## Wednesday, October 20, 2010

### Answers to Statistics and Probability Part 1: Mechanical Engineer Board Exam Review

Here's the solution and answers to the ME board questions in Mathematics (Statistics and Probability)

1. In how many ways can you arrange a group of 5 girls and 3 boys in 7 vacant chairs?
Solution:
The total number of persons to be seated is 8 and the number of chairs is only 7.
When words denoting “arrangement” are used, it most likely pertains to permutations.
So, the formula to be used is permutation of 8 taking 7 at a time.
n = 8, r = 7, nPr, then use your calculator or this formula
nPr = n!/(n-r)! = 8!(8-1)!=40320 permutations or arrangements
A. 40320

2. How many 3-digit numbers can you make out of the numbers 1 to 5 without repetition?
Solution:
Since the number varies as you rearrange the digits (example 1342 is different from 1423), then this problem is about permutation.
So, the formula is the permutation of 5 numbers taken 3 at a time.
n = 5, r = 3, nPr; use your calculator or this formula: nPr = n!/(n-r)!
nPr = 5!/(5-3)!=60 permutations
C. 60

3. There are 2 white, 3 red, and 4 blue balls inside a basket. If three balls are drawn randomly in succession without replacement, what is the probability that the first ball is white, and the next two balls are blue?
Solution:
The probability that the first ball is white is Pw = 2/(2+3+4)=2/9
The probability that the second ball is blue is
Pb1=4/8 =1/2; we changed the denominator since the total number of balls has been reduced.
The probability that the last ball drawn is blue again is
Pb2 = 3/7; we reduced the total to 7 and the number of available blue is down to 3.
The probability that all this happen in correct succession is:
Pwbb = (2/9)(1/2)(3/7)=1/21
D. 1/21

4. What is the mode of the following numbers: 54, 45, 75, 60, 65, 65, 60, and 57?
Solution: the mode of a given set of numbers is the most frequent number; in this case those numbers are 65 and 65, both occurring twice.
D. 60 and 65

5. From the given numbers on question number 4, what is the median?
Solution:
The media is the number that divides the upper and the lower half of the samples.
To find the media, arrange the data in order either increasing or decreasing
In this case: 45 54 57 60 60 65 65 75;
Since the number of samples is 8 (even), we have to get the average of the two middle samples
Median = (60+60)/2 = 60
B. 60

6. From the given numbers of question number 4, what is the variance?
Solution: variance is the measure of how much does the samples deviate from the mean or average. Variance (σ^2) is given by the formula σ^2= ∑[(x-ave)^2/(n-1)] or use your calculator
Average = 60.125
σ^2= ∑[(x-ave)^2/(n-1)] = 77.84
A. 77.84

7. Seven boys are to be seated around a circular table. How many arrangements can be made?
Since no indication as to the number of seats available, let’s assume that it is equal to the number of boys, 7
Again, this is a permutation problem, but this time, in circular arrangement
The formula for circular permutation is given by nPr (cyclic) = (n-1)!
So, nPr (cyc) = (7-1)! = 720
D. 720

8. In how many ways can you arrange 3 boys and 4 girls in a 7-seater bench supposing that the four girls want to be seated together?
Solution:
Since one group wants to be together, then we will use conditional permutation techniques.
In this case, because the girls can’t be separated, let’s treat the girls group as one person or block.
The permutation of the boys and the group of girls therefore has the total number of n= 4, that is 3 boys plus 1 girls block.
The number of arrangement possible is given by nPr = 4P4=24
Remember though, that the girls can be rearranged as long as they are together, therefore, the total permutation inside the girls block is nPr = 4P4=24.
Combining the two permutations we have nPr (total) = (24) (24) = 576
C. 576

9. The probability that you will arrive late is 35% and the probability that you will scolded by your boss is 15%. What is the probability that you will be both late and scolded by your boss?
Solution:
The problem doesn’t mention that the cause of being scolded is caused by you being late. With that we can treat those two events as independent from each other.
The probability that the two independent events will happen can be obtained by getting the product of each probabilities.
Pa&b = (Pa)(Pb) = (0.35)(0.15)=0.0525 = 5.25%
A. 5.25%

10. From question number 9, what is the probability that you will either be late or scolded by your boss?
Solution:
This is an OR condition (the condition is valid if only when one of the two conditions happened) and is not mutually exclusive (meaning the two outcomes may happen at the same time: being late and being scolded)
The formula to be used is: Paorb = Pa+Pb – Pa&b
Paorb = 0.35+0.15-(0.35)(0.15)=0.4475
Paorb = 44.75%
C. 44.75%

Note: Writing solution to a math problem in a webpage like these is extremely difficult so I expect that some of you may be confused by the way I wrote the solution. If you have questions and clarifications, don't hesitate to send me a message at admin@tupmechanical.com or post a comment below.

