Quantitative Reasoning Exam #1
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This is the Quantitative Reasoning Exam #1. Please read the following tips before beginning:
You will have 45 minutes to answer 40 questions.
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- Question 1 of 40
1. Question
The raw materials of a glass factory are made up of \( 10^3 \)g quartz and \( 10^{12} \)g borosilicate. These materials are mixed in a mixer and transferred to the furnace. Estimate the amount of material received by the furnace?
CorrectWhen two materials are mixed, they are added into the furnace, so,
\[ 10^3+10^{12}=1000,000,001,000\approx10^{12} \]
As we do not have the exact value, then the nearest option is the best solution.
A is the answer.
IncorrectWhen two materials are mixed, they are added into the furnace, so,
\[ 10^3+10^{12}=1000,000,001,000\approx10^{12} \]
As we do not have the exact value, then the nearest option is the best solution.
A is the answer.
- Question 2 of 40
2. Question
Which of the following is equal to \( 9a^6+9a^5-16a^4-16a^3 \)?
Correct\[ 9a^6+9a^5-16a^4-16a^3 = a^3(9a^3+9a^2-16a-16) \quad (I) \]
\[ (9a^3+9a^2-16a-16)=(9a^3-16a)+(9a^2-16) \]
\[ \begin{cases} 9a^3-16a=a(9a^2-16)=a(3a+4)(3a-4)\\ 9a^2-16=(3a+4)(3a-4) \end{cases} \quad (II) \]
\[ (I)\quad and \quad(II) \quad \implies \quad a^3(3a+4)(3a-4)(a+1) \]
A is the answer.
Incorrect\[ 9a^6+9a^5-16a^4-16a^3 = a^3(9a^3+9a^2-16a-16) \quad (I) \]
\[ (9a^3+9a^2-16a-16)=(9a^3-16a)+(9a^2-16) \]
\[ \begin{cases} 9a^3-16a=a(9a^2-16)=a(3a+4)(3a-4)\\ 9a^2-16=(3a+4)(3a-4) \end{cases} \quad (II) \]
\[ (I)\quad and \quad(II) \quad \implies \quad a^3(3a+4)(3a-4)(a+1) \]
A is the answer.
- Question 3 of 40
3. Question
The fuel consumption of two different cars, A and B, have a 3:5 ratio, while the total consumption of both cars is over 16 liters per 100 km. The producer of car B wishes to reduce its consumption to match car A. What is the needed improvement per 100 km?
Correct3:5 in 16 = 6 liters [A]: 10 liters [B]
1:1 A to B consumption = 6 liters [A]:6 liters [B]
10 liters [B] – 6 liters [A] = 4 liters must be reduced.C is the answer.
Incorrect3:5 in 16 = 6 liters [A]: 10 liters [B]
1:1 A to B consumption = 6 liters [A]:6 liters [B]
10 liters [B] – 6 liters [A] = 4 liters must be reduced.C is the answer.
- Question 4 of 40
4. Question
The temperature of the glacier is given as \( t^2+14t-50 \). If \( t \) is always negative and an integer, what is the maximum temperature of the glacier for a given value of \( t \)?
CorrectIf \( t \) is always negative and an integer, then the maximum value it can be is -1. Hence, the maximum temperature will be found by substitution the highest value of \( t \),
\[ t=-1 \]
\[ t^2+14t-50=1-14-50=-63 \]
B is the answer.
IncorrectIf \( t \) is always negative and an integer, then the maximum value it can be is -1. Hence, the maximum temperature will be found by substitution the highest value of \( t \),
\[ t=-1 \]
\[ t^2+14t-50=1-14-50=-63 \]
B is the answer.
- Question 5 of 40
5. Question
Which of the following functions is graphed below?
CorrectWe know that \( |f(x)|>0 \) is always true except for the zero. As depicted in the figure, the function crosses over the horizontal line at point 4.5, then results in \( f(x)=0 \). Thus, A is incorrect, and the same analysis can be done for B and C. Generally speaking, \( |f(x)|\ge0 \) is always correct so the answer is D.
D is the answer.
IncorrectWe know that \( |f(x)|>0 \) is always true except for the zero. As depicted in the figure, the function crosses over the horizontal line at point 4.5, then results in \( f(x)=0 \). Thus, A is incorrect, and the same analysis can be done for B and C. Generally speaking, \( |f(x)|\ge0 \) is always correct so the answer is D.
D is the answer.
- Question 6 of 40
6. Question
Which of the following is a solution to the equation \( \frac{7x-6y}{x+y}<-1 \)?
Correct\[ \frac{7x-6y}{x+y}<-1 \quad \implies \quad 7x-6y<-x-y \quad \implies \quad 8x<5y \]
C is the only case that meets the above-mentioned criterion.
