3-let - general education - mathematics 53-81

7/25/2019 3-Let - General Education - Mathematics 53-81

1/29

5GENERAL EDUCATION MATHEMATICS

MATHEMATICS 20%

COMPETENCE

1. MATHEMATICS 1 FUNDAMENTALS OF MATH 7%

1.1Use of four fundamental operations in problem solving involving: 4%1.1. Operations with whole numbers, decimals, fractions, and integers1.2 Prime, composite, denominate numbers1.3 Prime factorization1.4 LCM, GCF1.5 Divisibility rules1.6 Ratio and proportion1.7 Percentage, Base and Rate1.8 Measurement and units of measure 2%

1.8.1 Perimeter1.8.2 Area1.8.3 Volume

1.8.4 Capacity1.8.5Weight

1.9 Convertunits in the matrix system1%

2. MATHEMATICS 2 PLANE GEOMETRY 5%

2.1 Show mastery of basic terms and concepts in plane trigonometry 3%2.1.1 lines and curves, perpendicular and parallel lines2.1.2 angles, angle property2.1.3 special triangles and quadrilaterals2.1.4 polygons

2.2 Solve problems involving basic terms and concepts in plane trigonometry 2%

3. MATHEMATICS 3

ELEMENTARY ALGEBRA 5%3.1 Show mastery of basic terms and concepts in Elementary Algebra 3%

3.1.1 Algebraic expression3.1.2 Polynomials3.1.3 Linear equations3.1.4 Linear inequalities

3.2 Solve problems, evaluate and manipulate symbolic and numericalproblems in elementary algebra by applying fundamental rules, principlesand processes

4. MATHEMATICS 4 STATISTICS AND PROBABILITY 3%4.1 Show mastery and knowledge of basic terms and concepts in

statistics and probability 1%4.1.1 Counting techniques4.1.2 Probability of an event4.1.3 Measures of central tendency4.1.4 Measures of variability

4.2 Solve, evaluate, manipulate symbolic and numericalproblems in statistics and probability by applying fundamental rules 2%


2/29

54 LET Comprehensive Reviewer Based on NCBTS and Table of Specifications (TOS)

Read, analyze and reflect on these NOTES.

1. When evaluating mathematical expressions, always be guided by the order of operations:

a. Simplify all operations inside parenthesis.

b. Simplify all exponents

c. Perform all multiplications and divisions, working from left to right.d. Perform all additions and subtractions, working from left to right.

Remember the mnemonic P-E-M-D-A-S ( Parenthesis-Exponents- Multiplications/Divisions-

Subtractions-Additions).

2. Numbers may be classified as prime, composite and natural denominate:

a. Aprime numbersis a positive numberwhich may only divided by 1or itself.

b. A composite numberis a positive number which has a positive divisor other 1 or itself. All

even numbers are composite exceptthe number 2.

c. A natural numberlike 0 or 1 is neither prime nor composite.

d. A denominate numberis a number with an attached unit of measurement.

3. Prime factorization is the process of finding which prime numbers tou need to multiply together to

get the original number. There are 2 methods of Prime factorization.

a. In the factor treemethod, we use a pictorial method of finding factors, where the

number to be factorized is placed at the top and all its factors branch out one by one

till we get all prime factors, just like a tree.

24

12 2

3 4

2 2

3 2 2 2 = 24

b. In the continuousdivivision method, we rperform repeated divisionsusing prime

factors as divisor until the last dividend becomes 1.

2 24

2 12

2 6

3 3

1

4. The least common multiple (LCM) is the smalles multiple that 2 numbers have in common. The

greatest common factor (GCF) is the larges multiple that can exactly divide 2 numbers. To obtain the

LCM or the GCF, prform prime factorization of the 2 numbers and compare their prime factors.

a. For LCM, after you list down the 2 factors, mark similar prime factors as one pair. After you have

done so, multiply together the pairedand unpaired prime factors.

NOTES


3/29


b. For GCF, after you list down the prime factors, mark similar prime factors as one pair. After you

have done so, multiply together the paired prime factors ONLY.

1. Please be guided by the divisibility rules.

Divisor Divisibility condition Examples

2 The last digit is even (0, 2, 4, 6, or 8). 1294: 4 is even.

3 Sum of the digits is clearly divisible by 3. 405 => 4 + 0 + 5 = 94 The last two digits are divisible by 4. 40832: 32 is divisible by 4.

5 Thelast digit is 0 or 5. 495: the last digit is 5.6 It is divisible by 2 or by 3.

by 3 and the last digit is even, hence the1,458: 1 + 4 + 5 + 8 = 18, so it is divisiblenumber is divided by 6.

Subtracttwo times the last digit from the rest.The result should be divisible by 7

483: 48(3 2) = 42: 42 is divisible by 7.

8 The last three digits are divisible by 8. 34152: 152 is divisible by 8.Ad four times the hundreds digit to twice the tensdigit to the ones digit. The result should be divisibleby 8.

34152: (4 1) + (2 5) + 2 = 16

9 Sum of the digits is divisible by 9. 2,880: 2 + 8 + 8 + 0 = 1810 The last digit 0. 130: the las digit is 0.

2. Recall the concepts of ratio and proportion.

a.A ratio is an expression of the relative size of two quantities; it is usually expressed as the quotien

of one number divided by the other. The ratio of 1 to 2 is written as 1:2 or 1/2.

b.Aproportion is a statement of equality between two ratios. The ratio of 1:2 to 3:6 forms of the

proportion 1:2 = 3:6 or = 3/6.

3. Recall the concept of percentage, base and rate.a. Thepercentage is the fraction of the original number that is obtained by multiplying the rate and th

base. In problems, it is the number that comes before the word is.b. The baseis the number or quantity which represents the original number. It also represents the tota

It is obtained by dividing the percentage by the rate. In problems, , it is the number that comes afterthe word of.

c. The rate is the number represents the percent. It is obtained by dividing the percentage by the base.In problems, is the number that is attached to the word percent or to a % sign.

d. To facilitate recall of the formulate for percentage, base and rate, draw a PBR triangle.

