) + ab
● To find the r th term of the expansion ( x + y ) n :
METHODS OF SOLUTION
5) ( ax + b )( cx + d ) = acx2 + ( adx + bcx ) + bd
6) ( x +/- y ) 3 = x3 +/- 3x2y + 3xy2 + y3
r th term =
n!
x
n  r  1 ! (r  1)!
(n r 1)
y
1.
2.
3.
(r 1)
ILLUSTRATIVE PROBLEM
x 
● To find the sum of exponents in all terms in the expansion
1.
(Board Problem)
Solve for the value of x in the given equation.
4
2.
83 2
8x  2
Simplify the expression to its lowest possible term
a) 1 / a
b) a
c) 1
d) - a
1
a
a2 * a
1
a
3. Factor completely :
6 x + 3 y + 2 mx + my
4. Factor the expression : 4 m2 – n2 + 6 n - 9
5. In the expression 4 2x 1  1024 , solve for the value of x + 1
BINOMIAL EXPANSION
The Pascal’s Triangle
Is a triangular form of number which represent the coefficient of
the binomial expansion of ( x + y ) n.
-
FACTORING
COMPLETING THE SQUARE
QUADRATIC FORMULA
 b
b2  4ac
2a
S=(n+1) n
● To find the sum of numerical coefficients in the expansion
Substitute x and y = 1 , provided the binomial does not contain
any constant
Sample Problem
1. To find the rth term of the expansion ( x + y )
4
2. Find the sum of the coefficients of the expansion ( x + y )
a) 16
b) 18
c) 20
d) 24
METHODS OF SOLUTION
1.
2.
3.
ELIMINATION BY SUBSTITUTION
ELIMINATION BY ADDITION OR SUBTRACTION
DETERMINANTS ( CRAMER”S RULE )
franciscan2009@gmail.com
If
b2  4 ac  0 the roots are real and equal
b2  4 ac  0 roots are real and unequal
If
b2  4 ac  0 roots are imaginary or complex nos.
If
Sum of roots = 
4
3. Find the sum of all the exponents in the expansion ( x + y ) 3
a) 65
b) 81
c) 72
d) 12
SYSTEMS OF LINEAR EQUATION
Where : b2  4 ac is called the discriminant
b
a
Product of roots =
c
a
Sample Problem
1. Solve for the roots of the given equation : x 2 + 2x – 8 = 0
THE FACTOR THEOREM
- If ( x – r ) is a factor of f(x) then, f(r) is equal to zero and r is
one of the roots of the polynomial.
THE REMAINDER THEOREM
- If ( x – r ) is not a factor of f (x) then f(r) is not equal to zero
then, f(r) is the remainder of the polynomial.
MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
SEPTEMBER 2022
Sample Problem
RATE PROBLEM
Sample Problems
1. In the given function x3 + 3x2 – 5x + 2, is ( x – 2) a factor of the
function ?
DISTANCE = RATE X TIME
1. Laulix was 6 times as old as Reiand 6 years ago. Four years from
now Laulix will be 4 times as old as Reiand. How old is Laulix and
Reiand now ?
2. Find the value of k for which ( x – 2 ) is a factor of the function
3 x2 + 4k x – 5 .
WORDED PROBLEMS
NUMBER PROBLEM
CONSECUTIVE NUMBERS
x = first integer
x + 1 = second integer
x + 2 = third integer and so on . . .
CONSECUTIVE ODD OR EVEN NUMBERS
x = first odd / even integer
x + 2 = second odd / even integer
x + 4 = third odd / even integer and so on . . .
DIGIT PROBLEM
THE NUMBER = 100 h + 10 t + u
WITH THE NUMBER REVERSED = 100 u + 10 t + h
Where: h = hundred’s digit
t = ten’s digit
u = unit’s digit
Sample Problems
1. The square of a number increased by 16 is the same as 10 times
the number. Find the number.
a) 2
b) 4
c) 8
d) a and c
2. Of the 4 consecutive odd numbers the product of the second and
the fourth exceeds the product of the first and the third by 136.