For A: \( 8(1)<5(-1) \) – not correct
For B: \( 8(5)<5(7) \) – not correct
For C: \( 8(9)<5(16) \) – correct
For D: \( 8(-2)<5(-5) \) – not correct
For E: \( 8(-5)<5(-9) \) – not correct
C is the answer.
Incorrect\[ \frac{7x-6y}{x+y}<-1 \quad \implies \quad 7x-6y<-x-y \quad \implies \quad 8x<5y \]
C is the only case that meets the above-mentioned criterion.
For A: \( 8(1)<5(-1) \) – not correct
For B: \( 8(5)<5(7) \) – not correct
For C: \( 8(9)<5(16) \) – correct
For D: \( 8(-2)<5(-5) \) – not correct
For E: \( 8(-5)<5(-9) \) – not correct
C is the answer.
- Question 7 of 40
7. Question
Which of the following is the continuous exponential decay function for \( x\in(0,\infty) \)?
CorrectWe know that the continuous exponential function is a function that uses ‘e’ [Neper’s number] as the base. Choice C does not represent an exponential function because the base is an independent variable. In addition, for the decay function, we need to find the negative power which means A and D are incorrect. The figure of the function E is as follows,
Therefore, B is the continuous exponential decay function:
B is the answer.
IncorrectWe know that the continuous exponential function is a function that uses ‘e’ [Neper’s number] as the base. Choice C does not represent an exponential function because the base is an independent variable. In addition, for the decay function, we need to find the negative power which means A and D are incorrect. The figure of the function E is as follows,
Therefore, B is the continuous exponential decay function:
B is the answer.
- Question 8 of 40
8. Question
Which of the following is the solution of x given two equations and two unknowns \( \begin{cases} 2x-5y=6 \\ x+3y=1 \end{cases} \)?
Correct\[ \begin{cases} 2x-5y=6\\ x+3y=1 \end{cases} \quad \implies \begin{cases} \times3\\ \times5 \end{cases} \implies \begin{cases} 6x-15y=18 \quad (I)\\ 5x+15y=5 \quad (II) \end{cases} \]
\[ (I)+(II) \implies 11x=23 \implies x=\frac{23}{11} \]
B is the answer.
Incorrect\[ \begin{cases} 2x-5y=6\\ x+3y=1 \end{cases} \quad \implies \begin{cases} \times3\\ \times5 \end{cases} \implies \begin{cases} 6x-15y=18 \quad (I)\\ 5x+15y=5 \quad (II) \end{cases} \]
\[ (I)+(II) \implies 11x=23 \implies x=\frac{23}{11} \]
B is the answer.
- Question 9 of 40
9. Question
Half of a circular surface is filled by soil, while the other half is equally divided into two sections for water and oil. What is the ratio of soil to water?
CorrectThe area of a circle is \( \pi r^2 \), where \( r \) is the radius of the circle. We also know,
So the answer is \( \frac{\text{Area filled by soil}}{\text{Area filled by water}}=\frac{\frac{\pi r^2}{2}}{\frac{\pi r^2}{4}}=2 \).
E is the answer.
IncorrectThe area of a circle is \( \pi r^2 \), where \( r \) is the radius of the circle. We also know,
So the answer is \( \frac{\text{Area filled by soil}}{\text{Area filled by water}}=\frac{\frac{\pi r^2}{2}}{\frac{\pi r^2}{4}}=2 \).
E is the answer.
- Question 10 of 40
10. Question
The probability that each of the three independent fire alarms are triggered by the occurrence of the fire is 0.94. What is the probability that at least one of the three alarms goes off when there is a fire?
CorrectSince the probability that each of the three independent fire alarms are triggered by the occurrence of the fire is \( Pr(A_i)=0.94 \), the probability that none of the three independent fire alarms are triggered by the occurrence of the fire is \( 1-Pr(A_i)=1-0.94=0.06 \). Therefore,
\[ Pr(A_1\cup A_2 \cup A_3)=1-Pr(A_1^{\prime} \cap A_2^{\prime} \cap A_3^{\prime}) \]
Where \( Pr(A_i^{\prime}) \) is alarm \( i \) that does not go off.
As the alarms are independent: \( 1-Pr(A_1^{\prime} \cap A_2^{\prime} \cap A_3^{\prime})=1-Pr(A_1^{\prime})P(A_2^{\prime})P(A_3^{\prime})= 1- (0.06)^3 \)
D is the answer.
IncorrectSince the probability that each of the three independent fire alarms are triggered by the occurrence of the fire is \( Pr(A_i)=0.94 \), the probability that none of the three independent fire alarms are triggered by the occurrence of the fire is \( 1-Pr(A_i)=1-0.94=0.06 \). Therefore,
\[ Pr(A_1\cup A_2 \cup A_3)=1-Pr(A_1^{\prime} \cap A_2^{\prime} \cap A_3^{\prime}) \]
Where \( Pr(A_i^{\prime}) \) is alarm \( i \) that does not go off.
