Solution Set for Mathematics Proficiency 2

8/12/2019 Solution Set for Mathematics Proficiency 2

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BASIC ARITHMETHIC & ALGEBRA

Solutions Manual

1. D The part of the journey the bus traveled in fog: 2. C The profit of the dishonest shopkeeper:

3. D Method A: Algebraically

Let V = number of vanilla cones sold

C = number of chocolate cones sold

Given that and that ; solve for V in the system of equations.

Method B: Plugging In

Using option (A). If V = 42, then because vanilla cones sold to a 2:3 ratio, this means

C = 63. To confirm, adding 4 to V, then 46 over 63 should be 3:4. But 46/63 is about

0.73, so it isn't the correct answer.

Next, you must decide if the correct answer is more or less than 42. If you have good

math instincts, you should know it should be less than 42, but if you aren't sure thenjust pick a direction and you'll soon get the right answer anyway. Note that the

answer cannot be (C) because C is an odd number, indivisible by 2. Option (D) is the

correct answer, because if V = 32, then C should be 48 to get a 2:3 ratio. Then adding

4 to 32, 36/48 is indeed 3/4.

4. B To simplify 2*(3*4), start with the expression inside the parenthesis first.

3*4 = (3)(4)-42= -4. So now you have, 2*(-4) = (2)(-4)-(-4)2 = -24

5. B Note:

and

6. C You may be tempted to convert all the fractions into similar fractions using their

Least Common Denominator (LCD) of 81. However, in this particular case, it is

quicker and simpler to reduce the fractions instead since they all reduce to. So try

simplifying fractions first before instantly looking for the LCD.

So if , then 7. C Get the Least Common Multiple (LCM) of 18, 12 and 15 by prime factorization.

18 = 2,3,3 | 12 = 2,2,3 | 15 = 3,5

LCM = 2x2x3x3x5 = 180 minutes. So if they bark together at midnight, then the nexttime they will bark together would be at 12:00AM + 3 hours = 3:00 AM.

8. C This is a ratio problem, so we need to add the "parts" of our ratio to get the "whole"

required to form a fraction. Since our parts are 3 and 8, our whole is 11. The

fraction of the students that does not participate in the band or choral program is

thus. Now we can set up an equation , solving for x, x = 160.


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9. C Let N be the total number of seniors in this class.

Number of female = 0.6N. Number of male = 0.4N

The number of seniors going on the senior trip: 0.7(0.6N) + 0.9(0.4N) = 0.78N = 78%

10. C If 4 flights left the airport between 9 am and 10 am, then 8 left between 10 am and

11 am, and 16 left between 11 am and noon. Starting at noon, the flights begin to

double every two hours, so 32 left between noon and 2 pm, and 64 left between 2

pm and 4 pm.

11. D Let nbe the weight of the entire mixture in pounds.

Working equation:

Multiplying the LCD, So , and finally n= 180 pounds.

12. C Number of vehicles that are trucks: 30%(300) = 0.3(300) = 90

Number of vehicles that are diesel, including 15 trucks: 20%(300) = 0.2(300) = 60

where from 60 = 45 diesel + 15 trucks.

Percent of the motor pool that is composed of vehicles that are neither trucks nordiesel:

13. C Let nbe the original number of fishermen.

Working equation: 600 = 25(n-2)

Solving,

24 = n-2so n= 26.

14. C Letxbe the original amount of money Matt had.His money left after the first entry and exit: 2(x-$500)-$200 = 2x-$1200

His money left after the second entry and exit: 2[(2x-$1200)-$500]-$200 = 0

Solving, 2(2x-$1700)-$200=04x-$3400-$200=04x=$3600x= $900.

15. B Letxbe the total number of questions in the test.

Working Equation: Multiplying the LCD 10, you get 90+3x-30=5x, so x = 30.

16. C Let nbe the total estate. Each of the four sons receivedof the estate.

One died, and of his share was divided equally among 3 sons.Each of the remaining sons will receive

The fractional part that each of the remaining son receives is

17. B Let M be the number of men and B be the number of boys.

Working Equation: simplifying, Therefore, 9 boys are required to finish the job alone.