Find the second number.
a) 32
b) 33
c) 34
d) 35
3. In a two digit number, the units digit is twice the ten’s digit. If
36 is added to the number the order of the digits will be
reversed. Find the number.
answer : 48
OBJECT MOVING AGAINST THE CURRENT
( UPSTREAM / HEAD WIND )
DISTANCE = ( x - y ) ( time )
2. In how many years will the age of Asdie is four times the age of
Benny, if Asdie now is 58 years old and benny is 10 years old ?
a) 2
b) 4
c) 6
d) 8
OBJECT MOVING WITH THE CURRENT
( DOWNSTREAM / TAILWIND )
MIXTURE PROBLEM
DISTANCE = ( x + y ) ( time )
Where :
x = rate of the object in still water or air
y = rate of the current
“ SUM OF ALL VOLUMES OF AN INGREDIENT IN THE
INDIVIDUAL SMALL MIXTURES IS EQUAL TO THE TOTAL
VOLUME OF THE INGREDIENT IN THE FINAL MIXTURE “
%
%
%
+
Sample Problems
x
1. Two cars starting from the same point and at the same time are
moving towards the city. Five hours later the slower car, averaging
10 mi / hr. was 5 miles behind. What is the rate of the faster car ?
a) 9 mi / hr
b) 10 mi / hr
c) 11 mi / hr
d) 12 mi / hr
2. Two streams, has a rate of 6 mi / hr and 4 mi / hr, respectively. It
takes a man on a motorboat as long to travel 30 miles downstream
on the first stream as to travel 15 miles upstream on the second
stream. Find the speed of the boat in still water.
a) 12 mph
b) 14 mph
c) 16 mph
d) 18 mph
3. ( Board problem )
A speedboat goes 900 km against a current of 25 km / hr. It
took 10 minutes longer for his trip than it would have taken it
to travel with the current. What is the speedboat’s speed in
still water ?
answer : 520 km/hr
=
y
sum of individual mixtures
x+y
=
total volume final mixture
Sample Problems
1. How many gallons of a 25% solution of HCI should be added
to 5 gallons of 10% solution of the same acid to make a 15%
solution ?
answer : 2.5 gal.
2. How many grams of nickel must be removed from 1,500 grams
of alloy having 5 % nickel to make an alloy having 2 % nickel ?
a) 42 grams
b) 43 grams
c) 44 grams
d) 45 grams
3. ( Board Problem in EE )
An alcohol solution contains 80% alcohol in a liter container.
If a student uses half of it and fills up the container with water,
what will be the resulting concentration ?
a) 10 %
b) 20 %
c) 30 %
d) 40 %
AGE PROBLEM
“ THE DIFFERENCE BETWEEN THE AGES OF TWO PERSONS
REMAINS THE SAME “
PAST
●
ago
was
PRESENT
∙
now
is
franciscan2009@gmail.com
FUTURE
∙
from now
will be
WORK PROBLEM
“ THE RECIPROCAL OF TOTAL’S TIME IS EQUAL TO THE SUM
OF THE RECIPROCAL OF INDIVIDUAL’S TIME “
1
1
1
=
+
T
x
y
MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
“ A FRACTION OF WORK DONE BY X PLUS A FRACTION OF
WORK DONE BY Y IS EQUAL TO ONE COMPLETE WORK DONE
BY X AND Y “
1=
1
x
T 
+
1
y
T 
Sample Problems
LEVER PROBLEM
Sample Problems
- A lever is a rigid bar having a single point of support called the
fulcrum. Seesaw is an example of a lever.
1. A delivery boy collected P5.35, in nickel and dime coins. In the
number of nickels were 7 more than one-half the number of dimes,
how many dimes were there?
a) 10
b) 20
c) 30
d) 40
“ IF TWO OR MORE WEIIGHTS ARE ATTACHED TO A LEVER
AND THE LEVER IS IN EQUILIBRIUM, THE SUM OF MOMENTS
OF ONE SIDE IS EQUAL TO THE SUM OF MOMENTS OF THE
OTHER SIDE ABOUT THE FULCRUM “
1. John can do a piece of work in 2 days. Peter can do the same
piece of work in 4 days. How many days will it take to finish
the same job if the two work together ?