24 =

36 =

LCM = 2 x 2 x 3paired factors

x 2 3

= 72

2

2

2

2

2

2

2

2

24 =

36 =

GCF = 2 x 2 x 3paired factors

x = 12

2

2

2

2

2

2

2

2


4/29


P

B R

4. Measurment is the oprocess or result of determining the magnitude of a quantity.a. Perimeter is the total distance around any 2 dimensional shape. The formulae for for perimeter as

follows:

i. perimeter of triangle = a + b + c, whare a, b and c represens the lengths of the three sides

of the triangle

a b

c

ii. perimeter of a rectangle = 2l = 2w, where l and w represens the length and the widthrespectively

w

l

iii.perimeter of a square = 4s, where s represents the length of one side of a square

s

iv. perimeter of circle = 2 r, where represents the constant equal to 3.1416 and r represents the

radius of the circle

r

b.Area iWs the total amount of space that a 2 dimensional object occupies. It is measuresd in square

units (ie, square meters, square centimeters).

i. area of a triangle = b h, where b represents the base and h represents theperpendicular height.

h

b


5/29


ii. area of a rectangle = 1 w, where l and w represens the length and the width

respectively

w

l

iii. area of a square = s, where s represents the length of one side of a square

s

iv. area of a circle = r, where represents the constant equal to 3.1416 and r represents thradius of a circle

r

v. area of an isosceles trapezoid = x (b1 + b2) x h, where b1and b2 represent the length of th

2 parallel bases and h is the perpendicular height

b1

h

b2

c. volume is the total amount of space occupied by a three dimensional object. It is measured incubic units (ie, cubic meters, cubic centimeters).

i. volume of a cube = s3, where s is length of 1 side of the cube.

s

ii. volume of a rectangular prism = 1 w h, where l is the length, w is the width and h is

the height.

h

l w

iii. volume of a cylinder = r2h,where represents the constant equal to 3.1416, r representsthe radius of the circular base and h represents the height of the cylinder.

r

h


6/29


iv. volume of a cone = (1/3) x r2h, where represents the constant equal to 3.1416, rrepresents the radius of the circular base and h represents the height of the cone

h

s

v. volume of a pyramid = (1/3) x B x h, where B is the area of the base and h is the height

vi.volume of sphere = (4/3) x

r3

, where

represents the constant equal to 3.1416, rrepresents the radius

d. Capacityis the total amount of fluid that a 3-dimensional container can hold. It is used hand inhand with volume and is calculated using the same formulae.

e. Weight is a measure of the amount of gravitational pull exerted on a mass. Beyond the realm of

physics, weight and mass are used interchangeably. Conventional units include kilograms andpounds.

5. The standard system of measurement in the present day is the metric or SI system. The alternativesystem used in European countries is the English system.

a. Convertion of units in the SI system requires awareness of various key prefixes signifyingpowers of ten.

SI Prefixes for Multiples

Name deca hecto kilo mega giga tera peta exa zetta yotta

SymbolFactoer

Da101

H102

k103

M106

G109

T1012

P1015

E1018

Z1021

Y1024

SI Prefixes for Fractions

Name deci centi milli micro nano pico femto atto zepto yocto

Symbol d C m n p f a z yFactor 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 10-21 10-24

Dimensional analysisis done to determine equivalent SI units using different prefixes. For example, toconvert 8 kilometers into centimeters.

8 km 0 = 8 x103-(-2)cm = 8 105cm


7/29


b. Conversion of units from English system to the SI system requires memorization of convertionfactors.

1 kilogram = 2.2 pounds 1(lbs)2.54 centimeters = 1 inch

12 inches = 1 foot1 meter = 3.28 feet

3 feet = 1 yard1 mile = 5,280 feet

1 kilometer = 0.62 mile

6. Recall the following concepts in plane trigonometry.

a. A line is a seriesof points that extends in two opposite directions without end. Two points areneeded to define a line. It has no fixed length or width. It is considered infinitely long.

i. Thepoint of intersectionis the point where two lines meet or come together.ii.Perpendicular linesfrom right angles to the point of their intersection. They will never

intersect with each other.

b. A curve line is a line that representsa mathematical equation. IT may two-dimensional or three-dimensional.

c. An angle is the figure formed by two rays sharing a common endpoint, called the vertexof theangle. Angles are measured in degrees (0).

i. A right angle is an angle whose measure is exactly 90 degrees.ii. An acute angle is an angle whose measure is lees than 90 degrees.iii. An obtuse angle is an angle whose measure is more than 90 degrees but less than 180

degrees.iv. Astraight angle is an angle whose measure is is exactly 180 degrees.v. A reflex angle is an angle whose measure is more than 180 degrees but less than 360

degrees.

vi. Two angles are complementary if their sum is 90 degrees.vii. Two angles aresupplementary if their sum is 180 degrees.

d. A triangle is a plane geometric figure with three verticesand three sides. The sum of three internangles of a triangle is always equal to 180 degrees.

i. Triangles may be classified on the length of sides:1. An equilateral triangle has three sides of equal length. It is also called

equiangular triangles because all three angles measure exactly 60 degrees.2. An isosceles triangle has two sides of equal length. the two angles opposite the

two equal sides are also equal in measure.3. Ascalene triangle has three sides of unequal length. All three angles are also of

unequal measure.

ii. Triangles may also be classified based on the measure of internal angles:1. A right triangle has exactly one right angle among its internal angles.2. An acute triangle is composed of three acute internal angles.3. An obtuse triangle has exactly one obtuse angle among its internal angles.

ii. Special triangles have properties that aloow us to compute algebraically the lengths oftheir corresponding sides.

1. The dimension of the right trianglefollow thePythagorean theorem.

ca

b


8/29


a. The side opposite the right angle is called the hypotenuse (c).b. The two other sides are called the legs (a, b).c. For any right triangle, a2+b2= c2.

For a 45-45-90 right triangle:450

2

450

a. The length of the hypotenuse is equal to the length of one the legs

multiplied by 2.3. For a 30-60-90 right triangle:

a. The length of the hypotenuse is two time the length

of the shorter leg.b. The length of the longer leg is equal to the length of

the shorter leg multiplied by 3.

4. For equilateral triangle:

a. The height is defined by the following formula:

height = a2

The area is defined by the following formula:

area = a24

5. Herons theorem may be used to calculate the area of any triangle given the length of the three sides.a. First, calculate the semiperimeter:

s =++

b. Use the semiperimeter to calculate the area using Herons theorem:

area = ( )( )


9/29


c. A quadrilateral is a plane geometric figure with exactly four sides and four vertices. The sum of thmeasures of the interior angles of a quadrilateral is exactly 360o.

i. Aparallelogramis a quadrilateral with two pairs of parallel sides.1. The opposite angles of a parallelogram are equal in measure.2. The adjacent angles of a parallelogram are supplementary.3. The diagonals of a parallelogram bisect each other.

ii. A rectangleis aquadrilateral with four right internal angles.1. The diagonals are equal in length and bisect each other.