## Monday, October 18, 2010

### Statistics and Probability Part 1: Mechanical Engineer Board Exam Review

Here are some review questions in Mathematics, Statistics and Probability.

Mechanical Engineering Board Exam Review: Statistics and Probability part 1

1. In how many ways can you arrange a group of 5 girls and 3 boys in 7 vacant chairs?
a. 40320
b. 5040
c. 720
d. 8

2. How many 3-digit numbers can you make out of the numbers 1 to 5 without repetition?
a. 720
b. 10
c. 60
d. 120

3. There are 2 white, 3 red, and 4 blue balls inside a basket. If three balls are drawn randomly in succession without replacement, what is the probability that the first ball is white, and the next two balls are blue?
a. 32/729
b. 4/63
c. 8/243
d. 1/21

4. What is the mode of the following numbers: 54, 45, 75, 60, 65, 65, 60, and 57?
a. 65
b. 60
c. 62.5
d. 60 and 65

5. From the given numbers of question number 4, what is the median?
a. 65
b. 60
c. 62.5
d. 60 and 65

6. From the given numbers of question number 4, what is the variance?
a. 77.84
b. 60.125
c. 68.11
d. 8.82

7. Seven boys are to be seated around a circular table. How many arrangements can be made?
a. 7
b. 2520
c. 5040
d. 720

8. In how many ways can you arrange 3 boys and 4 girls in a 7-seater bench supposing that the four girls want to be seated together?
a. 24
b. 5040
c. 576
d. 48

9. The probability that you will arrive late is 35% and the probability that you will scolded by your boss is 15%. What is the probability that you will be both late and scolded by your boss?
a. 5.25%
b. 50%
c. 44.75%
d. 2.33%

10. From question number 9, what is the probability that you will either be late or scolded by your boss?
a. 5.25%
b. 50%
c. 44.75%
d. 2.33%

NOTE:
If you noticed a question with no correct answer in the option or the question itself seems to be erroneous, I would appreciate if you can send me an email or comment below. Computation-type review questions are very hard to do and requires extensive validation. I would appreciate any help for the improvement of this blog. Thanks!

Answers will be posted after a week.

## Tuesday, October 12, 2010

### Professional Mechanical Engineer (PME) October 2010 Results

The Professional Regulation Commission (PRC) announces that 22 passed the Professional Mechanical Engineer Examination given by the Board of Mechanical Engineering in Manila this October 2010.

The following are the successful examinees.

2 AGTARAP, FERDINAND OLIVAR
3 ALCORIZA, RODOLFO SANTOS
4 AMTO, JOEL CELMAR
6 BOTE, RICARDO JR BAUTISTA
7 CABRERA, WILLIAM MATURAN
8 CANSINO, CECILIO JR LOPEZ
9 CHAVIT, CECILIO BOMEDIANO
10 DE LA ROSA, FERDINAND MANGAT
11 DIASANA, SERGIO SR BABA
12 DOROTAN, ELMER VILLANCA
13 GULMATICO, LLOYD CASTROBERDE
14 LORETO, JORGE JR DELOS REYES
15 MALOLES, ROBERT DICK TELYAEN
16 PEREZ, RIOLANDO VILLANUEVA
18 SO, ROLANDO CHING
19 TAPU, BENITO TABUT
20 TIGARONITA, BEN JR CORDERO
21 VILLAREAL, REYNAN MORATA
22 YU, LAWRENCE RUSSIANA
NOTHING FOLLOWS----------------------

## Saturday, October 9, 2010

### ME Board Reviewer: Math, Algebra, Computation (ANSWER and SOLUTION)

1. Which of the following is the solution set of the equation x2-2x-15 = 0
Solution:
Think of factors of the last digit that when subtracted gives the coefficient of the middle term
In this case, -15 can only be (3, -5), (-3, 5), (15, -1) or (-15, 1). Among these factors, (3, -5) is the only one whose sum will be equal to -3. So the solution set of the equation will be
X-5=0; x1 = 5
X+3=0; x2 = -3
Alternative Solution:
By trial and error (best when using your calculator), try each choices and see which among them when substituted to x will make the equation true, that is, x2-2x-15 equals zero.