As the alarms are independent: \( 1-Pr(A_1^{\prime} \cap A_2^{\prime} \cap A_3^{\prime})=1-Pr(A_1^{\prime})P(A_2^{\prime})P(A_3^{\prime})= 1- (0.06)^3 \)
D is the answer.
- Question 11 of 40
11. Question
There are 4 red marbles and 5 blue marbles in a box. Michael drew 1 marble from the box, then he takes another marble from the box, both without replacement. What is the probability that the second marble he draws will be red?
Correct- The probability that first marble would be red: 4/9
In this case, the probability that the second one would be red: 3/8
- The probability that first marble would be blue: 5/9
In this case, the probability that the second one would be red: 4/8
Therefore, total probability is (4/9 * 3/8) + (5/9 * 4/8) = 32/72 = 4/9
B is the answer.
Incorrect- The probability that first marble would be red: 4/9
In this case, the probability that the second one would be red: 3/8
- The probability that first marble would be blue: 5/9
In this case, the probability that the second one would be red: 4/8
Therefore, total probability is (4/9 * 3/8) + (5/9 * 4/8) = 32/72 = 4/9
B is the answer.
- Question 12 of 40
12. Question
Suppose 10 people participate in a running competition and only three of them can win a prize. Jack and Jacob are two runners in this competition. In how many ways Jack and Jacob could reward winners?
CorrectIf Jack is one of the winners, then he achieves first, second, or third trophy. Thus, three states are possible for jack. Moreover, two states are possible for Jacob to achieve in two other trophies. Therefore, the other prize is for a runner among 8 other participants.
\[ 3\times2\times\binom{8}{1}=48 \]
A is the answer.
IncorrectIf Jack is one of the winners, then he achieves first, second, or third trophy. Thus, three states are possible for jack. Moreover, two states are possible for Jacob to achieve in two other trophies. Therefore, the other prize is for a runner among 8 other participants.
\[ 3\times2\times\binom{8}{1}=48 \]
A is the answer.
- Question 13 of 40
13. Question
How many four-digit numbers can be created by digits 0 to 9, if the last digit- ones – would be zero and repetition of digits are not allowed?
CorrectSince the ones is zero, we have one choice of it. In addition, we are not allowed to put zero into the thousands place as we need to have a four-digit number. Thus, we have 9 choices of thousands from 1 to 9. In this way, 8 choices are for hundreds and 7 choices are for tens with respect to No repetition!
Thousands Hundreds Tens Ones 9 8 7 1 Therefore, the number of four-digits is 9*8*7*1 = 504
C is the answer.
IncorrectSince the ones is zero, we have one choice of it. In addition, we are not allowed to put zero into the thousands place as we need to have a four-digit number. Thus, we have 9 choices of thousands from 1 to 9. In this way, 8 choices are for hundreds and 7 choices are for tens with respect to No repetition!
Thousands Hundreds Tens Ones 9 8 7 1 Therefore, the number of four-digits is 9*8*7*1 = 504
C is the answer.
- Question 14 of 40
14. Question
The average normal body temperature is generally accepted as \( 37^oC \). What is this temperature in Fahrenheit?
CorrectThe temperature T in degrees Fahrenheit (°F) is equal to the temperature T in degrees Celsius (°C) times 1.8 plus 32:
T(°F) = T(°C) × 1.8 + 32
T(°F) = 37 × 1.8 + 32 = 98.6
D is the answer.
IncorrectThe temperature T in degrees Fahrenheit (°F) is equal to the temperature T in degrees Celsius (°C) times 1.8 plus 32:
T(°F) = T(°C) × 1.8 + 32
T(°F) = 37 × 1.8 + 32 = 98.6
D is the answer.
- Question 15 of 40
15. Question
A driver drives the following route from A to B in 115 minutes. How long does it take for the driver drives a 200 Km road?
CorrectThe distance d in kilometers (km) is equal to the distance d in miles (mi) times 1.6:
d(km) = d(mi) × 1.6
d(km) = 45 × 1.6 = 72 km
72 km in 115 minutes, 200 km in ? minutes
(200 × 115) / 72 = 319.4 minutes
Therefore, it takes (319.4)/60 = 5.3 h
B is the answer.
IncorrectThe distance d in kilometers (km) is equal to the distance d in miles (mi) times 1.6:
d(km) = d(mi) × 1.6
d(km) = 45 × 1.6 = 72 km
72 km in 115 minutes, 200 km in ? minutes
(200 × 115) / 72 = 319.4 minutes
Therefore, it takes (319.4)/60 = 5.3 h
B is the answer.