2. The length of each diagonal is equal to iii. Asquare is a quadrilateralwith four equal sides and four right internal angles.

1. The diagonals of a square bisect each other and meet at 90 degrees.2. The diagonals of a square bisect its angles.3. The diagonals of a square are perpendicular.

iv. A rhombus is a quadrilateral with four equal sides.1. Oppositeangles of a rhombus are equal sides.2. The two diagonals of a rhombus are perpendicular.

d. Apolygonis a plane geometric figure bounded by a closed path or circuit, composed of a finitesequence of straight line segments.

i. The segments are called itssides, and the points where two edges meet are thepolygons vertices.

ii. Polygons are named based on the number of sides:

# of sides Name of polygon # of sides Name of polygon

3 Triangle 12 Dodecagon

4 Quadrilateral 13 Triskaidecagon

5 Pentagon 14 Tetradecagon

6 Hexagon 15 Pentadecagon

7 Heptagon 16 Hexadecagon

8 Octagon 17 Heptadecagon

9 Nonagon 18 Octadecagon10 Decagon 19 Nonadecagon

11 Undecagon 20 Icosagon

iii. A regular polygon has equal length of all sides and equal measure of all interiorangles.

iv. The sum of allthe interior angle of a regular polygon is equal to(n2) 180.

v. The measure of eac interior angle of a regular polygon is equal to(n 80

7.

Recall the following basic concepts in Elementary algebra:a. An algebraic expression is a mathematical expression made up of the signs and symbolsof algebra. These symbols include the Arabic numerals, literal numbers, the signs ofoperation, and so forth.

b. The components of algebraic expression are called terms. Based on the number of terms,special designation is given to algebraic expressions:

i. An expression containing only one term (ie, 2x) is called a monomial.ii. A binomial contains two terms (ie, 2x + 1).iii. A trinomial consist of three terms (ie, 2x2 + 3x + 4).iv. Any expressions containing two or more terms may also be called by the genera

name, polynomial.


10/29


8. A ppolynomial expression is a monomial or a sum of monomials.a. The degree of a polynomial expression with one variable is the value of the largest exponent

of the variable that appears in any term. For example, the degree of the binomial x2+ 4 is 2.b. A functional relationship between quantities that can be described by an equation where y

equals a polynomial expression of x is apolynomial function.i. A linear function is a polynomial function with a degree equal to 1 (ie, y = 5x + 3).

1. The graph of a linear function is a straight line.ii. A Quadratic function is a polynomial function with a degree equal to 2

(ie, y = 5x2+ 3x + 3).1. The graph of a quadratic function is a parabola.

iii. A cubic functionis a polynomial function with a degree equal to 3(ie, y = 5x3 + 3x2+ x + 3).

1. The graph of a cubic function is a curve.

9. Recall the principles in evaluation of polynomial expressions.a. Group like terms using cummutative and associative properties.b. Combine like terms using distributive property.c. Simplifying powers can also help you multiply monomials.

i. multiplying powers with like bases: amx an= am+n.

ii. Raising apower to a power: (am)n= amn.iii. Raising a product to a power: (ab)n= anx bn.iv. Zero power: ao= 1.

v. Negative power a-n=

vi. Dividing powers with like bases: = a

(m-n).

vii. Raising a quotient to a power:

=

10.Factoring a polynonmial means writing it as a product of 2 or more monomials.

a. Common monomial factor:i. Consider the trinomial 2x310x2 + 6x. The common monomial factor is 2x. So using thedistributive property in reverse, we factor this expression as: 2x (x2+ 5x + 3).

b. Grouping:i. Consider x3x2 + x1 = 0.

x3x2+ x1 = 0(x3x2) + (x1) = 0x2 (x- 1) + (x1) = 0(x1) (x2+1) = 0

c.square of a binomial (perfect square trinomial):i. (a +b)2= a2+ 2ab + b2.ii. (a +b)2= a2 - 2ab + b2.

d.Difference of two squares:ii. (a = b) (ab) = a2- b2.

e. Completing the square:i.Consider x2+ 6x + 5 = 0.

(x2+ 6x + 5 + 4)4 = 0(x2+ 6x + 9)4 = 0(X + 3)24 = 0(x + 3 + 2) (x + 3 + 2) = 0(x + 1) (x + 5) = 0


11/29


f. Sum of two cubes:i. (a+b)3 = (a + b) (a2 _ a + b2).

g. Difference of two cubes:i. (a - b)3= (a - b) (a2+ ab + b2)

15.A linear equation is an algebraic equation in which each term is either a constant or a product of aconstant and (the first power of) a single variable.These may be expressed in the following forms:

a. Standard form:Ax + By + C = 0, where,i. the x-intercept is (-C/A, 0).ii. the y-intercept is (0, -C/B).iii. the slope of a line isA/B.

b. Slope intercept form: y = mx + b, where, m is the slope and (0, b) is the y-intercept.y2-y1

c. Two point form: y-y1=

(x-x1) where, where (x1, y1) and (x2, y2)are two different poits on

the line.

d.Point slope form: yy1= m (xx1), where m is the slope of the line and (x1, y1) is any point onthe line.

e.Intercept form:+

= 1, where (a, 0) is the x-intercept and (0, b) is the y-intercept.

16.A linear inequality is an inequality which involves a linear function. The solution to a linearinequality is obtained by shading the corresponding half-space in the Cartesian plane after graphingthe expression as a linear equation.

17.Recall the various counting techniques:a. Fundamental Principle of Counting: In a sequence of events, the total possible number of way

all events can performed is the product of the possible number of ways each individual evecan be perform.

b. Factorial: n! = (n1) (n - 2) (3) (2) (1); for example, 5! = 5 x 4 x 3 x 2 x 1.c. Permutation: A permutation is an arrangement of objects without repetation where order isimportant. A permutation of n objects, arranged in groups of size r, without repetition, and order

being important is: nPr=!

()!d. Combination: A combination is an arranement of objects without repetation where order is not

important. A combination of n objects, arrangedin groups of size r, without repetation, and order

not being important is: nCr =!

()! !