2. A and B working together can finish a painting job in six days.
When A works alone, can finish it in five days less than B. How
long will it take each of them to finish the job ?
a) A = 10 & B = 5
b) A = 5 & B = 10
c) A = 10 & B = 15
d) A = 15 & B = 10
3. John and Carlo working together can finish a painting job in 10
days. After working for 4 days, John quits and Carlo finish the job
in 12 more days. Find the number of days that Carlo could finish
the painting work alone?
a) 10 days
b) 20 days
c) 30 days
d) 15 days
4. A tank can be filled by one pipe in 6 hours, by a second pipe in 4
hours, and emptied by a third in 8 hours. How long will it take to fill
the tank if all three pipes are open?
a) 3.43 hours
b) 2.43 hours
c) 5.43 hours
d) 6.43 hours
CLOCK PROBLEM
“ THE MINUTE HAND MOVES 12 TIMES FASTER THAN THE HOUR
HAND “
Let :
Then,
x = be the distance traveled by the minute hand starting
from 12.
x
= is the distance traveled by the hour hand
12
Sample Problems
1. what time after 4 o’clock will the hands of the clock are
a) perpendicular for the second time
b) opposite to each other
c) overlapping each other
d) forming 30o for the first time
SEPTEMBER 2022
2. A collection of coins has a value of 64 cents. There are two more
nickels than the dimes and three times as many pennies as dimes.
How many pennies are there?
a) 9
b) 11
c) 15
d) 7
PROGRESSION
Hence :
W 1 X1 = W 2 X2
- Set of things arranged in some definite order.
ARITHMETIC PROGRESSION ( A, P. )
Sample Problems
1. Where should a 100 kg weight be place in a teeterboard in order
to balance with 70 kg weight located 5 feet away from the fulcrum
a) 3.5 feet
b) 4.5 feet
c) 5.5 feet
d) none of these
2. Two children weighing 40 and 50 pounds respectively, are on
opposite ends of a seesaw. If the seesaw balances when the
fulcrum is 6 inches from the middle, find the length of the seesaw.
a) 13.5 feet
b) 55 inches
c) 106 inches
d) 9 feet
WEIGHT LOSS PROBLEM
- A sequence of things or element or numbers called terms each of
which after the first is formed from the preceeding one by adding a
fixed number called Common Difference.”
L  a  (n  1) d
S 
n
(a  L)
2
S 
n
 2a 
2 
a.m. 
a1  a2  ...  L
n
= last term of the sequence
= sum of all terms in the sequence
n  1 d  = sum of all terms
= arithmetic mean
TOTAL WEIGHT LOSS = ( % LOSS1 x WEIGHT1 ) + ( % LOSS2 x
WEIGHT2 )
HARMONIC PROGRESSION ( H.P.)
Sample Problem
1. An alloy of silver and gold weighs 15 oz. In air and 14 oz in water.
If silver losses 1 / 10 of its weight and gold losses 1 / 19. How
much of each metal is in the alloy ?
answer: 4.44 & 10.56
COIN PROBLEM
Name of Coin
Penny
Nickel
Dime
Quarter
Half-dollar
Dollar
Values in Cents
1 cents
5 cents
10 cents
25 cents
50 cents
100 cents
franciscan2009@gmail.com
- A sequence of things or element whose reciprocal of each term
forms an arithmetic progression.
1

n
s
2
1
 2a
 ( n  1)d

= sum of reciprocal terms
Sample Problems
1. A pile of creosoted poles used in electrification of mountain
barangays contains 1275 poles in layers so that the top layer
contain only one pole and each lower layer has one more pole
than the layer above. How many layers are there in the pile?
a) 10 layers
b) 30 layers
c) 50 layers
d) 70 layers
MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
2. In a pile of logs, there are 18 layers. Each layer contains one
more log than the layer immediately above. The top layer has
only one log. How many logs does the pile contain?
a) 169 logs
b) 170 logs
c) 171 logs
d) 172 logs
3. An electrician can choose between working for P 10 per day
and working for P 6 for the first day and an increase of P 0.50
per day each day thereafter. At the end of 30 days, how much
is the difference between the two pay-offs?
a) P 300.0
b) P 497.50
c) P 97.50
d) P 397.50
GEOMETRIC PROGRESSION ( G.P.)
- A sequence of things or elements or numbers called terms each of
which after the first is formed from the preceding term by multiplying a
fixed number called Common Ratio.
