Given that 1 mile equals 1609.34 meters, which of the following is the proper conversion of a 26.2 mile marathon to kilometers? (Round to the nearest hundredths.)
Rationale
26.2 miles converts to 42.16 kilometers.
To convert miles to kilometers, the conversion factor of 1 mile equals 1.60934 kilometers is used. Multiplying 26.2 miles by this factor provides the correct distance in kilometers, which rounds to 42.16.
A) 421.65 kilometers This option incorrectly multiplies the mile distance too low, resulting in a value that misrepresents the conversion. The correct calculation yields a value significantly higher than 421.65 kilometers.
B) 4216.47 kilometers This answer is an overestimation, stemming from a miscalculation that likely involved incorrect multiplication or unit conversion. The distance should be much lower than this inflated figure when properly converting 26.2 miles.
C) 42164.71 kilometers This choice is drastically incorrect, suggesting a misunderstanding of the conversion scale. The figure is excessively high and does not reflect a plausible distance for a marathon, indicating a complete miscalculation.
D) 42.16 kilometers This is the correct conversion of 26.2 miles to kilometers, as it accurately applies the conversion factor and rounds the result to the nearest hundredths. The calculation (26.2 miles * 1.60934 km/mile) equals approximately 42.164, which rounds to 42.16 kilometers.
Conclusion Converting miles to kilometers requires the application of the conversion factor of 1 mile to 1.60934 kilometers. For a marathon distance of 26.2 miles, the accurate conversion yields 42.16 kilometers. The other options reflect various degrees of miscalculation, underscoring the importance of precise mathematical conversion in unit changes.
Translate the phrase 'Five less than twice the number' into a mathematical expression.
Rationale
2x-5 is the mathematical expression that translates the phrase 'Five less than twice the number'.
In mathematics, the phrase 'Five less than twice the number' means we first multiply a number by two (twice the number) and then subtract five from the result. This operation can be represented as 2x-5, where x stands for the number.
A) 5x-2 This expression represents 'Two less than five times the number', not 'Five less than twice the number'. In this expression, the number is first multiplied by five (five times the number) and then two is subtracted from it, which is different from the original phrase.
B) 2x-5 The mathematical expression 2x-5 translates the phrase 'Five less than twice the number'. The term 'twice the number' is represented by '2x' and 'Five less than' is represented by '-5'. The expression is read as 'two times x, minus five', which matches the phrase.
C) 2-5x This expression represents 'Two less than five times the number', not 'Five less than twice the number'. In this expression, the number is first multiplied by five (five times the number), and then subtracted from two, changing the order of operations from the original phrase.
D) 5-2x This expression represents 'Five less than two times the number', not 'Five less than twice the number'. In this expression, two is multiplied by the number (two times the number) and then subtracted from five, which again changes the order of operations from the original phrase.
Conclusion In the given phrase 'Five less than twice the number', 'twice the number' translates to '2x' and 'Five less than' translates to '-5'. Therefore, the correct mathematical expression is 2x-5. All other expressions either misinterpret the phrase or change the order of operations.
Which of the following fractions is closest to 55.5%?
Rationale
{5/9} is closest to 55.5%.
To determine which fraction is closest to 55.5%, we can convert each fraction to a decimal and then to a percentage. Upon calculation, {5/9} converts to approximately 55.56%, making it the closest match to the target percentage.
A) {5/9} Calculating {5/9} gives approximately 0.5556, which converts to 55.56%. This percentage is very close to 55.5%, making it the best choice among the options given.
B) {6/10} The fraction {6/10} simplifies to {3/5}, which equals 0.6 or 60%. This percentage is above 55.5%, making it less suitable as the closest fraction.
C) {4/9} Calculating {4/9} results in approximately 0.4444, which converts to 44.44%. This value is significantly lower than 55.5%, indicating it is not a close match.
D) {5/6} The fraction {5/6} equals approximately 0.8333, which converts to 83.33%. This percentage is far greater than 55.5%, thus making it an unsuitable choice as well.