18.Probability is a measure of certainty or uncertainty that an event will happen.It arranges from 0 to 1.

a. The probability of an impossible event (an event that wil never occur) is 0.b. The probability of an certain event (an event that will surely happen) is 1.c. The probability (P) of an event (E) is expressed mathematically as:

P(E)=( )

( )


12/29


19.Measures of central tendency are numerical descriptive measures which indicate or locate the centerof distribution or data set.a. The mean of a set of values or measurements in the sum of all the measurements divided by the

number of measurements in the set.b. The meadian is the middle value of a given set of measurements, provided that the values or

measurements are arranged in an array. An array is an arrangement of values in increasing ordecreasing values.

c. The mode is the value occurs most frequently in a set of measurements or values.d. Measures of variability are measures of the average distance of each observation from the center

of distribution. They measure homogeneity or heterogeity of a particular group.a. The range is the difference between the highest and the lowest values. This is the

simpliest and most unreliable measure of variability since it uses only two values in thedistribution.

b. the mean obsolute deviation is the average of summationof the absolute deviation of eachobservation from the mean. The formula for the mean absolute deviataion is:

mean absolute deviation =

where, x is a value or score from the raw data, is the mean and n is the totalnumber of cases.

c. Variance (s2

) is the average of the squared deviation from the mean . The formula forfinding the variance is shown below: s2=

( )

where, x is a value or score from the raw data, is the mean and n is the totalnumber of cases.

d. Thestandard deviation is the square root of the average deviation from the mean. It ismathematically equal to the square root of the variance.

s = ( ) where, x is a value or score from the raw data, x is the mean and n is the total numberof cases.


13/29


PRACTICE TEST: Test your mastery of competencies. Choose the letter that correspond to the best

answer.

1. Evalute the following expression: 5 + 3 (42 + 7)7 (2 + 32 x 8)0A. 0 C. 72

B. 152 D. -444

2. Evaluate the following mathematical expression: +4 ( 6)+ (+4)++( 6) A. 16 C. 36

B. 24 D. 12

3. Which among the following is NOT a prime number?

A. 31 C. 51

B. 41 D. 61

4. How many prime numbersare there between 1 and 100?

A. 23 C. 25

B. 24 D. 26

5. What is the largest prime number less than 100?

A. 91 C. 95

B. 93 D. 97

6. What are the prime factors of 128?

A. 1 x 2 x 8 C. 2 x 2 x 2 x 4 x 4

B. 2 x 2 x 2 x 2 x 2 x 2 x 2 D.2 x 3 x 2 x 2 x 2 x 2 x 2 x 2


A. 3 x 3 17 C. 153 x 1

B. 3 x 3 x 7 x 9 D. 3 x 7 x 13


A. 3 x 3 x 7 x 7 C. 3 x 6 x 9 x 11

B. 3 x 17 x 11 D. 3 x 7 x 13

9. What is the least common multiple of 24 and 80?

A. 360 C. 240

B. 80 D. 480

I Math 1-Fundamentals of math

1. Use of four fundamental operations in problem solving involving:

1.1 Operation with whole numbers, decimals, Fractions, and integers1.2 Prime, composite, denominate numbers

1.3 Prime factorization

1.4 LCM, GCM

1.5 Divisibilty rules


14/29


10.One trip around a running track is 440 yards. One jogger can complete one lap in 8 minutes, the other

can complete it in 6 minutes. How long will it take for both joggers to arrive at their starting point

together if they start at the same time and maintain their jogging pace?

A. 12 minutes C. 36 minutes

B. 24 minutes D. 48 minutes

11.Josefa is making bead necklaces. She has 90 green beeds and 180 blue beads. What is the greatestnumber of identical necklaces she can make if she wants to use all of this beads?

A. 12 C. 16

B. 15 D. 18

12.Lisa bought a big bag of candy at a warehouse store. There are 102 pieces of candy in a bag. Lisa needs

to divide the candy up into smaller bags. She wants to put the same number of pices in each small bag.

How many small bags could lacey use?

A. 17 C.19

B. 18 D. 20

13.A dish company needs to ship an order of 117 glass bowls. The company will put the bowl into severalboxes. Each box contains the same number of bowls. How many boxes could the company use for the

order?

A. 11 C. 13

B. 12 D. 17

14.The number, 212115273999132, is NOT divisible by which of the following factors?

A. 2 C. 4

B. 3 d. 8

15.When 2,000 pounds of paper are recycled, 17 trees are saved. How many trees are saved if 5,000 pounds

of paper is recycled?

A. 41 C. 45

B. 42.5 D. 63

16.A recipe calls for 2 eggs for every 5 cups of flour. A local chief will use 35 cups of flour, how many eggs

must be have?

A. 12 C. 14

B. 13 D. 16

17.Five out of every seven households have cable TV. If 42,000 households in a certain city hyave a TV,

how many do not have cable TV?

A. 12,000 C. 30,000

B. 21,000 D. 32,000

18.Our school has 8 male teachers who comprise 25% of all our teachers. How many teachers do we have?

A. 24 C. 32

B. 28 D. 40

19.Miss Santiagos class has 20 boys and 15 girls in her English class. What percent of the students aregirls?

A. 28% C. 44%

B. 43% D. 50%


15/29


20.Jerome answered 80% of the 50 items correctly in math test. How many items did he answer correctly?

A. 40 C. 45

B. 44 D. 50

21.A business man had incurred the following expenses in his trips to the Visayanislands: P5100.00,

P4600.00, P3800.00, and P3200.00. What was his total for his trip?

A. P16000 C. P17000B. 16700 D. P17500

22.Ms. Tonelada weighed 60 kg. She lost 4 kg on her first week of exercise, gained 2 kg on her second

week, lost 6kg on her 3rdweek and and remained her weight on 4thweek. What was her weight on the 4

week?

A. 52 kg C. 68 kg

B. 58 kg D. 72 kg

23.A group of young people from four countries gathered together for an international conference: 40 from

Manila, 60 from Japan, 35 from Thailand and 45 from Singapore. The participants will form

discussiongroups with equal number of members from each country in each group. What is the greatestnumber of discussion groups that can be formed?

A. 5 C. 20

B. 15 D. 25

24.During summer, a lady visits Baguio 6 days, and his best friend every 4 days. If they visited Baguio las

April 11, what does the earliest date did both of them visit Baguio again

A. April 21 C. May 5

B. April 23 D. May 11

25.A recipe calls a of sugar. How much sugar should be used if only of the quantities given in the

recipe is to be prepared?

A.4 C.