L  ar
n1
a  a rn
S 
1  r
S 
a  rL
1  r
= last term of the sequence
= sum of all terms in the sequence
= sum of all terms
Geometric Mean 
n
product of the terms
Infinite Geometric Series
a
S 
1  r
= sum of infinite terms
Sample Problems
1. What is the sum of all numbers in series that starts from 1 and
decreases by ½ and so on . . . . . ?
RATIO, PROPORTION AND VARIATION WITH VENN DIAGRAM
RATIO – Is an indicated division or simply a fraction. The principles
that may apply to fractions apply likewise to ratios. The ratio of two
concrete quantities must be expressed in terms of the same unit of
measure. Thus, the ratio of 2 feet to 10 inches is not 2 / 10 but rather
24 / 10 or 24 : 10.
PROPORTION – Is a statement expression that two ratios are equal.
Thus; the two fractions a / b and c / d are equal and said to be
proportional and can be written either of the two ways a / b = c / d or
a : b :: c : d where a and d are called extremes while b and c are
called means.
Sample Problems
1. A volleyball team won 16 games and lost 4. What is the ratio
of the number of games lost to the number of games won ?
ans. 1:4
2. Find the mean proportion of 4 m and 9 m
a) 36 m
b) 4 m
c) 9 m
d) 6 m
4. Express 6 feet to 6 yards as a ratio
a) 1 : 1
b) 3 : 18
c) 6 / 3
d) 1 / 3
3. What is the sum of the first 7 terms of the progression 2, -4, 8 . .
. and so on?
a) 254
b) 86
c) 200
d) –90
3. Joint Variation
1. Direct Variation
2. Inverse Variation
1
; x  z
y
- An illustration or diagram usually made of circles that denotes
the pictorial relation of two or more sets of things or elements.
Sample Problems
1. In a survey of 100 persons revealed that 72 of them had eaten
at a restaurant and that 52 of them had eaten at a sea foods
bar. How many of them had eaten at both food house ?
 z 
or x  k 

 y 
Sample Problems
1. ( EE Bd.March ’98 )
The electric power which a transmission line can transmit is
proportional to the product of its design voltage and current
capacity and inversely to transmission distance. A 115 kilovolt line
rated at 1000 amperes can transmit 150 Megawatt over 150 km.
How much power in Megawatt can a 230 kV line rated at 1,500
franciscan2009@gmail.com
“ ANALYZE, UNDERSTAND AND WRITE WHAT ARE THE
GIVEN AND THE REQUIRED IN THE GIVEN PROBLEMS “
Sample Problems
1. A farmhouse has a total of 36 cows and chicken. The
animals have a total of 96 feet. How many cows are there ?
a) 24
b) 12
c) 36
d) 16
x  z or x = k y
1
1
x 
or x = k
y
y
x 
VENN DIAGRAM
MISCELLANEOUS PROBLEM
5. A linotype operator can set up a book of 400 pages in 10 days.
How many days will it take him to set-up two books, one of 300
pages and the other of 500 pages?
a) 10 days
b) 15 days
c) 20 days
d) 25 days
LANGUAGE OF VARIATION
2. ( ME board exam 2000 )
The force exerted by a lever varies directly as the length of
the arm. If a lever 3 ft in length is capable of lifting a weight of
56 pounds, what weight could a lever 4 and ½ feet long can lift?
a) 37.3 lbs.
b) 84 lbs.
c) 94.4 lbs.
d) none of these
2. In a club of 40 executives, 33 like to smoke Marlboro,
and 20 like to smoke Philip Morris if there are two who
doesn’t smoke, how many executives smoke both
brands of cigarette?
a) 15
b) 13
c) 11
d) 8
3. What is the fourth proportional of 4, 20, & 60 ?
a) 80
b) 24
c) 640
d) 300
2. A rubber ball is dropped from a height of 15 meters. 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 meters
b) 96 meters
c) 100 meters
d) 85 meters
4. Laulix bets $1 on the first poker hand, $2 on the second, $4 on the
third and so on. If laulix loses nine hands in a row, and wins on the
tenth hand. What is his net profit or net lose?
a) $ 511
b) $ 3
c) $ 514
d) $ 1
SEPTEMBER 2022
amperes transmit over 100 k m?
a) 675
b) 485
c) 595
d) 785
2. A vendor sells balot on the condition that when you buy half
of the total number of balot inside his basket you’ll get 1 free.
