1. If n is a multiple of 5 and n = p^2q, where p and q are prime numbers, which of the following must be a multiple of 25A. p^2
B. q^2
C. pq
D. p^3q
E. p^2q^2
n=5k and n=p^2p, (p and q are primes).
Q 25m=?
Well obviously either p or q is 5. As we are asked to determine which choice MUST be multiple of 25, right choice must have BOTH p and q in power of 2 or higher to guarantee the divisibility by 25. Only E offers this.
Answer: E.
2. Alice’s take-home pay last year was the same each month, and she saved the same fraction of their take-home pay each month. The total amount of money that she has saved at the end of the year was three times the amount of that portion of her montly take-home pay that she did not save. If all the money that she saved last year was from her take-home pay, what fraction of her take-home pay did she save each month?A. 1/2
B. 1/3
C. 1/4
D. 1/5
E. 1/6
Let Alice's monthly take-home pay be \(p\) and her monthly savings be \(s\). Total savings will be \(12s\) and we know that this is 3 times the amount she spends in month which is: \(p-s\). So we have:
\(12s=3(p-s)\) --> \(s=p\frac{1}{5}\)
Answer: D.
3. The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 100%, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?A 100% decrease
B 50% decrease
C 40% decrease
D 40% increase
E 50% increase
NOTE: Put directly proportional in nominator and inversely proportional in denominator.
\(RATE=\frac{A^2}{B}\), (well as it's not the exact fraction it should be multiplied by some constant but we can ignore this in our case).
We are told that B increased by 100%, hence in denominator we have 2B. We want the rate to be the same. As rate is directly proportional to the SQUARE of A, A should also increase (nominator) by x percent and increase of A in square should be 2. Which means x^2=2, x=~1.41, which is approximately 40% increase. \(R=\frac{A^2}{B}=\frac{(1.4A)^2}{2B}\)
Answer: D.
4. On his drive to work, Leo listens to one of three radio stations A, B or C. He first turns to A. If A is playing a song he likes, he listens to it; if not, he turns it to B. If B is playing a song he likes, he listens to it; if not, he turns it to C. If C is playing a song he likes, he listens to it; if not, he turns off the radio. For each station, the probability is 0.30 that at any given moment the station is playing a song Leo likes. On his drive to work, what is the probability that Leo will hear a song he likes?A. 0.027
B. 0.090
C. 0.417
D. 0.657
E. 0.900
Favorable scenario is the sum of individual probabilities:
A is playing the song he likes - 0.3;
A is not, but B is - 0.7*0.3=0.21;
A is not, B is not, but C is - 0.7*0.7*0.3=0.147
P=0.3+0.21+0.147=0.657
Answer: D.
Or: 1-the probability that neither of the stations is playing the song he likes=1-0.7*0.7*0.7=0.657
5. An certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: blue, green, yellow, or pink. The store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size of 3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?A. 6
B. 8
C. 16
D. 24
E. 32
Notepads of the same color = 4 (we have 4 colors). As we have two sizes then total for the same color=4*2=8
Notepads of the different colors = 4C3=4 (we should choose 3 different colors out of 4). As we have two sizes then total for the different color=4*2=8
Total=8+8=16
Answer: C.
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