B. D.

8

26.In a bundle of new 100 bils, The bills are consecutively numbered RTC3432260 to RTC3432280. How

much is the total amount of the bills?

A. 3200 C. 3400

B. 3300 D. 3500

27.The sum of three consecutive integers is 96. What are the integers?

A. 31, 32, 33 C. 30, 32, 34

B. 32, 33, 34 D. 33, 34, 35

28.A farmer can plow of a hectare in 1 hr. At this rate, in how many hours will 5 farmers plow the same

fields?

A. C.

B.0 D.10


16/29


29.If the exchange rate of the US dollars to pesos is $1 = Php47.90 what is the value of Php1 in American

cents?

A. 02 C. 2

B. 479 D. 4.79

30.In a card game, a player got the following scores: 35, -60, -40, 80, -100, 25, -25. Whatis his final score?

A. -115 C. 85B. -85 D. 115

31.The first 5 numbers in a sequence are 2, 3, 5, 8, and 12. What is the 7thnumber in the sequence?

A. 15 C. 23

B. 19 D. 25

32.A teacher wants to group his pupils into groups of 3 or 5 or 6. However, if she found out that if she do

that there will always be 1 pupil left. What is the least possible number of pupils in the class?

A. 81 C. 21

b. 31 d. 11

33.Twelve pesos more than twice Micos allowance is at most 600 pesos. What is her maximum allowance?A. 300 pesos C. 588 pesos

B. 634 pesos D. 294 pesos

34.What integer should be added to 11 to get the sum of at least 37?

A. At least 15 C. At least 26

B. At least 25 D. At least 32

35.What are the largest two consecutive odd integers whose sum is at most 60?

A. 25, 27 C. 29, 31

B. 27, 29 D. 31, 33

36.Two numbers are in ratio of 3:4. If there sum is 84, what is the smallest number?

A. 24 C. 48

B. 36 D. 54

37.The numerator of a fraction is 3 less than the denominator. If the numerator and denominator are each

increased by 1, the value of the fraction becomes . What is the original fraction?A. 7/12 C. 8/11

B. 8/12 D. 6/13

38.Mr. Lucido deposited 225 pesos in the bank. If his deposit consisted of 29 bills, consisting of 5 peso and

10 peso bills, how many 10 peso bills did he deposit?

A. 13 C. 15

B. 14 D. 16

39.Brenda has saved 300 coins, consisting of 25 centavo and 10 centavo coins.if the total value of her

savingsis 4 pesos, how many 10 centavocoins did she save?

A. 100 C. 300

B. 200 D. 400


17/29


40.Julia can finish a project in 10 hours while Elenita can do the same project in 8 hours. How will it take

them to finish the project together?

A. 3.44 hours C. 4.44 hours

B.4.00 hours D. 18 hours

41.If a liter of chemical X is 95% pure, how many liters of water must be added to make a 50% solution?

A. 0.80% L C. 0.94% LB. 0.90 L D. 0.09 L

42.Arthur has blended coffee worth 95 pesos per kilogram with coffee worth 115 pesos per kilogram to

make 50 kilograms of coffee that will be sold at 107 pesos per kilogram. How many kilograms of each

kind did he blend?

A. 20 kilos of 95/kg and 30 kilos of 115kg

B. 30 kilos of 95/kg and 30 kilos of 115kg

C. 30 kilos of 95/kg and 50 kilos of 115kg

D. 60 kilos of 95/kg and 30 kilos of 115kg

43.A Victory liner bus traveling at a rate 70 km/h leaves the station after a freight truck has left andovertakes it in 5 hours. At what ratewas the freight truck traveling?

A. 20km/h C. 40km/h

B. 30 km/h D. 50 km/h

44.Don Antonio invested part of 30,000 pesos at 5% interest and the remaining interest at 6% interest at

BPI. If his investment yields annual income of 1,620 pesos, how much did he invest at 6% interest?

A. 12,000 pesos C. 16,000 pesos

B. 14, 000 pesos D. 18,000 pesos

45.The units digit of a two-digit number exceeds the tens digit by 2. Find the number if it 4 times the sum o

its digits.

A. 24 C. 48

B. 42 D. 82

46.Using t6he integers 4, 7, 9, 8 and 5, how many two digit numberscan be formed if repletion is NOT

allowed.

A. 20 C. 25

B. 23 D. 28

47.In how many ways can 5 boys be seated in a row of 5 seats?

A. 72 C. 102

B. 9 D. 120

48.In how many ways can a term of 10 basketball players be chosen from 12 players?

A. 62 C. 66

B. 64 D. 72

49.If a picture frame is 27 cm long and 18 cm wide, what is the ratio of its length to its width?

A. 3:2 C. 3:5

B. 2:3 D.5:3

MATH 1-1.6 Ratio and proportion


18/29


50.Five bananas weigh as much as 3 star apples. Inthis rate, how many star apples will weigh as much as 45

bananas?

A. 27 C. 33

B. 30 D. 36

51.It takes 20 men to build a building for 60 days. Assuming that all men work at this rate, how many men

will be needed to build the same building in 15 days?A. 5 C. 100

B. 80 D. 120

52.The ratio of the numbers of carabaos, goats and cows in a farm is 5:1:2. If there are 48 animals of these

kinds in his backyard, how many of them are goats?

A. 2 C. 6

B. 4 D. 8

53.The ratio of the numbers of red, green and blue balls in a box is 2:5:6. How many green marbles are

there if there are 52 marbles in all?

A. 4 C. 20B. 8 D. 24

54.In a university, the ration of the female professors to the male professors 8:5. If there are 75 male

professors, how many are female professors?

A. 120 C. 225

B. 180 D. 375/8

55.A meter stick is cut into 2 at the 25 cm mark. What is the ratio of the smaller piece to the larger pice?

A. 1:3 C.3:4

B. 2:5 D. 4:5

56.In asurvey to determine the reaction of people about having a new GSIS card, 80% of the 2,400 people

voted in favor of the new card. How many of the voters did not vote for the new card?

A. 1920 C. 800

B. 1600 D. 480

57.Lulu spends 15% of her monthly income for house rental, 10% for electric bill and 25% for food and

other miscellaneous expenses, she still has P6,000 left. How much does she earn every month?

A. P15,000.00 C. P9,000.00

B. P12,000.00 D. P8,000.00

58.If 500 or 25% of a graduating class are girls, how many are graduating?