If this agreement were made on three of his customers until all
his balot were sold out, how many balot did the vendor originally
have ?
DIAPHANTUS PROBLEM
“ A PROBLEM IN WHICH THE NUMBER OF UNKNOWNS IS
GREATER THAN THE EQUATION FORMED, PROVIDED THE
PROBLEM INVOLVED DEALS WITH WHOLE NUMBER OR
UNIT IN WHICH CASE CAN BE SOLVE ONLY BY TRIAL AND
ERROR “
MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
Sample Problems
1. A merchant has three items on sale, namely a radio for Php 50, a
clock for Php 30 and a flashlight for Php 1. At the end of the day,
the merchant has sold a total of 100 items and has taken exactly
Php 1000 on the total sales. How many radios did he sale ? 16
2. The sum of the digit of a 3 digit number is 17. The hundred’s digit
is twice the unit’s digit. Find the number. ans. 683 and 854
PRACTICE PROBLEMS
1.
Two cars approaches each other in a 1000 meters straight road.
One has a constant rate of 36 kph and the other at 54 kph. How
long they will meet?
a) 20 seconds
b) 30 seconds
c) 40 seconds
d) 50 seconds
2. A speedboat goes 900 km. against a current of 25 km/hr. It took
10 minutes longer for his trip than it would have taken it to travel
with the current. What is the speedboat’s speed in still water?
a) 62.3 km per hr.
b) 25.4 km per hr.
c) 519 km per hr.
d) 425 km per hr.
3.
It takes approximately 4.95 hrs to travel from Bacolod to
Dumaguete. If travelling 11 kph faster the trip is reduced by an hour.
What is the approximate distance between the two cities?
a) 220 km
b) 215 km
c) 225 km
d) 210 km
4.
John can ran around a circular track in 20 seconds and Eddie in
30 seconds. Two seconds after Eddie starts, John starts from the
same place in opposite direction. When will they meet?
a) 18.50 seconds after
b) 11.20 seconds after
c) 20.82 seconds after
d) 15.67 seconds after
5.
The sum of two numbers is 30. What is the larger number if 3
times the smaller equals twice the larger?
a) 12
b) 14
c) 16
d) 18
6.
What is the first of three consecutive numbers if 6 times the first
minus twice the second is equal to twice the third?
a) 3
b) 5
c) 7
d) 9
7. The sum of the digits of a two-digit number is 14. If 8 be added to
the number, the order of the digits will be reversed. Find the original
number.
a) 68
b) 77
b) 59
d) none of these
8. The sum of the first and third digit of a three digit number is 5,
and the middle digit is twice the first digit. Adding 99 to the
number reverses the order of the digits. What is the number?
a) 263
c) 362
b) 144
d) 401
9. The sum of the digits of a 3 digit number is 18. If the number is
divided by the sum of the units and tens digits, the quotient is 69
with remainder 6. If 198 is subtracted from the number, the digits
will be reversed. Find the number.
a) 675
b) 467
c) 647
d) 765
10. John is twice as old as his sister. 4 years ago he was 3 times as
old as his sister. How old is John at the present time?
a) 15 years old
b) 16 years old
c) 17 years old
d) 18 years old
11. The sum of our ages is 40 years. In 5 years I will be 4 times as
old as you. What is your age?
a) 35 years old
b) 20 years old
c) 5 years old
d) 8 years old
12. In how many years will Mr. Go’s age be 7 times that of his
grandson’s if he is now 69 years old and his grandson is 3?
a) 8 years
b) 6 years
c) 4 years
d) 4 years
13. Harry is 3 years younger than his brother Cris, and Cris is twice
as old as his little sister Gina. The sum of their ages equal to that
of their father, who is 42 years of age. How is old is Gina next
year?
a) 10 years old
b) 12 years old
c) 9 years old
d) 8 years old
14. How old is the man if his present age is 5 / 8 of what it will be 18
years hence?