- Question 16 of 40
16. Question
John cuts grass at $0.25 per square meter. How much does John earn cutting the shown triangular field:
CorrectThe area of the shown triangle is (800 × 600) / 2 = 240,000 sq. in. To convert square inches to square meters, we can use the formula below:
\( m^2 = in^2 / 1550 \)
Thus, 240,000/1550 = 154.8 sq. m.
Therefore, John could earn 154.8 * $0.25 = $38.7A is the answer.
IncorrectThe area of the shown triangle is (800 × 600) / 2 = 240,000 sq. in. To convert square inches to square meters, we can use the formula below:
\( m^2 = in^2 / 1550 \)
Thus, 240,000/1550 = 154.8 sq. m.
Therefore, John could earn 154.8 * $0.25 = $38.7A is the answer.
- Question 17 of 40
17. Question
Let the area of a square be equal to the area of a rectangle. If perimeter of the square and rectangle are 36 and 60, respectively, then which of the following is the ratio of the length to width of the rectangle?
CorrectConsider the square and rectangle below:
Where, the perimeter of the square and rectangle above are 4*x = 36 and 2*(y + z) = 60, respectively. We want to know what the value of z/y is. Of the perimeter of the square, x is equal to 9, and we know that x2 = yz so yz = 81. Thus,
\[ \begin{cases} y+z=30\\ yz=81 \end{cases} \implies \begin{cases} y+(81/y)=30 \implies y^2-30y+81=0\\ z=81/y \end{cases} \\ y^2-30y+81=0 \implies (y-27)(y-3)=0 \implies \begin{cases} y=27 \implies z=3\\ y=3 \implies z=27 \end{cases} \]
We know that z is the length of the rectangle and is more than y that is assigned to width. Hence, y = 3, z = 27, and z/y = 9.
C is the answer.
IncorrectConsider the square and rectangle below:
Where, the perimeter of the square and rectangle above are 4*x = 36 and 2*(y + z) = 60, respectively. We want to know what the value of z/y is. Of the perimeter of the square, x is equal to 9, and we know that x2 = yz so yz = 81. Thus,
\[ \begin{cases} y+z=30\\ yz=81 \end{cases} \implies \begin{cases} y+(81/y)=30 \implies y^2-30y+81=0\\ z=81/y \end{cases} \\ y^2-30y+81=0 \implies (y-27)(y-3)=0 \implies \begin{cases} y=27 \implies z=3\\ y=3 \implies z=27 \end{cases} \]
We know that z is the length of the rectangle and is more than y that is assigned to width. Hence, y = 3, z = 27, and z/y = 9.
C is the answer.
- Question 18 of 40
18. Question
Which of the following is equal to the length of BC side of the triangle below?
CorrectBy definition, the sum of the angles in a triangle is 180o:
As shown in figure above, \( BC\times cos(15^o+30^o)=BA\times cos(30^o)=BO \) and \( BA=1 \). By substituting, we have
\[ BC=\frac{1\times cos(30^o)}{cos(15^o+30^o)}=\frac{\sqrt{3}}{\sqrt{2}}=\frac{\sqrt{6}}{2} \]
B is the answer.
IncorrectBy definition, the sum of the angles in a triangle is 180o:
As shown in figure above, \( BC\times cos(15^o+30^o)=BA\times cos(30^o)=BO \) and \( BA=1 \). By substituting, we have
\[ BC=\frac{1\times cos(30^o)}{cos(15^o+30^o)}=\frac{\sqrt{3}}{\sqrt{2}}=\frac{\sqrt{6}}{2} \]
B is the answer.
- Question 19 of 40
19. Question
A rectangular plot has a width of 15 feet and a length of 30 feet. The plot is tiled with squared \( 0.5 \times 0.5 \) ft2 ceramics. What is the total cost of the tiles used in the plot given the cost of $2 per tile?
CorrectConsider the plot as follows:
The area of the plot is 15*30 = 450. In order to answer this question, we must know how many tiles can include in this plot, so we divide the area of the plot to the area of the tile: \( \frac{450}{0.5\times 0.5}=1800 \). Each tile is $2, so 1800 tiles are $3600.
D is the answer.
IncorrectConsider the plot as follows:
The area of the plot is 15*30 = 450. In order to answer this question, we must know how many tiles can include in this plot, so we divide the area of the plot to the area of the tile: \( \frac{450}{0.5\times 0.5}=1800 \). Each tile is $2, so 1800 tiles are $3600.
D is the answer.
- Question 20 of 40
20. Question
A tent that is 5-feet tall is in the shape of a triangular prism has a rectangle base with a length of 9 feet and a width of 3 feet. If half of the tent is filled with water bottles, what is the volume of the empty half?