A. 2,000 C. 10,000

B. 5,000 D. 20,000

59.At 25% discount, Ms. Barat paid P150.75 for a bag. What was the original price of the bag?

A. P37.69 C. P201.00

B. P150.75 D. P603.00

MATH 1-1.7 Percentage, Base and Rate


19/29


60.A man accepts a position at P14,250 basic salary with an agreement that he will receive a 2% increase

every year for 3 years. What will his salary be at the end of 3 years?

A. P14,950.00 C. P15,122.21

B. P15,105.00 D. P16,500.00

61.A man invested Php 100000. He put part of it in a bank at 5% interest. On the other hand, he invested th

remainder in bonds with a 9% yearly return. How much did he put in a bank if his yearly income fromthe two investments was Php 7,400?

A. Php40,000.00 C. Php60,000.00

B. Php50,000.00 D. Php 70,000.00

62.Max is planning to take a leisurely stroll around their rectangular patio, which measures 27.7 m long an

21.5 m wide. Howfar does Max have to walk?

A. 96.4 m C. 98.4 m

B. 120.4 m D. 88.4 m

63.Which among the following has the largest perimeter?

A. Square pizza with perimeter of 80 cm

B. Circular pizza with radius of 13 cm

C. Rectangular pizza with dimension 10 cm x 14 cm

D. Circular pizza with radius 8.5 cm

64.You own a small rectangular box measuring 3 cm x 2 cm. If the dimensions of this box are increased by

10%, what is the area of resulting box?

A. 6.26 cm2 C. 7.12 cm2

B. 6.22 cm2 D. 7.26 cm2

65.What is the area of the rhombus whose diagonals measures10 m and 12 m respectively?

A. 120 m2 C. 60 m2

B. 100 m2 D. 360 m2

66.Find the volume of a toy ball whose radius is 2 cm.

A. 33. 49 cm3 C. 51.76 cm3

B. 48.56 cm3 D. 50.24 cm3

67.A tin can has a radius of 6 cm and a height of 20 cm. What is the volumeof milk that this container can

hold?

A. 3, 00cm3 C. 1,689 cm3

B. 2,261 cm3 D. 2,200 cm3

68.The total surface area of a cubic box is 600 cm2. What is the length one side of this box?

A. 6 cm C. 9 cm

B. 8 cm D. 10 cm

69.After a stone is dropped into a cylindrical container filled with 100 cm3of water, the water rises and the

new reading is 106.5 cm3. What is the volume of the stone?

A. 6.5 cm3 C. 60.5 cm3

B. 60.65 cm3 D. 10.65 cm3

MATH 1-1.8 Measurement and units of measure


20/29


70.Which of the following length is the longest?

A. 555 cm C. .005 km

B. 5.5 m D. 5555 mm

71.One-fifth of the width and one-fourth of the length of a rectangular cardboardis cut off. What part of the

original cardboard is the area of the remained piece?

A. 30% of the original area C. 50% of the original areaB. 40% of the original area D. 60% of the original area

72.If the area of a triangle is 1 sq unit and its height is unit, what is the length of its base?

A. 1 unit C. 3 units

B. 2 units D. 4 units

73.What is the radius of a circle whose area is 25 cm2?

A. 25 cm C. 5 cm

B. 25 cm D. 5 cm

74.Refer to the figure. Given: m 2 = 55

o

and m 3 = 80. What is m 4?

3

4 1 2 5

A. 90o C. 115o

B. 105o D. 135o

75.Mrs. Dina Ta Tandadivided her lot among her 4 children. The first got 3 ha, the second 3

ha, the

third 3

4 ha and the fourth 3

ha.

How big is Ms. Tandas lot?

A. 12

4 ha C. 13960 ha

B. 13 ha D. 14 ha

76.An elevator can carry a maximm load of 605 kg. How many passengers of weight50.5 kg each can theelevator hold?

A. 12 C. 11B. 11.9 D. 10

77.A room is 30 ft. long, 25 ft wide and 14 ft high. If 42 ballons are inside the room, how many cubic feet ofspace does this allow for each balloon?

A. 25 C. 250B. 69 D. 690

78.What is the volume of air in an atmospheric balloon with a diameter of 24 cm?A. 144 cm3 C. 570 cm3B. 240 cm3 D.2304 cm3 \


21/29


79.A photograph measuring 7 cm by 5 cm is enlarged so that the longer side is 24 cm. What is the length(in cm) of the shorter side?

A. 36 C. 6B. 16 D. 1.6

80.How many cm are there in 2 m and 550 mm?A. 75 C. 2055B. 255 D. 2550

81.How much liquid containing 6% boric acid should be mixed with 2 quarts of a liquid that is 15% boricacid in order to obtain a solution that is 12% boric acid?

A. 1 quart C. 3 quartsB. 2 quarts D. 5 quarts

82.A defective ruler was found found 11.5 in long. Using this ruler, Samuel was found to be 4 ft tall. Whatis Samuls actual height?

A. 4 ft 2 in C. 3 ft 11.5 inB. 4 ft 4 in D. 3 ft 10 in

83.How many grams are there in 1 petagram?A.5 C.1015B.5,000 D.5 x 1015

84.If an apple weighs about 170grams, about how many apples are in 3.5 kilogram bags of apple?A. 20 C. 22B. 21 D. 23

85.A new supercomputer measures 462 lbs in weight. How much does it weigh in grams?

A. 21 g C. 21,000 gB. 2,100 g D. 210,000 g

86.How many liters are there in 353 quarts?A. 353 L C. 326 LB. 334 L D. 324 L

87.Paul and his son participated in a marathon. Paul traveled 3 km 50 m while his son ran 502 m 36 cm.What is the total distance that the father and tandem covered?

A. 3,760 m C. 3,552 mB. 5,452 m D. 3550 m

88.A slow moving snail traveled 4,800 mm. How far did it travel in kilometers?A. 0.0048 km C. 0.48 kmB. 4.8 km D. 0.048 km

89.For the fruit punch, 3050mL of fruit juice is needed. How much fruit juice is needed in dekaliters?A. 0.305 daL C. 5.03 daLB. 3.05 daL D. 0.053 daL

90.The measure of an angle is 25 more than its supplement. What is the measure of the larger angle?A. 102.5 degrees C. 90 degreesB. 77.5 degrees D. 110 degrees

MATH 1-1.9 Convert units in the metric system


22/29


91.If the measure of angle is twice the measure of its complement, what is the measure of the angle?A. 30 degrees C. 90 degreesB. 60 degrees D. 120 degrees

92.Which among the following statements is ALWAYS TRUE?A. The supplement of an angle is acute.B. The supplement of an angle is obtuse.C. The supplement of any acute angle is acute.D. Two supplementary angles are congruent.