a) 25 years old
b) 30 years old
c) 35 years old
d) none of these
15. How old is a man last year, if 3 / 4 of his age 6 years ago equals
3 / 8 of his age 18 years from now?
a) 29 years old
b) 25 years old
c) 30 years old
d) 26 years old
16. A grocer has in stock one brand of tea selling 50 cents a pound
and another brand selling for 45 cents a pound. How many
pounds of 50 cents a pound tea must he take to make a mixture
of 75 pounds worth 48 cents a pound?
a) 30 pounds
b) 35 pounds
c) 40 pounds
d) 45 pounds
17. How many liters of water must be added to 45 liters of
solution which is 90% alcohol in order to make the resulting
solution 80% alcohol?
a) 2.63 liters
b) 3.63 liters
c) 4.63 liters
d) 5.63 liters
franciscan2009@gmail.com
SEPTEMBER 2022
18. An 8 – gallon radiator is filled with a 20% alcohol solution.
How much of the solution must be drained off and replaced
by pure alcohol to obtain 35% alcohol solution?
a) 1.2 gallons
b) 1.5 gallons
c) 1.7 gallons
d) 1.9 gallons
19. A tank contains 36 gallons of pure alcohol. Some gallons of
it are drawn off and replaced by the same number of gallons
of pure water. Then the same number of gallons of the mixture
is drawn off leaving only 25 gallons of pure alcohol in the tank.
How many gallons are drawn each time?
a) 6 gallons
b) 5 gallons
c) 11 gallons
d) none of these
20. At what time after 12 noon will the hour and minute hands of
a clock first form an angle of 300 ?
a) 12:05.45 p.m
b) 1:05.45 p.m
c) 12:10.91 p.m
d) 12:21.82 p.m
21. How many seconds after 5 o’clock will the hands of the clock
be perpendicular to each other for the first time?
a) 10.91
b) 100.82
c) 565.4
d) 654.5
22. How many times in one complete day will the hands of the
clock (minute & hour) coincides with each other?
a) 24 times
b) 23 times
c) 22 times
d) none of these
23. Abner and Bosing together can do a piece of work in 4 hours.
Bosing alone takes 9 hours to do it. How long does Abner
take alone?
a) 34 / 5 days
b) 36 / 5 days
c) 2 / 3 days
d) none of these
24. A tank can be filled by one pipe in 6 hours, by a second pipe
in 4 hours, and emptied by a third in 8 hours. How long will it
take to fill the tank if all three pipes are open?
a) 3.43 hours
d) 2.43 hours
c) 5.43 hours
d) 6.43 hours
25. Suzy and Carla can decorate a certain room in 4 and
6 hours respectively. Suzy started decorating at 7:30 a.m.
and was joined by Carla at 9:00 a.m. What time did
the two girls finish decorating the room?
a) 10:00 a.m
b) 11:30 a.m
c) 10:30 a.m
d) 12:00 a.m
26. Jay can do piece of work in 4 days, Belen can do the same job in
6 days and Cris can do it in 8 days. How long will it take to do the
job if Jay and Belen work for one day and Belen and Cris finish
the job?
a) 1 day
b) 3 days
c) 2 days
d) 4 days
MATHEMATICS – ALGEBRA
FRANCISCAN ENGINEERING REVIEW PROGRAM
27. The prizes in a raffle are $ 50, $ 100, $ 150, and so on, and the
highest prize is $ 500. what is the total value of the prizes?
a) $ 1,500
b) $ 3,750
c) $ 2,500
d) $ 2,750
28. Ten potatoes are placed at distances of 5, 10, 15, . .feet from a
basket for a potato race. Each runner is required to start from the
basket and bring back each potato one at a time. How far does
the winner of this race travel?
a) 275 feet
b) 550 feet
c) 375 feet
d) none of these
29. Laulix bets $1 on the first poker hand, $2 on the second, $4 on
the third and so on.If laulix loses nine hands in a row, and wins
on the tenth hand. What is his net profit or net lose?
a) $ 511
b) $ 3
c) $ 514
d) $ 1
30. In a survey of 100 persons revealed that 72 of them had eaten at
a restaurant and that 52 of them had eaten at a sea foods bar.