CorrectConsider the tent as follows:
The volume is B * h, where B is the area of the triangle that is (5 * 3)/2 and h is the length of the rectangle that is 9 ft. Thus, the volume is 7.5 * 9 = 67.5 ft3 and half of it is empty 67.5/2 = 33.75.
A is the answer.
IncorrectConsider the tent as follows:
The volume is B * h, where B is the area of the triangle that is (5 * 3)/2 and h is the length of the rectangle that is 9 ft. Thus, the volume is 7.5 * 9 = 67.5 ft3 and half of it is empty 67.5/2 = 33.75.
A is the answer.
- Question 21 of 40
21. Question
Which of the following is equal to \( (\sin(30^o)+cos(45^o))\times(sin(45^o)-cos(60^o)) \)?
Correct\[ (\frac{1}{2}+\frac{\sqrt{2}}{2})\times(\frac{\sqrt{2}}{2}-\frac{1}{2})=(\frac{\sqrt{2}}{2})^2-(\frac{1}{2})^2=\frac{2}{4}-\frac{1}{4}=\frac{1}{4} \]
C is the answer.
Incorrect\[ (\frac{1}{2}+\frac{\sqrt{2}}{2})\times(\frac{\sqrt{2}}{2}-\frac{1}{2})=(\frac{\sqrt{2}}{2})^2-(\frac{1}{2})^2=\frac{2}{4}-\frac{1}{4}=\frac{1}{4} \]
C is the answer.
- Question 22 of 40
22. Question
What is the height ‘h’ of the figure shown below?
CorrectOf the geometry we know that if we multiply hypotenuse by the sinus of the angle between hypotenuse and the side of triangle we can get the h; so h is as follows:
\[ h=2sin(45^o)=\sqrt{2} \]
C is the answer.
IncorrectOf the geometry we know that if we multiply hypotenuse by the sinus of the angle between hypotenuse and the side of triangle we can get the h; so h is as follows:
\[ h=2sin(45^o)=\sqrt{2} \]
C is the answer.
- Question 23 of 40
23. Question
Which of the following have the limited values?
CorrectLet us review all the options above:
A is not correct because \( cot(180^o) \) is infinity.
C is not correct because \( tan(90^o) \) is infinity.
D is not correct because \( 1/sin(180^o) \) is infinity.
E is not correct because \( tan(90^o) \) is infinity.
B is the answer.
IncorrectLet us review all the options above:
A is not correct because \( cot(180^o) \) is infinity.
C is not correct because \( tan(90^o) \) is infinity.
D is not correct because \( 1/sin(180^o) \) is infinity.
E is not correct because \( tan(90^o) \) is infinity.
B is the answer.
- Question 24 of 40
24. Question
Which of the following is equal to \( tan(\theta)-\sqrt{\frac{1}{cos^2(\theta)}-1} \) if \( \theta \) is in Quadrant II?
Correct\[ tan(\theta)-\sqrt{\frac{1-cos^2{(\theta)}}{cos^2(\theta)}}= tan(\theta)-\sqrt{\frac{sin^2{(\theta)}}{cos^2(\theta)}}= tan(\theta)-\sqrt{tan^2{(\theta)}} \]
Since we know that \( \theta \) is in Quadrant II, then \( tan(\theta)<0 \). Therefore,
\[ tan(\theta)-(-tan(\theta))=2tan(\theta) \]
D is the answer.
Incorrect\[ tan(\theta)-\sqrt{\frac{1-cos^2{(\theta)}}{cos^2(\theta)}}= tan(\theta)-\sqrt{\frac{sin^2{(\theta)}}{cos^2(\theta)}}= tan(\theta)-\sqrt{tan^2{(\theta)}} \]
Since we know that \( \theta \) is in Quadrant II, then \( tan(\theta)<0 \). Therefore,
\[ tan(\theta)-(-tan(\theta))=2tan(\theta) \]
D is the answer.
- Question 25 of 40
25. Question
Two people with the weights of 180 pounds and 140 pounds have a plan for going to the forest. The foods they take has a total of 6500 Kcal of energy. If each person needs at least twice his weight to the energy needed daily (in Kcal), then which of the following is the maximum days that they can survive into the forest?
CorrectEach person needs at least double his weight to the energy so:
– The consumed energy of a person with a weight of 180 pounds per day: 180*2=360 Kcal.
– The consumed energy of a person with a weight of 140 pounds per day: 140*2=280 Kcal.
– The total consumed energy per day: 360+280=640 Kcal.
As they took food with 6500 Kcal then maximum days are:6500/640 = 10.15
C is the answer.
IncorrectEach person needs at least double his weight to the energy so:
– The consumed energy of a person with a weight of 180 pounds per day: 180*2=360 Kcal.
