PS 1

Bergen Academies Practice problems 1

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Bergen Academies Math Practice Problems 1. Solutions are provided below.
1. How many positive factors including 1 and itself does 72 have?
  A. 4    B. 6    C. 8    D. 10    E. 12
 Correct Answer:  E. 12   Explanation: {1,2,3,4,6,8,9,12,18,24,36,72}  That totals 12 factors. You can approach this by trial and error counting the factors as you move from 1 to 72, but the risk is you miss some. One easier, foolproof method is to factor 72 as follows. 72= 2x2x2x3x3. All the combinations of these like 2 and 2x3 and 2x2x3 and 3x3 etc. will make up all the factors, but again you do not need to find all the factors to know how many there are.  \[72 = 2 \times 2 \times 2 \times 3 \times 3 = {2^3} \times {3^2}\] The easiest approach is to add 1 to the power of each base. 3 becomes 4 and 2 becomes 3. Multiply those numbers, 4 x 3=12. That is the number of positive factors. Another example to make it clear is 96 which also has 12 factors.  \[96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times  = {2^5} \times {3^1}\] (5+1) x (1+1)=6 x 2= 12. You do not need to know that the actual factors are {1,2,3,4,6,8,12,16,24,32,48,96}
2. If the sides of a square are increased by 20%, by what percentage does the area of the square increase??
  A. 20%    B. 30%    C. 36%    D. 40%    E. 44%
 Correct Answer:  E. 44%   Explanation: Do not be fooled into answering 20%. The side may increase 20%, but the area increases by the length squared. If the new length is 120% then 120% x 120%= 144% units squared is the new area of the square. The original area was 100% x 100%=100% units squared. The increase is 144%-100%=44%.
3. Jeff answered all 25 questions on his chemistry test. For each right answer he got 4 points and for each wrong answer he lost 2 points. If he got a score of 70 points how many questions did he get right?
  A. 18    B. 19    C. 20    D. 21    E. 22
 Correct Answer:  C. 20   Explanation: There are a couple ways to approach this problem and trial and error between available answers is as quick as any method. You might note that given the 70 score, the number of right answers must be less than 25 and at least 18 because 18 right answers x 4pts=72pts creates the first potential score above 70. In fact, 18 can easily be eliminated after noting 7 wrong answer deductions will bring the figure below 70. Somewhere in the middle is a good place to start. 20 right answers x 4pts- (25-20) wrong answers x 2pts= 80-10=70. That works! A more rigorous approach is to apply algebra to this word problem and that is quick too.                                 R=Right Answers, W=Wrong Answers            R+W=25 based on total questions       4R-2W=70 based on scoring methodology            Solving...W=25-R then plugging W into 2nd equation...                            4R-2(25-R)=70                           4R+2R-50=70                           6R=120                             R=120/6=20     *Don't get too excited yet. There will be harder                                                       problems of this ilk coming.   Unlike this problem there are no deductions for wrong answers on the real BCA math test so you can always benefit by eliminating some answers and taking educated guesses if you are not certain of the one correct answer. There is no benefit to leaving an answer blank.
4. A military clock has the numbers 1 through 24 evenly spaced around a circular face. What number is directly across from 19?
  A. 3 or 4    B. 5 or 6    C. 7 or 8    D. 9 or 10    E. 11 or 12
 Correct Answer:  B. 5 or 6   Explanation: The military clock doesn't go from 1 to 12 like a normal clock. It goes from 1 to 24 as shown where the 6,12,18,24  are positioned where the 3,6,9,12 would be on a normal clock. 19 would be in the upper left quadrant as shown. "Directly Across" implies the horizontal chord as shown to the number 5. Some may claim 7 is on the opposite side of the face of the clock and answer C, but "directly across" implies 5. I know. The wording is a little ambiguous in my opinion too, but it is what it is.

5. If a race began at 3:43 PM and ended at 5:27 PM the race was half over at what time? 
  A. 4:05    B. 4:15    C. 4:25    D. 4:35    E. 4:45
 Correct Answer:  B. 5 or 6   Explanation: There are 17 minutes between 3:43 PM and 4:00 PM, 60 minutes from 4:00 to 5:00 PM. and 27 minutes from 5:00 to 5:27 PM. The total minutes for the race is 17+60+27=104 minutes. The race was half over after 104/2=52 minutes. What time is that? Counting forward from 3:43 PM it will be 52-17=35 minutes after 4:00. The answer is 4:35 PM.
6.  \[\sqrt {\frac{1}{4} = } \]

  A. 1/2    B. 1/4    C. 1/6    D. 1/8    E. 1/16
 Correct Answer:  A. 1/2   Explanation: Do not be confused by another square like 1/16. If it was the square root then 1/16 x 1/16  would equal 1/4, but it does not. The simple solution is...                                    \[\sqrt {\frac{1}{4} = } \frac{1}{{\sqrt 4 }} = \frac{1}{2}\]
7. Which of the following fractions is equal to 1 1/4%?     A. 5/4    B. 5/40    C. 1/80    D. 4/500    E. 4/50
 Correct Answer:  C. 1/80   Explanation: 1 1/4% is the same as .0125 or 1.25/100. The answers are expressed in fractions where the numerator and denominator both are whole numbers. To get that multiply by (4/4)=1 to get the numerator to a whole number without changing the fraction. (1.25/100) x (4/4)=5/400. Be careful! Many will jump to write down D, 4/500, quickly without realizing it is not the correct fraction 5/400. It is designed to trap people rushing through the exam. 5/400 is not shown among the answers. We must reduce the fraction 5/400 further. (5/5) /(400/5)=1/80.
8. Which of the following is closest to 0.25?
  A. 9/40    B. 5/16    C. 9/32    D. 0.28    E. 15/64
 Correct Answer:  E. 5/64   Explanation: You can start doing long division and converting fractions to decimals, but I caution against all that brute force. Leave brute force to the future engineers! You mathematicians may see a more elegant solution. All of the fractions are of the form 1/X away from 1/4=0.25. X is the denominator. The closest fraction to 0.25 will be the one with the largest denominator X and that is 15/64. One exception is answer D which is expressed in decimal form. How close is that? 0.28-0.25=0.03 away from 0.25. We can express 0.03 as a fraction of form 1/X as well. It is 1/33.333. The denominator is still less than 64 so 15/64 is still the closest to 0.25.