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18. C Let P be the original price of the item.

Spring Price: 100%P - 20%P = 80%P

Summer Price: 80%P-20%(80%P)=64%P

19. C Rate of 20 lumbermen: 50 trees/5 days = 10 trees/day

so Rate of 1 lumberman: 1/2 tree/day

Therefore, the rate of 50 lumbermen is 25 trees/day or 100 trees/4 days

20. D The red marbles are the hardest to pick. Exhaust the white and blue marbles first.

At least 10 marbles must be removed from the bag to ensure 1 marble of each color.

21. B Let N be the total number of students.

Since 50% in the class are majoring in chemistry, 0.5N = 30N = 60.

The number of students majoring in industrial engineering is 0.2(60) = 12.

22. C Minimum number of typists in a department: 8 x 10 x 40 x 10% = 320.

23. B Let G be the # of green balls, B of blue balls, Y of yellow balls, and W of white balls.

3G = 6B | 2Y = 5B | 6B = 4WG = 2B | 2Y = 5B | 3B = 2WWorking Equation: 4G + 2Y + 2W = 4(2B) + 5B + 3B = 16 blue balls.

24. C Let tbe the time from starting, in hours, that they will be 5 miles apart.

Using the Pythagorean Theorem for Right Triangles, 25. D Average Speed =

26. D Rate ng eroplano A, sa km/oras: n-15

Rate ng eroplano B, sa km/oras: n

Tamang Tumbasan: 3(n) + 3(n-15) = 945 6n-45 = 945n = 165

27. D Representasyon: t= bilang ng oras na sabay na magtatagpo and dalawang pasahero

sa kanilang patutunguhan

Tamang Tumbasan:

Distansya ng kanilang pinuntahan mula Maynila patungong Bicol, 28. C Tulin ng motorsiklo (km/oras): r

Tulin ng paglalakad (km/oras): r-10

Tamang Tumbasan:

4t

3t

5


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29. C Rate ng kotse (km/oras) = r

Tamang Tumbasan:

30. B Distansya na kanyang nilakbay (km) : d

Tamang Tumbasan:

31. B Bilang ng mga lalaki: bBilang ng mga babae: g

[Eqn 1] | [Eqn 2]

Mula sa [1], . Ihalili ang halaga ng b na ito sa [2], makukuha ang halaga ng .Ang halaga ng .

32. C Ang suma ng mga numero sa set ng mga odd na integer ay sero kapag ang

additive inverse ng bawat numero sa set ay elemento din ng set. Kaya, and

pahayag III ay palaging totoo.

Gayundin, ang average ng mga integer sa set ay sero kapag ang suma ng odd

na integer sa set ay sero. Kaya, ang II ay palaging totoo rin.

Ang pahayag I ay hindi palaging totoo. Isipin na lamang ang {1,-1}.

33. A 34. C Representasyon: h, magiging haba ng picture frame kung ang lapad ay cm

Lapad ng picture frame: 16 cm

Haba ng picture frame: 24 cm

Tamang tumbasan: 16:24 = : h 35. D Ang numerong rasyonal ay pwedeng terminating decimal o kaya nama'y

nonterminating repeating decimal. Ang numerong nasa (D) ay hindi ganito kaya

hindi ito rasyonal.

36. B Puhunan sa 6% interest: n

Puhunan sa 5% interest: 12000 - n

Tamang Tumbasan:

37. C


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38. D Savings account ni Toni: (may simple interest rate na 10% bawat taon)

Interest pagkatapos ng unang taon: Php 10,000(0.1) = Php 1,000

Interest pagkatapos ng ika-2 taon: Php 10,000(0.1) = Php 1,000

Kabuuang halaga pagkalipas ng dalawang taon: Php 10,000 + Php 1,000 + Php 1,000

ay Php 12,000.

