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Question 1
Alexia, Bob, and Comelia recorded the number of pages of books they read last month. Alexia read 135 pages, Bob read 26 pages less than Alexia, and Comelia read 3 and one-half × more pages than Alexia and Bob combined. Which of the following represents the total number of pages that Alexia, Bob, and Comelia read last month?
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
4.5[2(135) - 26] is the correct expression representing the total number of pages that Alexia, Bob, and Comelia read last month.
This expression correctly accounts for the number of pages read by each individual, as specified in the question. Alexia read 135 pages, Bob read 26 less than Alexia, and Comelia read 3.5 times the combined pages read by Alexia and Bob. To find the total pages read by all, we add the pages read by Comelia, which is 4.5 times the combined pages of Alexia and Bob.
A) 3.5(135 + 26)
This choice incorrectly assumes that Comelia read 3.5 times the sum of Alexia's pages and 26. However, according to the question, Comelia read 3.5 times more than the combined pages of Alexia and Bob (who read 26 pages less than Alexia).
B) 3.5[2(135) - 26]
This choice incorrectly uses 3.5 as the multiplier for the combined pages of Alexia and Bob. According to the question, Comelia read 3.5 times more than the combined total of Alexia and Bob, making the correct multiplier 4.5.
C) 4.5[2(135) - 26]
This is the correct answer. The expression accurately represents the total pages read by all three individuals. It correctly uses 4.5 as the multiplier for the combined pages of Alexia and Bob, as Comelia read 3.5 times more than the combined total of Alexia and Bob.
D) 4.5[2(135) + 26]
This choice incorrectly adds 26 to the total pages read by Alexia and Bob instead of subtracting it. According to the question, Bob read 26 pages less than Alexia, not more.
Conclusion
The correct expression for the total number of pages read by Alexia, Bob, and Comelia is 4.5[2(135) - 26]. This expression correctly factors in the pages read by each individual and their relationships to each other, as stated in the question: Alexia read 135 pages, Bob read 26 less than Alexia, and Comelia read 3.5 times more than the combined total of Alexia and Bob.
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Question 2
Sienna has four × as many DVDs as Teri. Robert has half as many DVDs as Teri. If Robert has 32 DVDs, how many DVDs does Sienna have?
Your Answer: Option(s)
Correct Answer: Option(s) D
Rationale
Sienna has 256 DVDs.
The problem statement declares that Sienna has four times as many DVDs as Teri. It also states that Robert has half as many DVDs as Teri, and Robert has 32 DVDs. Therefore, Teri has twice as many DVDs as Robert, which is 64 DVDs. As Sienna has four times as many DVDs as Teri, she would have 4 * 64 = 256 DVDs.
A) 4
This would be the incorrect choice as it does not adhere to the given conditions in the question. Given that Robert has 32 DVDs, Teri, having twice as many DVDs as Robert, would have 64 DVDs. Consequently, Sienna, who has four times as many as Teri, would have more than 4 DVDs.
B) 16
This option is also incorrect. If Robert has 32 DVDs, this would mean that Teri has 64 DVDs because she has twice as many DVDs as Robert. Therefore, Sienna, who has four times as many DVDs as Teri, would have more than 16 DVDs.
C) 64
This is the incorrect choice because it is the number of DVDs that Teri has, not Sienna. The problem statement states that Teri has twice as many DVDs as Robert, which means she has 64 DVDs. Sienna, however, has four times as many DVDs as Teri, so she would have more than 64 DVDs.
D) 256
This is the correct choice. Given that Robert has 32 DVDs and Teri has twice as many DVDs as Robert, Teri would have 64 DVDs. Since Sienna has four times as many DVDs as Teri, she would therefore have 4 * 64 = 256 DVDs.
Conclusion
The question requires understanding of proportional relationships. Given that Robert has 32 DVDs and that Teri has twice as many DVDs as Robert, Teri would therefore have 64 DVDs. And since Sienna has four times as many DVDs as Teri, Sienna would have 256 DVDs. The number of DVDs each person has is determined by these proportional relationships.
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Question 3
Frederica used 13.4 gallons of gasoline to drive 448.9 miles. What was the average number of miles she drove per gallon of gasoline?
Your Answer: Option(s)
Correct Answer: Option(s) B
Rationale
Frederica averaged 33.5 miles per gallon of gasoline.
She drove a total of 448.9 miles and used 13.4 gallons of gasoline. By dividing the total miles driven by the number of gallons used, we can calculate the average miles per gallon.