9. The advertisement read "Buy 3 tires at the regular price and get a fourth tire for only $3." Carol paid $240 for a set of 4 tires. What is the regular price of a tire?
  A. 5/4    B. 5/40    C. 1/80    D. 4/500    E. 4/50
 Correct Answer:  D. $79   Explanation: 3 tires x P regular + $3 for the 4th tire= $240 total spent on all 4 tires. Solving 3P+3=240 results in P=237/3=$79. 

10. Which of the following is closest to 19.9 times 199?
  A. 4000    B. 3990    C. 3980    D. 3970    E. 3960
 Correct Answer:  E. 3960   Explanation: After crunching the numbers for the multiplication, the careful engineers among you arrived at 3,960.1. The mathematicians are already working one or two problems ahead because they made a quick and close estimate. One way to do that is to note 20x199=3980. The answer must be less since the number in the question is less than 20. It is 19.9. That reduces our options to answer D or E. Good start, but how much less? 20-19.9=0.1 less or expressed as a percent of 20 it is 0.1/20=0.5% less. If the solution is roughly 4,000 then 0.5% less is another 0.5%*4000=20 lower than our first estimate of 3980. 3980-20=3960. 

10. The average of 6 numbers is 4. A seventh number is added and the new average is 5. What is the seventh number? 
  A. 1    B. 5    C. 6    D. 11    E. 12
 Correct Answer:  D. 11   Explanation: There is no need to compute an average to solve this problem. Recall from Sample Test 1 Average=(Sum of Values)/(Number of Values) or Sum of Values =Average x Number of Values. The last or 7th number will just be the increase in the sum from the 6th to the 7th result. The sum of the first 6 numbers is 6 numbers x 4 average= 24. The sum after 7 numbers grows to 5 average x 7 numbers= 35. The difference 35-24=11 is the value of the 7th number.

12. Suppose the Egyptian Pharaoh Khufu was born 4300 years ago. Approximately when did his twelfth birthday occur?
  A. 2300 BC    B. 2700 BC    C. 3000 BC    D. 4300 BC    E. 6300 BC
 Correct Answer:  A. 2300 BC   Explanation: AD represents a positive number on the annual timeline and BC represents a negative number. The current year is approximately 2000 AD so 2000-4300= -2300 or approximately 2300 BC. The big differences in answers do not require us to adjust for the actual current year of 2013 or the 12 years until his twelfth birthday. Move on to the next problem and pocket those extra minutes for a real brain buster!

13. A clock loses 1 second every minute. It is set to the correct time at 10 AM on February 4. In which month is the next day on which it shows the correct time? (Note: A person can see whether it is AM or PM on this clock.)
  A. February    B. March    C. April    D. May    E. June
 Correct Answer:  C. April   Explanation: This problem will get the heart rate for many of you rising, but it turns to be a simple solution in the end. The clock will tell the right time again after 1 day worth of total errors. While the engineers in the room are trying to calculate the number of 1 second errors needed and arriving at a figure like 60sec/min x 60min/hr x 24hr/day, the elegant mathematicians in the room have solved the problem and once again moved onto the next problem. They realized right away that the ratio of the real time that passes to the time in each error is the only thing that matters! If 1 minute passes in real time for 1 second of lag on the clock then the real time will be 60 times the cumulative error no matter what that error is. If the cumulative error is one day on the clock as it must be in this problem then the actual time elapsed is simply 60 days. Likewise, 60 hours would elapse if the clock were 1 hour behind and so on.
60 days after February 4 takes us into April even in a leap year when it is still only 25+31=56 days until April begins. That wasn't too hard was it?
14. For approximately how many million seconds have you been alive?
  A. 4    B. 40    C. 400    D. 4000    E. 40000
 Correct Answer:  C. 400   Explanation: Let's get that calculator out. Oops! You can't do that, but we can estimate quickly. The actual calculation would be something like 
13 yr old x 365 day/yr x 24 hr/day x 60 min/hr x 60 sec/min= number of seconds alive
That is roughly 1.3 x 10 x 400 x 20 x 4000= 1.3 x 320 million~ 400 million seconds.
15. I want to cut out a piece of wood into five shorter pieces. The first cut takes 2 seconds and each subsequent cut takes twice as long as did the previous one. How many seconds will the job require?
  A. 8    B. 12    C. 114    D. 30    E. 62
 Correct Answer:  D. 30   Explanation: Each cut takes twice as long as the 1st which was 2 seconds. The next is 4 seconds. Then 8 seconds and so on. Many will calculate 5 iterations and choose answer D 62. That is wrong. The real question is how many cuts are required to get 5 pieces. It only takes 4 cuts to get 5 pieces. Make sure you agree. If I make one cut in half, for example, I end up with 2 pieces. 2 cuts make 3 pieces etc. N cuts make N+1 pieces. The answer then becomes...
2+4+8+16= 30 seconds elapse while making 4 cuts.