Conclusion Among the choices presented, {5/9} stands out as the closest fraction to 55.5%, with its calculated percentage of 55.56%. The other options either exceed or fall short of the target percentage, confirming {5/9} as the optimal answer. Accurate fraction analysis is essential for precise comparisons in mathematical contexts.
A temperature gauge reads 95 F. Which of the following is the correct conversion to degrees Celsius? (Note: C = (F - 32) x 5/9)
Rationale
The correct conversion of 95 F to degrees Celsius is 35C.
The formula to convert Fahrenheit to Celsius is (F - 32) x 5/9. Substituting 95 for F in the formula results in (95 - 32) x 5/9 = 35C, which is the correct conversion.
A) 34C This is incorrect because when you use the formula to convert Fahrenheit to Celsius, (95 - 32) x 5/9, the result is 35C, not 34C.
B) 63C This is incorrect because when you use the formula to convert Fahrenheit to Celsius, (95 - 32) x 5/9, the result is 35C, not 63C.
C) 113C This is incorrect because when you use the formula to convert Fahrenheit to Celsius, (95 - 32) x 5/9, the result is 35C, not 113C.
D) 35C This is the correct answer. When you use the formula to convert Fahrenheit to Celsius, (95 - 32) x 5/9, the result you get is 35C.
Conclusion The Fahrenheit scale and the Celsius scale are two different temperature scales used to measure temperature. The formula for converting Fahrenheit to Celsius is (F - 32) x 5/9. When this formula is applied to convert 95F to Celsius, the result is 35C. The other options, 34C, 63C, and 113C, are incorrect as they do not match the result obtained by using the conversion formula.
Which of the following expressions has the greatest value?
Rationale
A) 34 + 96 has the greatest value.
The sum of 34 and 96 is 130, which is greater than the other choices provided.
A) 34 + 96 The sum of 34 and 96 is 130. This is the largest value amongst the options provided.
B) 0.372 0.372 is a decimal number and less than 1, making it much smaller than the sum of 34 and 96.
C) 03-Aug 03-Aug seems to be a date, and as such, it does not have a numerical value that can be compared to the other choices.
D) 37% 37% is equivalent to 0.37 when converted to decimal form, which is significantly less than 130, the sum of 34 and 96.
Conclusion The comparison of the values of the given expressions shows that the sum of 34 and 96, which is 130, is the greatest value. The other options either have much smaller numerical values (0.372 and 37%), or do not represent a numerical value that can be compared to the others (03-Aug). Therefore, A) 34 + 96 is the correct answer.
How many milliliters are in 0.5 liters?
Rationale
There are 500 milliliters in 0.5 liters.
The conversion factor between liters and milliliters is 1 liter equals 1000 milliliters. Therefore, to convert 0.5 liters to milliliters, multiply 0.5 by 1000, which results in 500 milliliters.
A) 5,000 This answer would be correct if the conversion was for 5 liters instead of 0.5 liters. 5 liters equals 5000 milliliters, not 0.5 liters.
B) 5 This answer would be correct if the conversion was for 0.005 liters instead of 0.5 liters. 0.005 liters equals 5 milliliters, not 0.5 liters.
C) 500 This is the correct answer. 0.5 liters equals 500 milliliters. This is calculated by multiplying 0.5 liters by the conversion factor of 1000 milliliters per liter.
D) 50 This answer would be correct if the conversion was for 0.05 liters instead of 0.5 liters. 0.05 liters equals 50 milliliters, not 0.5 liters.
Conclusion To convert liters to milliliters, we multiply the number of liters by 1000. Therefore, 0.5 liters equals 500 milliliters, not 5,000, 5, or 50 milliliters. Understanding the correct conversion factor is essential to accurately convert between different units of measurement.
What is the length of the unknown leg of a right triangle that has one leg measuring 8 feet and a hypotenuse of 12 feet? (Round to the nearest tenth)
Rationale
The length of the unknown leg of a right triangle that has one leg measuring 8 feet and a hypotenuse of 12 feet is approximately 8.9 feet.
