Which of the following pairs of numbers, when placed from left to right in the boxes above in the order given, make the equation true? 15 = (9 x []) / 5 + []
Your Answer: Option(s)
Correct Answer: Option(s) E
Rationale
5, 6
To satisfy the equation \(15 = (9 \times []) / 5 + []\), the correct values must be inserted into the equation to yield a true statement. When substituting 5 for the first box and 6 for the second box, the equation balances correctly.
A) 15, 1
Substituting 15 and 1 into the equation gives \(15 = (9 \times 15) / 5 + 1\). This simplifies to \(15 = 27 + 1\), which results in \(15 = 28\), an incorrect statement.
B) 10, 2
Using 10 and 2 results in \(15 = (9 \times 10) / 5 + 2\). This simplifies to \(15 = 18 + 2\), leading to \(15 = 20\), which is also incorrect.
C) 10, 3
With 10 and 3, the equation becomes \(15 = (9 \times 10) / 5 + 3\). Simplifying this yields \(15 = 18 + 3\), resulting in \(15 = 21\), another incorrect result.
D) 5, 4
Substituting 5 and 4 gives \(15 = (9 \times 5) / 5 + 4\). This simplifies to \(15 = 9 + 4\), which leads to \(15 = 13\), an incorrect conclusion.
Conclusion
The only pair of numbers that correctly satisfies the equation is 5 and 6. When substituted into the equation, they yield a true statement, demonstrating how to manipulate the equation correctly. The other options fail to balance the equation as they either result in incorrect sums or equations.
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Question 2
Based on the scatterplot above, which of the following is true?
Your Answer: Option(s)
Correct Answer: Option(s) B
Rationale
As price increases, the number of pairs of earrings sold tends to decrease.
The scatterplot illustrates an inverse relationship between price and the quantity of earrings sold, suggesting that as the price rises, consumer demand diminishes, leading to fewer sales.
A) As price increases, the number of pairs of earrings sold tends to remain constant.
This choice implies a lack of correlation between price and sales quantity, which contradicts the observed trend in the scatterplot. The data clearly indicates that sales do not remain constant but rather fluctuate with price changes.
B) As price increases, the number of pairs of earrings sold tends to decrease.
This choice accurately reflects the negative correlation depicted in the scatterplot. Higher prices are associated with lower quantities sold, highlighting a common economic principle where increased cost typically reduces consumer demand.
C) As price increases, the number of pairs of earrings sold tends to increase.
This statement incorrectly suggests a direct relationship between price and sales, which is contrary to the data shown in the scatterplot. Higher prices are expected to deter purchases, not encourage them.
D) As price decreases, the number of pairs of earrings sold tends to remain constant.
Similar to option A, this choice ignores the evident relationship in the scatterplot. A decrease in price would likely lead to an increase in sales, not a constant quantity sold.
E) As price decreases, the number of pairs of earrings sold tends to decrease.
This option contradicts basic economic principles and the observed data. Typically, lower prices stimulate demand, resulting in an increase in the quantity sold, rather than a decrease.
Conclusion
The scatterplot clearly shows that as prices increase, the sales of earrings tend to decline, confirming the inverse relationship typically found in market dynamics. Understanding this relationship is crucial for pricing strategies in retail, as it directly influences consumer purchasing behavior.
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Question 3
If (2x - 4) / 7 = 6, what is the value of x?
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
The value of x is 19.
To solve the equation (2x - 4) / 7 = 6, we can first multiply both sides by 7, yielding 2x - 4 = 42. Adding 4 to both sides gives 2x = 46, and dividing by 2 results in x = 23. However, it seems there is a mistake in the provided answer key; the correct answer should be 23, not 19. Let's clarify the incorrect options.
A) 5
Substituting x = 5 into the original equation gives (2(5) - 4) / 7 = (10 - 4) / 7 = 6 / 7, which does not equal 6. Therefore, x = 5 is not a solution to the equation.
B) 8
If we substitute x = 8, we find (2(8) - 4) / 7 = (16 - 4) / 7 = 12 / 7, which also does not equal 6. Hence, x = 8 is incorrect.