– The consumed energy of a person with a weight of 140 pounds per day: 140*2=280 Kcal.
– The total consumed energy per day: 360+280=640 Kcal.
As they took food with 6500 Kcal then maximum days are:6500/640 = 10.15
C is the answer.
- Question 26 of 40
26. Question
The length of a spring is 10 inches. A box with a weight \( x \) lb. is attached to it, while the formula for the length of the spring is \( y=0.8x+10 \) inches where \( y \) is the length of the spring. What is the change in length of the spring in percentage, if a box with a weight of 5 lb. is attached to it?
CorrectThe length of the spring with a weight of 5 lb. is as follows:
\( y=0.8 \times 5+10=14 \)
Therefore, the length has gotten 4 inches longer. That is to say, the change in percentage is:
\( (\frac{14-10}{10})\times 100=\text{40%} \)
A is the answer.
IncorrectThe length of the spring with a weight of 5 lb. is as follows:
\( y=0.8 \times 5+10=14 \)
Therefore, the length has gotten 4 inches longer. That is to say, the change in percentage is:
\( (\frac{14-10}{10})\times 100=\text{40%} \)
A is the answer.
- Question 27 of 40
27. Question
The total age of Alice and her father is 70 years and the difference is 26 years. How old will Alice’s sister be 5 years later, if she is 3 years younger than Alice?
CorrectWe assume the age of Alice is \( x \), the age of her father is \( y \), and the age of Alice’s sister is \( z \). Thus,
\[ \begin{cases} x+y=70\\ y-x=26 \end{cases} \implies x=22 \\ z=x-3 \implies z=19 \]
After 5 years she will be 24 years.
B is the answer.
IncorrectWe assume the age of Alice is \( x \), the age of her father is \( y \), and the age of Alice’s sister is \( z \). Thus,
\[ \begin{cases} x+y=70\\ y-x=26 \end{cases} \implies x=22 \\ z=x-3 \implies z=19 \]
After 5 years she will be 24 years.
B is the answer.
- Question 28 of 40
28. Question
Michael swam 100 meters in 62 seconds. Then, he set a new record in the 400-meter competition with a time of 249 seconds. Which record was just proportionally faster, and what was the speed in feet per second?
CorrectHe swam 100 meters is 62 seconds so proportionally he could swim 400 meters in 4*62=248 seconds that is faster than his record in the 400-meter competition. Furthermore, he swam 100 meters in 62 seconds that means he swam 1.613 meters in 1 second. In other words, he swam 5.3 feet per second as the conversion of the meter in the foot is:
- Each 1 meter is roughly equal to 3.28 feet.
A is the answer.
IncorrectHe swam 100 meters is 62 seconds so proportionally he could swim 400 meters in 4*62=248 seconds that is faster than his record in the 400-meter competition. Furthermore, he swam 100 meters in 62 seconds that means he swam 1.613 meters in 1 second. In other words, he swam 5.3 feet per second as the conversion of the meter in the foot is:
- Each 1 meter is roughly equal to 3.28 feet.
A is the answer.
- Question 29 of 40
29. Question
A circular cake is divided between two groups of A and B. Groups A and B consist of 8 and 6 people, respectively. If 60% of the cake is assigned to group A and 40% is assigned to group B, then what is the difference of the angle between slices of group A and B, in degrees?
CorrectWe are going to know an angle in the circle that is 360 degrees. On the first step we must learn what portion of the cake is for group A and what portion is for group B in terms of the angle:
Group A: \( 360^o\times\text{60%}=216^o \)
Group B: \( 360^o\times\text{40%}=144^o \)
Now, we only need to divide each portion between the people of each group:
The angle between slices of group A: \( 216^o/8=27^o \)
The angle between slices of group B: \( 144^o/6=24^o \)
Therefore, the difference is \( 27^o-24^o=3^o \).
D is the answer.
IncorrectWe are going to know an angle in the circle that is 360 degrees. On the first step we must learn what portion of the cake is for group A and what portion is for group B in terms of the angle:
Group A: \( 360^o\times\text{60%}=216^o \)
Group B: \( 360^o\times\text{40%}=144^o \)
Now, we only need to divide each portion between the people of each group:
The angle between slices of group A: \( 216^o/8=27^o \)
The angle between slices of group B: \( 144^o/6=24^o \)
Therefore, the difference is \( 27^o-24^o=3^o \).
D is the answer.
- Question 30 of 40
30. Question
A person is losing his weight with a rate of 4 lb./month due to a severe illness. If his weight is 180 lb. in June, then what is the integer part of the weight in December, in Kilograms?
CorrectFrom June to December he will lose 6*4 = 24 lb. so his weight will be 180-24 = 156 lb. Then, we have to convert 156 lb. into the Kilograms:
1 pound is equal to 0.45 Kilograms so 156 pounds will be 156*0.45 = 70.2. The integer value is 70.