93.Which among the following statements is ALWAYS TRUE?A. Two adjacent right angles are supplementary.B. Complements of congruent angles are congruent.C. Two intersecting lines form two pairs of vertical angles.

D. Angles that form a linear pair are complementary.

94.What is the measure of an angle if the measure of its supplements is 39 degrees more than twice themeasure of its complement?

A. 29 degrees C. 49 degreesB. 39 degrees D. 59 degrees

95.Consider the triangle illustrated below. Find the measure of angle x.

3x - 5

5x + 5 3x15


96.Consider the triangle illustrated below. Find the value of x.

68

x 125

A. 55 C. 60

B. 35 D. 25

97.Two angles of a triangle measure 4 cm and 7 cm. What is the range of values for the possible lengths ofthe third side?

A. 4 < x < 7 C. 7 < x < 11B. 3 < x < 11 D. 11 < x < 15

98.Which of the following statements is TRUE?A. Arectangle is a square.B. A parallelogram is a trapezoid.C. A rhombus is a rectangle.D. A square is a rhombus.

2. PLANE GEOMETRY

2.1 Show mastery of basic terms and concepts in Plane Geometry

2.2 Solve problems involving basic terms and concepts in Plane Geometry


23/29


99.Consider the quadrilateral below. Find the value of x.

A. 22 C. 112B. 56 D. 54

100. If the sum of the interior angles of a regular polygon is 1980 degrees, how many sides does ithave?

A. 11 C. 13B. 12 D. 14

101. What is the sum of the measures of the interior angles of an icosagon?A. 3100 C. 3240B. 3140 D. 2850

102. What is the measure of an interior angle of a dodecagon?


103. How many unique diagonals can be drawn in the pentagon?A. 5 C. 7B. 6 D. 10

104. Nine unique diagonals can be drawn in a regular polygon.How many sides does it have??A.9 C.7B.8 D.6

105. A triangle has a perimeter of 50.If 2 of its sides are equal and the third side is 5 more than the equal

sides , What is the length of the third side?A. 5 C. 15B. 10 D. 20

106. A rectangle is4times as long as it is wide. If the length is increased by 4 inches and the width isdecreased by by 1 inche, the area will be 60 square inches. What were the dimensions of theoriginal rectangle?

A. 2 x 32 C. 3 x 14B. 4 x 16 D 5 x 12

107. In a quadrilateral two angles are equal. The third angle is equal to the sum of the two equal angles.The fourth angle is 60olees than twice the sum of the other three angles. Find the measures of the

angles in the quadrilateral.A. 25o, 35o, 90oand 220o C. 35o, 35o, 70oand 220o

B. 25o, 45o, 70oand 70o D. 35o, 35o, 70oand 210o

108. If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, tharea of the resulting rectangle is 96 square centimeters larger than that of the original square. Findthe dimensions of the rectangle.

A. 17 x 24 C. 10 X 24B. 24 x 24 D. 6 x 16


24/29


109. The smallest angle of a triangle is two-third the size of the middle angle, and the middle angleisthree-sevenths of the largest angle. Find all three angle measures.

A. 30o, 60o, 90o C. 35o, 45o, 110oB. 45o, 45o, 90o D. 30o, 445o, 105o

110. If the height of a triangle is 5 inches less than the length of its base, and if the area of the triangle is52 square inches, find the base and the height.

A. Base = 13, height = 8 C.Base = 11, height = 7B. Base = 12, height = 9 D. Base = 13, height = 4

111. A wood frame for pouring concrete has an interior perimeter of 14 meters. Its length is 1 metergreater than its width. The frame is to be braced with twelve-gauge steel croos wires. Assuming anextra half meter of wire is used at either end of a cross-wire for anchoring, what length of wireshould be cut for its brace?

A. 6 m C. 8 mB. 7 m D. 12 m

112. Find the largest possible rectangular area you can enclose, assuming you have 128 meters of fencing.A. 256 m2 C. 1024 m2

B. 512 m2 D. 2048 m2

113. In a photograph, Bianca is 9 cm tall and his brother Tristan is 10 cm tall. Biancas actual height is153 cm. What is Tristans actual height?

A. 148 cm C. 168 cmB. 156 cm D. 170 cm

114. The measures of the angles of a triangle are in a ratio of 2:3:4. Find the measure of the middle angle.A. 30o C. 60oB. 40o D. 80o

115. The graph shows the number of socks, belts, handkerchiefs, neckties sold by a store in one week.

The names of the items are missing on the graph. Socks were the most often sold, and fewer necktiesthan any other item were sold. More belts than handkerchiefs were sold. How many belts were sold?

A. 80 C. 120B. 90 D. 140

150

100

50

0

Numberofitems

Items Sold by Store


25/29


116. An empty box weighs 1.3 kilos. A Math book weighs 1.5 kilos. Which expression gives the weightof the box when filled with theyMath books?

A. 1.3y + 1.5 C. 1.3 + 1.5y

B. 1.5y- 1.3 D. 1.3y+ 1.5y

117. A builbing 25 m tall casts a shadow 10 m long. How long is the shadow of a 5-foot girl standing

beside the buiding?

A. 2 ft C. 10 ft

B. 2.5 ft D. 250 ft

118. What is the maximum number of books, each 1.4 cm thick that can be put vertically in a shelf

which is 64 cm long?