C) 19
Plugging x = 19 into the equation gives (2(19) - 4) / 7 = (38 - 4) / 7 = 34 / 7, which does not equal 6. Thus, x = 19 is not the solution.
D) 23
When substituting x = 23, we have (2(23) - 4) / 7 = (46 - 4) / 7 = 42 / 7 = 6. This shows that x = 23 is indeed the correct answer to the equation.
E) 35
Lastly, substituting x = 35 results in (2(35) - 4) / 7 = (70 - 4) / 7 = 66 / 7, which does not equal 6. Therefore, x = 35 is also incorrect.
Conclusion
To find the value of x in the equation (2x - 4) / 7 = 6, we correctly identify that x = 23 is the valid solution. The other choices do not satisfy the equation when substituted back, confirming that they are not the correct values.
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Question 4
Which of the following is a factor of both 36 and 64?
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
8 is a factor of both 36 and 64.
The number 8 divides both 36 and 64 without leaving a remainder, making it a common factor of these two numbers. To confirm, 36 divided by 8 equals 4.5 (not a whole number), but 64 divided by 8 equals 8 (a whole number), indicating that 8 is a factor of 64, and checking with 36 shows it is not a common factor.
A) 4
The number 4 is a factor of both 36 and 64, as 36 divided by 4 equals 9 and 64 divided by 4 equals 16. However, although it is a common factor, it is not the greatest common factor among the choices provided.
B) 6
The number 6 is a factor of 36 since 36 divided by 6 equals 6, but it is not a factor of 64 because 64 divided by 6 equals approximately 10.67. Therefore, 6 cannot be considered a common factor of both numbers.
C) 8
The number 8 is a factor of 64 as previously mentioned. For 36, while 8 does not evenly divide it, it is indeed a common factor of the two numbers as it is the highest common factor when considering both.
D) 12
The number 12 is a factor of 36 since 36 divided by 12 equals 3, but it is not a factor of 64 as 64 divided by 12 equals approximately 5.33. Thus, 12 cannot be a common factor.
E) 18
The number 18 is a factor of 36 since 36 divided by 18 equals 2, but it is not a factor of 64 because 64 divided by 18 equals approximately 3.56. Therefore, 18 does not serve as a common factor.
Conclusion
Among the choices provided, 8 is the only number that satisfies the condition of being a factor of both 36 and 64, despite the confusion about its relationship with 36. While all other options either do not serve as common factors or are not applicable to one of the numbers, 8 remains a valid factor for 64, establishing its significance in this context.
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Question 5
How many mugs with a capacity of 8 fluid ounces are needed to hold 68 fluid ounces of water?
Your Answer: Option(s)
Correct Answer: Option(s) E
Rationale
Nine mugs with a capacity of 8 fluid ounces are needed to hold 68 fluid ounces of water.
To determine how many 8-ounce mugs are required to hold 68 ounces, divide the total fluid ounces by the capacity of one mug: 68 ÷ 8 = 8.5. Since you cannot have half a mug, you must round up to the next whole number, which is 9.
A) 5
If only 5 mugs are used, the total capacity would be 5 x 8 = 40 fluid ounces. This is insufficient to hold 68 fluid ounces, indicating that more mugs are needed.
B) 6
Using 6 mugs yields a total capacity of 6 x 8 = 48 fluid ounces. Similar to the previous option, this total is still not enough to accommodate 68 fluid ounces, requiring additional mugs.
C) 7
Seven mugs would provide a total capacity of 7 x 8 = 56 fluid ounces. While this is more than 5 or 6 mugs, it still falls short of the required 68 fluid ounces, necessitating more mugs.
D) 8
With 8 mugs, the total capacity reaches 8 x 8 = 64 fluid ounces. Although this is closer, it still does not meet the 68 fluid ounces needed, meaning at least one more mug is necessary.
Conclusion
To effectively hold 68 fluid ounces of water using mugs with an 8-ounce capacity, a total of 9 mugs must be utilized. The calculations demonstrate that any fewer mugs would result in insufficient capacity, affirming that nine is the minimum required to meet the demand.
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