Savings account ni Janelle: (may compounded interest rate na 10% bawat taon)

Interest pagkatapos ng unang taon: Php 10,000(0.1) = Php 1,000

Interest pagkatapos ng ika-2 na taon: (Php 10,000 + Php 1,000)(0.1) = Php 1,100

Kabuuang halaga pagkalipas ng dalawang taon: Php 10,000 + Php 1,000 + Php 1,100

ay Php 12,100

Konklusyon: Mas mataas ng Php100 ang savings account ni Janelle kaysa kay Toni.

39. B 9 oras : 12 katao : 180 metro

9 oras : 20 katao : 300 metro

12 oras : 15 katao : 300 metro

40. C 1 lata : 3 upuan1 lata : 2 mesa

Kaya't ang 8 lata : 24 upuan

Kung may 15 upuan na napinturahan na, 24-15 = 9 upuan pa pwedeng pinturahan

Sa unang pagtatala sa 1 lata, 3 upuan = 2 mesa. Mula dito, masasabing 9 upuan = 6

mesa.


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ALGEBRA

Solutions Manual

86. C If , then 87. D

88. A 89. D Statement I is not always true. For example, if a = -2 and b = 3, then b > 2a, but if a

= 2 and b = -3, then b < 2a.

Statement II is false. Where ,

Statement III is also false, because 1 is a factor (NOT A MULTIPLE) of every integer.

90. D In exponential equations, when the bases of both sides of the equation are

identical, their exponents are equal. 91. D Start by translating the two initial equations: and . If ,

then and . Thus and . The difference between thetwo values is 5.

92. B 93. A Let n, n+1, n+2, n+3 and n+4be the first 5 consecutive integers in Set X.

Working Equation: , so , the first integer. Therefore the last 5 consecutiveintegers in Set X are 56, 57, 58, 59 and 60. Their sum is 290.

94. C Let R be the present age of Romeo, in years

Let F be the present age of his father, in yearsGiven and that , solving the system of equations: , years old.

95. B


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96. D and and . So , and substituting the value of in terms of, 97. A Let be the tens digit of the original number and be the ones digit of the original

number, so

.

Working Equation: By Elimination Method: . So , and the original number is 72.98. C

99. D Let M be the total number of movies rented during the two-week period.Let A be the number of action movies rented, C be the number of comedies rented

and D the number of dramas rented.

0.7M = C

0.3M = A + D, and D = 5A so 0.3M = A + 5A = 6A, so M = 20A

To get C in terms of A, since C = 0.7M = 0.7(20A) = 14A

100. D Let be the number of liters of acid that must be added.Working equation:

101. B Let be the number of milliliters of water that must be added.Working equation:

102. D Let be the number of pounds of water that must be evaporated.Working equation:

103. B Let be the number of grams of the 80% alloy used.Working equation: 104. B Let be the number of liters of pure water that must be added. Since 20% of the 5

liter container was removed, was left.Working equation:

, so


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105. B Let be the number of students and be the cost of each share, in pesos.

. The nonnegative value of x is considered for the number of students.

106. C Let be the denominator and be the numerator of a certain fraction. | Substitute the expression of in terms of to the second equation. Therefore, the fraction is 7/18.

107. D 108. D Strategic Advice: What is the x-coordinate of D? What do you know about the

functions at that value?

Getting to the Answer: Parabolas are symmetric, so each half of is 2. Thereforethe x value at point D is 2. But the functions also intersect here, so plug in toeach equation and set the results equal.

Since the graphs intersect at , their coordinates should also be equal. 109. B To solve for x and y,

Simplifying the system, where 110. B Let be the number of hours that Brooklyn and Victoria will take to finish the job.

Name Rate Time

Brooklyn 1/6

Victoria 1/4 Working equation: 111. A Let be the number of days that Edward can do the work by himself.

Name Rate Time

Bella 1/10 5

Edward 3Working equation:


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112. C Let be the number of days the slower man can finish the job alone.Name Rate Time

Slower 20Faster 20

Working equation:

To choose between 45 and 4, note that if x is 4, then the faster man can completethe job in (4-9)=-5 days, which is not possible because there is no negative amount

of days. So the answer is x=45.