A is the answer.
IncorrectFrom June to December he will lose 6*4 = 24 lb. so his weight will be 180-24 = 156 lb. Then, we have to convert 156 lb. into the Kilograms:
1 pound is equal to 0.45 Kilograms so 156 pounds will be 156*0.45 = 70.2. The integer value is 70.
A is the answer.
- Question 31 of 40
31. Question
Alex lives in a society where the height of men is 60% proportional to their fathers and 40% proportional to their mothers. He is currently 180cm tall. What is the height of Alex’s father, if the difference in height between his parents is 20cm?
CorrectIf we assume the height of Alex’s father and mother are \( x \) and \( y \), respectively, then we have:
\[ \begin{cases} 0.6x+0.4y=180cm\\ x-y=20cm \end{cases} \implies x=188cm \]
B is the answer.
IncorrectIf we assume the height of Alex’s father and mother are \( x \) and \( y \), respectively, then we have:
\[ \begin{cases} 0.6x+0.4y=180cm\\ x-y=20cm \end{cases} \implies x=188cm \]
B is the answer.
- Question 32 of 40
32. Question
Tom borrowed $3000 for 2 years at a 6% annual interest rate. He receives 0.9% interest per month on his account with a balance of $1500. If he would like to pay a refund through the interest he gets off of his account, what is the difference between the interest he earns and the interest he has to pay in 2 years?
CorrectThe Interest of his loan is:
\( $3000\times\text{6%}\times2\text{years}=$360 \)
The interest on his account is:
\( $1500\times0.9\text{%}\times12\text{months}\times2\text{years}=$324 \)
So the difference is \( 360 – 324 = $36 \).
A is the answer.
IncorrectThe Interest of his loan is:
\( $3000\times\text{6%}\times2\text{years}=$360 \)
The interest on his account is:
\( $1500\times0.9\text{%}\times12\text{months}\times2\text{years}=$324 \)
So the difference is \( 360 – 324 = $36 \).
A is the answer.
- Question 33 of 40
33. Question
Which of the following is a bargain?
CorrectTo compare all the options, we convert them in liters:
A: 0.2 liters of cream at $2.20, means 0.2/2.20 = 0.090 liters per dollar.
B: 2.5 liters of cream at $32.2, means 2.5/32.2 = 0.077 liters per dollar.
C: 0.35 liters of cream at $3.8, means 0.35/3.8 = 0.092 liters per dollar.
D: 1.7 liters of cream at $18.48, means 1.7/18.48 = 0.091 liters per dollar.
E: 0.8 liters of cream at $8.72, means 0.8/8.72 = 0.091 liters per dollar.
C is the answer.
IncorrectTo compare all the options, we convert them in liters:
A: 0.2 liters of cream at $2.20, means 0.2/2.20 = 0.090 liters per dollar.
B: 2.5 liters of cream at $32.2, means 2.5/32.2 = 0.077 liters per dollar.
C: 0.35 liters of cream at $3.8, means 0.35/3.8 = 0.092 liters per dollar.
D: 1.7 liters of cream at $18.48, means 1.7/18.48 = 0.091 liters per dollar.
E: 0.8 liters of cream at $8.72, means 0.8/8.72 = 0.091 liters per dollar.
C is the answer.
- Question 34 of 40
34. Question
The amount of energy gained by eating 20 oz. of sweets is 200 calories. A loaf of beard makes up 8 times of this amount of energy. If the amount of energy consumed by a person who has 160 lb weight is 150 calories for a two-mile walk, then how many miles does this person need to walk in order to consume the energy of 3 loaves of bread?
?
CorrectThe amount of energy for a loaf of bread is 8*200 calories so this amount for the 3 loaves of bread is 8*200*3 calories. We know the consumed energy by the mentioned person is 150 calories for two miles walk. Therefore,
\[ \frac{2\text{miles}}{150\text{calories}}=\frac{?}{4800} \implies \frac{4800\times2}{150}=64\text{miles} \]
E is the answer.
IncorrectThe amount of energy for a loaf of bread is 8*200 calories so this amount for the 3 loaves of bread is 8*200*3 calories. We know the consumed energy by the mentioned person is 150 calories for two miles walk. Therefore,
\[ \frac{2\text{miles}}{150\text{calories}}=\frac{?}{4800} \implies \frac{4800\times2}{150}=64\text{miles} \]
E is the answer.
- Question 35 of 40
35. Question
Quantity A Quantity B The greatest prime number less than 42 The least common multiple of 5 and 8 CorrectA prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number.To find the quantity A we should count down from 42. therefore, the greatest prime number is 41.