How many of them had eaten at both food house ?
a) 20
b) 24
c) 30
d) 50
BRING HOME PROBLEMS
6.
At what time after 12:00 noon will the hour hand and the minute
hand of the clock first form an angle of 120 o ?
a) 12:21.818 *
b) 12:31:132
c) 12.18.818
d) 12:22.828
7. The gasoline tank of a car contains 50 liters of gasoline and
alcohol, the alcohol comprising 25%. How much mixture must be
drawn off and replaced by alcohol so that the tank contain a
mixture of which 50% is alcohol ?
a) 18.67 liters
b) 12.57 liters
c) 16.67 liters *
d) 14.57 liters
8.
How many liters of water must be removed from a 50 liter
containing 3% salt solution so that the remaining solution will be
5% salt ?
a) 10 liters
b) 15 liters
c) 20 liters *
d) 5 liters
9. Mammoth, deposited 1 peso n d piggy bank on the first day, two
pesos on the second day and 4 pesos on the third day until it
becomes full on the 9th day. In what day will the piggy bank is half
– full ?
a) on the fifth day
b) on the fourth day
c) on the eight day *
d) on the last day
SEPTEMBER 2022
of 16 psi and having a volume of 500 ft3 is compressed to a
volume of 25 ft3. What is the final pressure if the temperature
remains constant?
a) 310 psi
b) 315 psi
c) 320 psi *
d) 325 psi
16. A vendor has 12 fruits in his basket consisting of mango,
apple and papaya. If mango sells at 8 pesos each, and
apple sells at 10 pesos each and papaya at 12 pesos each,
the total sales would be 118 pesos. How many mango is
inside the basket ?
ans. 4
17. Find the sum of infinite geometric progression 6, - 2, 2/3 ….
and so on ?
a) 9/2 *
b) 5/2
c) 7/2
d) 11/2
18. What is least common factor of 10 and 32 ?
a) 320
b) 2
c) 180
d) 90
19. What is the 100th term of the sequence1.01, 1.00, 0.99 . . .
a) 0.02
b) 0.03
c) 0.05
d) 0.04
20. Ten less than four times a certain number is 14. Determine
the number.
a) 4
b) 6
c) 7
d) 5
1.
Find the coefficient of the term involving x 7 y3 from the expansion
of ( x + y ) 10.
a) 120
b) 240
c) 60
d) 180
10. The sum of the ages of Paulo and Gringo is 35. When Paulo was
two thirds her present age and Gringo was 3 / 4 of his present
age, the sum of their ages was 25. How old is Paulo now ?
a) 20
b) 10
c) 25
d) 15 *
2.
An airplane flying with the wind, took 2 hours to travel 1,000 km,
and 2.5 hrs in flying back. What was the wind velocity in kph ?
a) 50 kph
b) 150 kph
c) 70 kph
d) 80 kph
11. Solve for x if
a) 3
c) -1
3.
The ten’s digit of a number is 3 less than the units digit. If the
number is divided by the sum of the digits, the quotient is 4 and the
remainder is 3. What is the original number ?
a) 47
b) 36
c) 58
d) 69 *
12. A can do a piece of work in 9 days, B in 12 days and C in 18
days. A and B work for 3 days after which C replaces B. How
long must A and C work together to finish the job?
a) 6.5 days
b) 4.5 days
c) 2.5 days
d) 8.5 days
22. For a particular experiment you need 5 liters of a 10% solution.
You find 7% and 12% solution on the shelves. How much of the
7% solution should you mix with 12% solution to get 5 liters of a
10% solution?
a) 1.5
b) 2
c) 3
d) 2.5
4.
The time required by an elevator to lift a weight varies directly with
the weight and the distance through which it is to be lifted and
inversely as the power of the moter. If it takes 30 seconds for a 10
Hp motor to lift 100 lbs through 50 ft, what size of motor is required
to lift 800 lbs in 40 seconds through a distance of 40 feet ?
a) 48 hp
b) 50 hp
c) 56 hp
d) 58 hp
13. On a trip, Laulix noticed that his trip averaged 21 miles per
gallon of gas except for the days he used the air
conditioning, and then it averaged only 17 miles per gallon. If
laulix used 91 gallons of gas to drive 1,751 miles, on how
many of those miles did he use the air-conditioning ? 680 miles
23. The denominator of a certain fraction is three more than twice
the numerator. If 7 is added to the both terms of the fraction, the
resulting fraction is 3/5. Find the original fraction.
a) 8/3
b) 5/3
c) 13/5
d) 5/13
14. Decos can paint a fence 20% faster than peter. Together
they can paint a given fence in 4 hours. How long will it
take Decos to paint the same fence if he had to work
alone ? 7 1/3 hrs.