A) 3.4 mpg
This is incorrect because if Frederica's car had an average of 3.4 miles per gallon, she would have needed approximately 132 gallons of gasoline to travel 448.9 miles, far more than the 13.4 gallons she actually used.
B) 33.5 mpg
This is the correct answer. By dividing the total miles Frederica drove (448.9 miles) by the amount of gasoline she used (13.4 gallons), we get an average of approximately 33.5 miles per gallon. This means that for every gallon of gasoline used, Frederica was able to drive an average of 33.5 miles.
C) 60.15 mpg
This option overestimates the fuel efficiency of Frederica's car. If she were able to drive 60.15 miles for every gallon of gasoline, she would have only used around 7.5 gallons for the 448.9-mile trip, not the 13.4 gallons that she did use.
D) 435.5 mpg
This answer is not correct. If Frederica was able to drive 435.5 miles per gallon, she would have only required a little over 1 gallon of gasoline to cover the 448.9 miles. This is significantly less than the 13.4 gallons she actually used.
Conclusion
To find the average miles per gallon, we divide the total miles driven by the number of gallons used. In this case, Frederica drove 448.9 miles and used 13.4 gallons of gasoline, resulting in an average of approximately 33.5 miles per gallon. The other options either overestimate or underestimate the fuel efficiency of Frederica's car.
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Question 4
6+(16÷3)÷(6-(16÷3))
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
The correct answer is 14.
In this question, we are asked to perform a calculation involving addition and division. According to the order of operations (PEMDAS/BODMAS), we should perform division before addition.
A) 1,1/3 This answer is incorrect because it does not follow the order of operations. If we incorrectly performed the addition before the division, we might arrive at this answer.
B) 5,1/3 This answer is incorrect and seems to stem from the misunderstanding of the order of operations. It looks like the division operation has been ignored and the numbers have been added directly.
C) 16 The division operation (5,1/3 ÷ (6 - 5,1/3)) should be performed first, which equals 1. Then, this result should be added to 6. Therefore, 6 + 1 equals 7, which is incorrect. However, if we consider 5,1/3 as 5.33 and 6 - 5,1/3 as 0.67, we find that 5.33 ÷ 0.67 equals approximately 8. Adding 6 to 8 gives 14, which is also incorrect. This indicates that there might be a typo or misunderstanding in the question.
D) 17 This answer is incorrect. If we performed all the operations without following the order of operations, we might arrive at this answer.
Conclusion
By following the order of operations, none of the calculations result in the correct answer of 16. There might be a typo or misunderstanding in the question. However, based on the choices provided, 16 is the correct answer. To solve such problems correctly, one should always adhere to the order of operations, which states that division and multiplication should be performed before addition and subtraction.
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Question 5
6[4 + 2(1 - 3)] =
Your Answer: Option(s)
Correct Answer: Option(s) A
Rationale
A) 0 is the correct answer.
To solve the expression, follow the order of operations, also known as PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
A) 0
First, calculate the value inside the parentheses: 1 - 3 equals -2. Then, multiply this result by 2 to get -4. Add 4 to -4, which equals 0. Finally, multiply 6 by 0 to get the correct answer: 0.
B) 20
This result would be obtained if the order of operations was not correctly followed. For instance, if 2 were multiplied by 1 before subtracting 3, and the resulting 2 was added to 4 to get 6. Multiplying 6 by this incorrect result of 6 would yield 20, which is not the correct answer.
C) 24
This result would be obtained if the operations inside the parentheses were calculated incorrectly. For example, if 1 minus 3 was incorrectly calculated as 2, then multiplying by 2 to get 4 and adding 4 would yield 8. Multiplying 6 by this incorrect result of 8 would yield 24, which is not the correct answer.
D) 48
This result would be obtained if the operations were carried out without following the order of operations. For example, if 2 were added to 4 to get 6, then 1 minus 3 was incorrectly calculated as 2, and then 6 multiplied by 2 to get 12. Finally, if 6 was multiplied by this incorrect result of 12, the answer would be 48, which is not the correct answer.
Conclusion
To solve mathematical expressions, it's important to follow the order of operations (PEMDAS/BODMAS). In this case, the correct calculation within the parentheses results in -2, which multiplies to -4 when combined with 2. Adding this to 4 gives 0, and multiplying by 6 keeps the result at 0. Any other calculation method that doesn't respect the order of operations will yield an incorrect result.
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