B. (5, -3)

2. The next term in the progression 2, 8, 32, 128 is …
Solution:
The progression above is a Geometric progression since there is a common ratio (r). In this case, the common ratio is 4 and the next terms are obtained by multiplying the previous term by 4. So, next to 128 will be 128X 4 = 512

C. 512

3. Solve for x in the equation 3 lnx = 8
Solution:
Using properties of logarithm: nlna = lna^n
In this case, lnx^3=8
By making both sides an exponent of e (Euler’s number), we can cancel the ln.
e^(lnx^3)=e^8
x^ 3 = e^8
x= e^ (8/3)

A. e^ (8/3)

4. If x + y = 1 and x2+y2=2. Find x4+y4
Solution:
Squaring both sides of x+y =1 we get (x+y) ^2=1^2; x2+2xy+y2=1;
But x2+y2=2 so 2+ 2xy=1 or xy=-1/2;
Square both sides, 4x2y2=1; x2y2=1/4
Let’s square both sides of x2+y2=2; (X2+y2) ^2=2^2 then x4+2x2y2+y4 = 4;
But x2y2=1/4 so, x4+y4+2(1/4) =4
Solving the equation will give us X4 +y4 =7/2

D. 7/2

5. If lnx=5 and lny=2, lnx3/lnx4
Solution:
Using properties of logarithm; lnx3=3lnx and lny4=4lny
So lnx3/lny4 will be 3lnx/4lny = 3(5)/4(2) =15/8

B. 15/8

6. Joan can type the whole document in 3 hrs. On the other hand, James, being new to the job can type the same document in 5 hours. If they work together, how long can they finish 2 documents?
Solution:
rate of Joan 1/3; rate of James 1/5
Rate=no job/time time = no of job/rate
If they work together, their rate will be added
t= 2 jobs/ (1job/3hrs+1job/5hrs)
t=3.75 hours

D. 3.75 hrs

7. Three numbers are in the ratio 1: 2: 4. Find the numbers if their sum is 42.
Solution:
add the ratios 1+2+4 = 7
Divide the sum of the numbers by the sum of the ratio; 42/7=6
Multiply 6 by each ratio. The numbers will be 6, 12, and 24

A. 6, 12, 24

8. Find the sum of the coefficients in the expansion of (5x-4)^5
Solution:
To find the sum of the coefficients of an expansion, just replace the variable by 1 in the expression.
In this case, (5x-4) ^5; [(5(1)-3] ^5=32

D. 32

9. Evaluate the expression log 3+ log 2+ 4log7
Solution:
Note: When there’s no base mentioned, it is automatically considered to be 10.

D. 4.16

10. The sum of the infinite geometric progression is 3 and the common ratio is 1/3. What is the first term?
Solution:
For an infinite Geometric progression, the sum is given by the formula S=a1/r where S is the sum and r is the ratio.
In this case, the Sum and the ratio were given so using the formula we can get the first term.
a1=Sr; a1=9(1/3) =3

A. 3

Shortcut to questions.
Note: Writing solution to a math problem in a webpage like these is extremely difficult so I expect that some of you may be confused by the way I wrote the solution. If you have questions and clarifications, don't hesitate to send me a message at admin@tupmechanical.com or post a comment below.

## Sunday, October 3, 2010

### Mechanical Engineering Board Exam Reviewer: Algebra, Mathematics, Computation Part 1

Here’s a set of question on the topic Mathematics, specifically on Algebra. These questions are similar to those coming out of the board exam.

1. Which of the following is the solution set of the equation x^2-2x-15 = 0
a. (3, -5)
b. (5, -3)
c. (15, -1)
d. (-15, 1)

2. The next term in the progression 2, 8, 32, 128 is …
a. 128
b. 256
c. 512
d. 1024

3. Solve for x in the equation 3 lnx = 8
a. e^ (8/3)
b. e^ (3/8)
c. e^ (4/3)
d. e^ (3/4)

4. If x + y = 1 and x^2+y^2=2. Find x^4+y^4
a. ½
b. 3/4
c. 15/4
d. 7/2

5. If lnx=5 and lny=2, lnx3/lnx4
a. 10/3
b. 15/8
c. 125/16
d. 5/2

6. Joan can type the whole document in 3 hrs. On the other hand, James, being new to the job can type the same document in 5 hours. If they work together, how long can they finish 2 documents?
a. 3 hrs
b. 3.5 hrs
c. 2.75 hrs
d. 3.75 hrs

7. Three numbers are in the ratio 1: 2: 4. Find the numbers if their sum is 42.
a. 6, 12, 24
b. 4, 12, 36
c. 4, 16, 22
d. 3, 15, 24

8. Find the sum of the coefficients in the expansion of (5x-4)^5
a. 32768
b. 25
c. 243
d. 32

9. Evaluate the expression log 3+ log 2+ 4log7
a. 9.58
b. 4.79
c. 3.43
d. 4.16

10. The sum of an infinite geometric progression is 3 and the common ratio is 1/3. What is the first term?
a. 3
b. 2
c. 1
d. 6