113. C 114. C Representasyon:

, bilang ng oras na mapupuno ang pool kung parehong bukas and

pasukan at labasan ng tubig

Pangalan Rate Time

Pasukan ng Tubig 1/6 Labasan ng Tubig 1/9

Tamang tumbasan:

115. D

116. A So Therefore,

117. D

118. C


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119. DI-represent ang

Ang halaga ng

ay positibo, kaya't 2 ang sagot.

120. C 121. C

Pero hindi maaaring maging halaga ng ang 4 dahil magiging undefined andtumabasan. Ang set ng solusyon ng tumbasan ay .122. B 123. A

Dahil ang , at 124. D

125. C


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SEQUENCES, SERIES & STATISTICS

Solutions Manual

1. C 2. A

3. B 4. A

5. D

6. C Integers divisible by 7 are from 14 to 224, so

7. B

8. A


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9. B 10. C In increasing order, the numbers in the set are 25,30,30,40,45,60.

11. C 12. B 13. C

14. D Tens digit can be even or odd while ones digit should always be even.

Odd tens ones Even tens Ones

# ofpossibilities of

even numbers

3 3 2 2

No. of possibilities of even numbers = 15. C

16. B Product is 6 = (1,6), (6,1), (2,3), (3,2) 17. D Total number of tickets is equal to 101, so P(one of your tickets will be drawn)=

C NT Not C & Not NT = 9


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18. B Let and 19. C Assume combination problem.

20. C 21. D 22. A Let be the event of drawing an ace card and be the that of a face card 23. B Let be the event of drawing an spades and be the that of an ace 24. A

25. B If Matt obtained 99 percentile in a national examination, this means that 99% of

those who took the same national examination scored lower than he did.

26. C The data set will have a median equal to 7 if x is less than 6.27. A Example:

Average: 67 Average: 67

73 73

67 (average) 67

66 66(median)

65 65

64 64

28. C ,

40% 15% 30%


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29. A 30. B = number of ways of arranging distinct objects in a circle 31. A Note that the question is asking for how many different groups, thus it is a

combination problem. 32. C Sa tens digit and pwede lang ay 6,7,8 at 9

Sa ones digit, pwede na rin ang zero 33. C

34. A [kung lahat ng posibleng sagot ang hinahanap dahil may apat na

sagot na pagpipilian sa bawat tanong]

35. D 36. C : ang pangyayaring may nabunot na alas at

: ang pangyayaring may nabunot na dos

37. C dahil sa hundreds digit, ang pwede lamang ay 1,2,3,4,5 at 638. B

(1,1), (2,2), (3,3), (4,4), (5,5),(6,6)

39. A

(1,1), (2,1), (1,2), (2,2),(1,3),(3,1)

40. D


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GEOMETRY

Solutions Manual

1. A Let = radius of cone, & = height of cone , and

2. C Angles of each corner is equal to 360, so sum of all angles is . The sum

of interior angles of a triangle is 180, therefore, angles shown is equal to sum of all

angles minus the interior angles, .3. B and

4. B

Since and , So the area of the shaded region is .

5. C Let be the distance (in feet) the ladder will slide

, distance from the top of the ladder to the base of the wall. , we consider the nonnegative root for the distance. The ladder will slide8 feet.

6. A Interior angles of a triangle equals 180, so , So interior angles are Two angles are equal so it is an isosceles.

7. C Since , the triangle is an isosceles

a

b c

d

7 x

20

4

25


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8. C Since the shortest cord length is needed, find the least common multiple of the

number of sides of each shape. LCM of 3,4 and 6 is 12.

9. B The diagonals of a rectangle divide the area into 4 equal parts. Opposite angles are

equal so shaded region is 1/4 of the total area or 25%

10. C Let , so the largest angle is

11. D From the figure, and are corresponding angles. So they are equal.12. D Conditional: pq

Inverse:~p~qa. qp

b. p-q

c. ~q~p

d. ~p~q

13. B In the given diagram, 14. A The radius of a circle from center to point of tangency is perpendicular to the line oftangency.