A. 44 C. 46

B. 45 D. 64

119. Factor completely the expression: a2x5by5a2y + b2x.A. (a2+ b2)(x5y) C. (a + b)(ab)(x5y)

B. (a2

+ b2

)(x + 4y + 3) D. (a2

+ b2

)(x2

5y2

)

120. Which among the following is NOT a perfect square trinomial?

A. x2+ 8x + 16 C. 49x2+ 70x + 36

B. 9x2+ 12x + 4 D. x2+ 6x + 9

121. Factor completely the expression: 27a354a2b + 36ab28b3.A. (3a3b)3 C. (4a3b)3

B. (a3b)3 D. (3a2b)3

122. What is the greatest monomial factor of the expression: -13abc39bc +26ab?

A. 3b C. 13abcB. -13b D. 26b

123. Which factoring technique will best help you to factor the expression: x2 + 6x7 = 0?

A. Difference of two cubes C. GroupingB. Common monomial factor D. Completing the square

124. Find the general equation of the line which passes through the points: (2, -1) and (-3, 5).

A. 6x + 5y7 = 0 C. 6x + 7y5 = 0B. 5x6y7 = 7 D. 6x + 6x5 = 0

2.2 Solve problems involving basic terms and concepts in plane geometry

ELEMENTARY ALGEBRA

3.1 Show astery of basic terms and concepts in

3.1.1 Polynomials

3.1.2 Linear equation

3.1.3Linear inequalities

3.2 Solve, evaluate, and manipulate symbolic and numerical problems in eelementary

al ebra b


26/29


123. What is the equation of the line with x-intercept of 4 and y-intercept of 3?A. 3x + 4y = -12 C. 6x8y = 12B.3x4y = 12 D. 4x3y = 12

124. Find the equation of the line with a slope of 4 and passing through the point (-5, 3).A. x4y = -23 C. 4xy = -23B. 4x - 4y = 23 D. xy = -4

125. Find the slope of the line describe by the following table of values:

x -2 0 2 4

y -7 -4 -1 2

A.1.50 C. 2.50B. 0.66 D. 3.50

126. What is the equation of the line whose slope is -2 and whose y-intercept is 3?A. 2x + 3y = 6 C. 2x +2y = 4B. x + 2y = 3 D. 2x + y = 3

129. The length of rectangle is 18 cm. What are the possible widths that will give a perimeter lessthan 150 cm?

A. 3 < width < 54 C. 18 < width < 36B. 0 < width < 57 D. 12 < width < 57

130. What is the simplest form of the expression?A. -2x + 4y17 C. 2x + 4y17B. 2x + 4y + 17 D. 2x + 4y

131. If x = 1 and y = -2, what isthe value of the expression

4x+?

A. 9 C. 7B.152 D.-

74

132. If -3x < 6, which of the following statement is TRUE?A. x < - 2 C. x > 2B. x < 2 D. x > -2

133. If a die is rolled, what is the probability of getting a number divisable by 2?A. 1/6 C.1/4

B.1 2 D. 1/3134. Which among the measures of central tendency is NOT influenced by outliers?

A. mean C. medianB. weightend mean D. mode

4. STATISTICS AND PROBABILITY

4.1 Show mastery and knowledge of basic terms and concepts in

statistics and probability

4.2 Solve, evaluate, and manipulate symbolic and numerical problems in statistics and

probability by applying fundamental rules, principles and processes.


27/29


135. Monica obtained the following results from her mathematics exams: 80, 82, 83, 91. Whatscore must she get on the next exam so that her average score is 85?

A. 92 C. 89B. 93 D. 8

136. In the Filipino test, eight students obtained the following scors: 10, 15, 12, 18, 16, 24, 12,14. What is the median score?

A.14 C. 15B. 14.5 D. 15

137. The following table summarizes the scores of Section A on the recent periodic test in socialstudies. What is the median score interval?

Score Frequency

16-23 2

24-31 4

32-39 6

40-47 12

48-45 1056-62 8

A. 24-31 C. 40-47B. 32-39 D. 48-55

138. The following measurements were obtained from the caliper: 20, 15, 20, 14, 18, 15, 6. What is themode?

A. 15 C. 14B. 20 D. 15 and 20

139. The following are Joselitos grades for the 3rdquarter. Find his general weighted average.

Subject Units Grade

Math 3 89

English 2 84

Science 3 90

Filipino 2 86

HEKASI 2 84

A. 87.08 C. 86.36B. 89.50 D. 87.45

For the numbers 96-100, consider the following situation.The grades in math of the students in

section B are as follows: 70, 95, 60, 80, 100.

140. What is the mean absolute deviation of their group?A. 11.7 C. 14.6B. 13.2 D. 15.9

141. What is the population variance of their group?A. 224 C. 264B. 250 D. 280


28/29


142. What is the population standard deviation of their group?A. 16.73 C. 1.58B. 1.41 D. 14.97

143. What is the range of their group?A. 60-95 C. 60-100B. 70-100 D. 80-95

144. What can you infer from the measures of variability obtained from this population?A.The population is very homogeneous.B. The measures are very unstable.C. The grades are very scattered.D.The range of scores is a very reliable measure of variability.

145. What measure of central tendency can best describe the size of t-shirts commonly used byteen-agers

A. mean C. modeB. median D. both A and C

146. The following aree the results of the recent achievement test in mathematics of fourdivisions.

Division Mean Standard deviation

I 34 4.5

II 34 3.0

III 23 1.0

IV 20 2.0

Which division performed best?A. I C. IIIB. II D. IV

147. What is the probability of getting a multiple of 3 when a die is tossed?A. 1/6 C. 1/3B. 1/4 D. 1/2

148. In how many ways can a 5 basketball players be choosen from a group of 9 players?A. 126 C. 15,120B. 212 D. 362,880


29/29


ANSWER KEY:

MATHEMATICS

1 B 54 A 106 B2 A 55 A 107 C3 C 56 D 108 C4 C 57 B 109 D5 D 58 A 110 A6 B 59 C 111 A7 A 60 C 112 C8 D 61 A 113 D9 C 62 C 114 C10 B 63 B 115 C11 D 64 D 116 C12 A 65 C 117 A13 C 66 A 118 B14 D 67 B 119 A15 B 68 D 120 C16 C 69 A 121 D17 C 70 D 123 B

18 C 71 D 124 D19 B 72 C 125 A20 A 73 D 126 C21 B 74 D 127 A22 A 75 D 128 D23 A 76 C 129 B24 B 77 C 130 C25 D 78 D 131 C26 B 79 B 132 D27 A 80 B 133 B28 A 81 A 134 C29 A 82 D 135 C30 B 83 C 136 B

31 C 84 A 137 C32 B 85 D 138 D33 D 86 B 139 A34 C 87 C 140 B35 C 88 A 141 A36 B 89 A 142 C37 C 90 A 143 C38 D 91 B 144 C39 B 92 C 145 C40 C 93 D 146 B41 B 94 B 147 C42 A 95 A 148 A43 D 96 D44 A 97 B45 A 98 D46 A 99 B47 D 100 C48 C 101 C49 A 102 D50 A 103 A51 B 104 D52 C 105 D53 C 105 B

3-let - general education - mathematics 53-81

Documents