24. Find the common ratio of the infinite geometric progression
where the sum is 2 and the first term is ½ .
a) 1/3
b) ½
c) ¾
d) ¼
15. If the temperature of a gas remains constant, the
Pressure varies inversely as the volume. A gas at a pressure
25. A club of 40 executives, 33 like to smoke Marlboro and 20 like to
smoke Philip Morris. How many like both ?
5.
From the time 6:15 P.M. to the time 7:45 P.M. of the same day, the
minute hand of the standard clock describes an arc of how many
degrees ?
a) 180 o *
b) 540 o
o
c) 360
d) 220 o
x+1 +
2x + 3 = 1
b) 1
d) a and c
franciscan2009@gmail.com
21. What is the equation of the quadratics whose roots are reciprocal
of the roots of the equation 2x2 – 3x – 5 = 0 ?
a) 2x2 – 5x – 3 = 0
b) 5x2 + 3x – 2 = 0
c) 5x2 – 2x – 3 = 0
d) 3x2 – 5x – 2 = 0
FRANCISCAN ENGINEERING REVIEW PROGRAM
a) 10
b) 11
c) 12
d) 13
26. A farmhouse has a total of 36 cows and chicken. The animals
have a total of 96 feet. How many cows are there ?
a) 24
b) 12
c) 36
d) 16
27. A carpenter bought 24 boxes of three different screws for P2, 200.
Screw A costs P300 a box, screw B costs P150 per box and screw C
costs P50 per box. How many boxes of screw B did he buy?
A. 6
B. 9
C. 8
D. 5
MATHEMATICS – ALGEBRA
D O N’ T
QUIT
When things go wrong as they sometimes will,
When the road you’re trudging seems all uphill,
When the funds are low and the debts are high;
When you want to smile, but you have to sigh,
When care is pressing you down a bit
rest if you must, but DON’T you QUIT
28. If 19 kilos of gold loses 1 kilo; and 10 kilos of silver loses 1 kilo
when weighed in water, find the weight of gold and silver weighing 106
kilos in air and 99 kilos in the water?
*A. 76 kilos
C. 84 kilos
B. 55 kilos
D. 68 kilos
Life is queer with its twists and turns,
As everyone of us sometimes learns
And many a fellow turns about,
When he might have won had he stuck it out,
Don’t give up though the pace seems slow;
You may succeed with another blow.
29. Eight men can excavate 15 m3 of drainage open canal in 7 hrs.
Three men can backfill 20 m3 in 4 hrs. How long will it take 10 men to
excavate and back fill 20 m3 in the project?
*A. 9.87 hrs.
C. 8.76 hrs.
B. 7.65 hr.
D. 10.98 hrs.
Often the goal is nearer than
It seems to a faint and faltering man;
Often the struggler has given up,
When he might have captured the victor’s cup;
And he learned too late when the night came down,
How close he was to the golden crown.
30. A geometric progression is 1+ z + z²+ z3 + . . . + zn where z<1
Determine the sum of the series as n approaches infinity.
A. 1/ (1- 2z)
*C. 1/ (1- z)
B. 1/ (2- z)
D. 2/ (1- z)
31.The sum of the numerical coefficient in the expansion of (x- 2y+ 3z)6
A. 32
C. 38
B. 68
*D. 64
Success is failure turned inside out,
The silver tint of the clouds of doubt;
And you never can tell how close you are,
It may be near when it seems so far;
So, stick to the fight when you’re hardest hit,
It’s when things seem worst that you
MUSTN’T QUIT.
franciscan2009@gmail.com
SEPTEMBER 2022
FRANCISCAN ENGINEERING REVIEW PROGRAM
MATHEMATICS – ALGEBRA
franciscan2009@gmail.com
SEPTEMBER 2022