The least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.To find the quantity B:
Multiples of 5: 5*1, 5*2, 5*3, 5*4, 5*5, 5*6, 5*7, 5*8, … : 5, 10, 15, 20, 25, 30, 35, 40, …
Multiples of 8: 8*1, 8*2, 8*3, 8*4, 8*5, … : 8, 16, 24, 32, 40, …Therefore, the least common multiples of 5 and 8 is 40.
A is the answer.
IncorrectA prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number.To find the quantity A we should count down from 42. therefore, the greatest prime number is 41.
The least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.To find the quantity B:
Multiples of 5: 5*1, 5*2, 5*3, 5*4, 5*5, 5*6, 5*7, 5*8, … : 5, 10, 15, 20, 25, 30, 35, 40, …
Multiples of 8: 8*1, 8*2, 8*3, 8*4, 8*5, … : 8, 16, 24, 32, 40, …Therefore, the least common multiples of 5 and 8 is 40.
A is the answer.
- Question 36 of 40
36. Question
\( \begin{cases} x+y=70\\ x-y=26 \end{cases} \)
Quantity A Quantity B \( 3y \) \( 11x/8 \) CorrectIn order to answer this question, we need to find both x and y:
\[ 2x=96 \implies x=48 \implies y=22 \]
Therefore, Quantity A is equal to 3*22 and Quantity B is equal to 66.
C is the answer.
IncorrectIn order to answer this question, we need to find both x and y:
\[ 2x=96 \implies x=48 \implies y=22 \]
Therefore, Quantity A is equal to 3*22 and Quantity B is equal to 66.
C is the answer.
- Question 37 of 40
37. Question
Quantity A Quantity B The length of AC The length of BC CorrectThe length of HC: \( \sqrt{3^2-(\sqrt{5})^2}=2 \)
The length of HB: \( \sqrt{(\sqrt{6})^2-(\sqrt{5})^2}=1 \)
Therefore, the length of BC will be 2 + 1 = 3 similar to the length of AC.
C is the answer.
IncorrectThe length of HC: \( \sqrt{3^2-(\sqrt{5})^2}=2 \)
The length of HB: \( \sqrt{(\sqrt{6})^2-(\sqrt{5})^2}=1 \)
Therefore, the length of BC will be 2 + 1 = 3 similar to the length of AC.
C is the answer.
- Question 38 of 40
38. Question
Quantity A Quantity B a \[ -\frac{b-6}{5} \]
CorrectWe definitely know that a is greater than 1.5. If we assume b is -1.5 at the least, then Quantity B is still less than Quantity A.
A is the answer.
IncorrectWe definitely know that a is greater than 1.5. If we assume b is -1.5 at the least, then Quantity B is still less than Quantity A.
A is the answer.
- Question 39 of 40
39. Question
Quantity A Quantity B Area of the larger circle \( 2\times \)area of the smaller circle CorrectConsider the length of the square side is a. To find the length of the diagonal of the square, multiply the length of one side by the square root of 2: \( \sqrt{(a)^2+(a)^2}=a\sqrt{2} \)
As can be seen, the half of the diagonal of the square is the radius of the larger circle and the length of the square side is equal to the radius of the smaller circle. Therefore,
The area of the larger circle: \( \pi \times (radius)^2=\pi(\frac{a\sqrt{2}}{2})^2=\frac{\pi}{2}a^2 \)
The area of the smaller circle: \( \pi \times (radius)^2=\pi(\frac{a}{2})^2=\frac{\pi}{4}a^2 \)
C is the answer.
IncorrectConsider the length of the square side is a. To find the length of the diagonal of the square, multiply the length of one side by the square root of 2: \( \sqrt{(a)^2+(a)^2}=a\sqrt{2} \)
As can be seen, the half of the diagonal of the square is the radius of the larger circle and the length of the square side is equal to the radius of the smaller circle. Therefore,
The area of the larger circle: \( \pi \times (radius)^2=\pi(\frac{a\sqrt{2}}{2})^2=\frac{\pi}{2}a^2 \)
The area of the smaller circle: \( \pi \times (radius)^2=\pi(\frac{a}{2})^2=\frac{\pi}{4}a^2 \)
C is the answer.
- Question 40 of 40
40. Question
The average salary of employees in company B is $35,000 a year, while mangers in company A earn $70,000 a year.
Quantity A Quantity B The salary for employees in company A which is increased by 40% The salary for managers in company B which is decreased by 10% CorrectBe careful! We know the average salary for the employee in company B, but we do not know this amount for the employee in company A. The same procedure is followed for the manager so we do not have the right information.
D is the answer.
IncorrectBe careful! We know the average salary for the employee in company B, but we do not know this amount for the employee in company A. The same procedure is followed for the manager so we do not have the right information.
D is the answer.