15. B The point of intersection of the two lines is (2,-1).

The y-intercept of is -3.


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16. C

17. B Let and 18. C Let

be the length of each side of the cube

, so 156% increase19. B Two identical isosceles triangles joined at their bases forms a rhombus.

20. B

Width of original square =

21. A Working equation: Take the value of 2 because cannot exceed 7.

2

x

2

2 x 2

2 x 2

2

x

2


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22. D

23. C 24. B Identify the similar triangles.

25. D is a 3-4-5 right triangle. Therefore, BC is 4.

26. D , so for the inner circle.The large circle radius is then

27. C 28. B

29. A

So

r

2-r


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30. C 31. C 32. B

So , so 33. B Representasyon: =ang sukat ng angulong hinahanap =supplement ng =complement ng

Working equation: 34. C

35. C , 36. A 37. C


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38. A 39. B 40. C Representasyon: = sukat ng gilid (sa sentrimetro) ng bawat parisukat


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INTEGRATED MATHEMATICS

Solutions Manual

61. C Only two elements, .62. C Lines that are parallel have no solution, because they don't intersect.

63. C

The slope of the line that passes through the points (1,-1) and (-1,1) is 64. C (A). There is no known relationship between y and z

(B). not true if the value of x is negative(C). True for all real numbers

(D). x can be less than zero

65. D All values of x will result to zero except 2


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70. A

71. D The slope of is: . Since the line isperpendicular to this, . 72. C Equilateral triangles are also equiangular (60-60-60).

73. C

74. D Let . Squaring both sides of the equation, 75. B Distance from and .

76. C Sketch the figure.

s/2

s

s/2

h

(0,3)

(3,0)

(3,6)

(5,3)

3

23

3


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77. C So .

78. B Sketch the figure. 79. C 80. A Graph is shaded below y=3, and line y=3 is a broken line, indicating that y=3 is not

included.

81. A

82. C 83. B (A)

(B) (C) (D)

84. C 85. B 86. D

. Square both sides,

. Square both sides, but upon checking, 1 is not a solution. only.


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87. D

88.

89. B 90. A Coordinates of the circle's center would be the midpoint of the diameter

91. D

92. B Representasyon:


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93. B Ang suma ng haba ng alin mang dalawang gilid ng isang tatsulok ay higit sa haba ng

pangatlong gilid. (The sum of the lengths of any two sides of a triangle is greater

than the length of the third side.) (Also, the difference of the lengths of any two

sides of a triangle is less than the length of the third side.) 94. A

o

B

Gamiting ang Pythagorean Theorem,

kung saan ang

ang

pinakamahaba o hypotenuse.

*A and B can both be answers 95. B Representasyon:

96. B are the critical points

Interval Test Value -5 - - +

0 + - -

8 + + +97. B 98. D

The domain is the set of all numbers in which the function is defined.

para sa numerator

para sa denominatorTherefore, [lahat ng numerong real maliban sa mga numerong


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INTEGRATED MATHEMATICS SET B

Solutions Manual

61. D

62. C , 63. C 64. B , so 65. B

. They intersect at (2,2).66. C The sum of the interior angles of any triangle is always 180.

67. B

68. B 69. C Let = Noel's income , so

He wants to save 70. B Let = smaller angle, and = larger angle

, so


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71. C

name now In 2 years

Boy Father

72. D 73. C The supplement is 74. C 75. C

76. C 77. C Graph the points.

78. B

79. D The opposite angles of a parallelogram are equal.


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80. C

Let be the length of the side of the square

81. C Let be the original radius of the circle

If we increase it by 100%, the new radius will be Area is increased by 300%.

82. C

Equating from both equations,

83. D Let be the speed at which Steve ran. 84. D Let

be the unknown number.

, so


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91. A 92. C Using 3 as an odd integer,

(A) , even(B)

, even

(C) , odd(D) , even93. D

94. C 95. A I. AB

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Solution Set for Mathematics Proficiency 2

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