This length was found using the Pythagorean theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can use this theorem to find the unknown leg's length by subtracting the square of the known leg from the square of the hypotenuse, and then taking the square root of the result.
A) 40 feet This answer is incorrect because it significantly exceeds the length of the hypotenuse. In a right triangle, the hypotenuse is always the longest side. If the unknown leg were 40 feet, this would make it longer than the hypotenuse, which contradicts the properties of a right triangle.
B) 4 feet This answer is incorrect. By using the Pythagorean theorem, the square of the hypotenuse (144 square feet) minus the square of the known leg (64 square feet) equals 80 square feet. The square root of 80 is approximately 8.9, not 4 feet.
C) 14.4 feet This answer is incorrect because it also exceeds the length of the hypotenuse. The hypotenuse is always the longest side in a right triangle, so the length of the unknown leg cannot be greater than that of the hypotenuse.
D) 8.9 feet This is the correct answer. Using the Pythagorean theorem, the square of the hypotenuse (144 square feet) minus the square of the known leg (64 square feet) equals 80 square feet. The square root of 80 is approximately 8.9 feet, which is the length of the unknown leg.
Conclusion The Pythagorean theorem allows us to calculate the length of the unknown leg in a right triangle if we know the lengths of the other two sides. In this case, the unknown leg is approximately 8.9 feet, making choice D the correct answer. The other options are inconsistent with the properties of a right triangle and the application of the Pythagorean theorem.
A teacher has asked all the students in the class which days of the week they get up after 8 a.m. Which of the following is the best way to display the frequency for each day of the week?
Rationale
Bar graph is the best way to display the frequency for each day of the week.
A bar graph effectively represents categorical data, such as the frequency of students getting up after 8 a.m. on different days of the week. Each day can be distinctly labeled on the x-axis, while the y-axis indicates the frequency, making it easy to compare the data visually.
A) Pie graph A pie graph is used to represent parts of a whole and is best suited for categorical data that sums to 100%. While it can show the proportion of students getting up after 8 a.m. across the week, it does not effectively display the individual frequencies for each day, making comparisons difficult.
B) Histogram A histogram is designed for displaying the distribution of numerical data and requires continuous data, divided into intervals or "bins." In this case, since the question concerns categorical data related to specific days of the week, a histogram is not appropriate for showcasing frequency.
C) Bar graph The bar graph excels in illustrating categorical data frequencies. Each day of the week can be represented by a separate bar, allowing for straightforward comparisons of how many students report getting up after 8 a.m. on each specific day. This visual clarity is why it is the best choice.
D) Scatterplot A scatterplot is mainly used to display relationships between two continuous variables. In this scenario, where we are dealing with categorical days of the week and their corresponding frequencies, a scatterplot is not suitable as it does not convey the necessary information effectively.
Conclusion To summarize, a bar graph is the most effective method for displaying the frequency of students getting up after 8 a.m. on different days of the week. It provides a clear visual comparison of categorical data, unlike pie graphs, histograms, or scatterplots, which are ill-suited for this type of analysis. By using a bar graph, the teacher can easily interpret the data and identify patterns in students' routines.
A gardener is planning to plant a border of flowers around a rectangular garden that measures 15 meters in length and 10 meters in width. Which of the following is the total length of flower border needed for the garden?
Rationale
50 meters of flower border is needed for the garden.
To find the total length of the flower border around the rectangular garden, we calculate the perimeter using the formula: Perimeter = 2 × (length + width). For this garden, the perimeter is 2 × (15 m + 10 m) = 2 × 25 m = 50 m.
A) 150 meters This option incorrectly suggests that the flower border length is much larger than it actually is. A perimeter of 150 meters would imply a much larger garden than the given dimensions of 15 meters by 10 meters.
B) 50 meters This choice correctly represents the calculated perimeter of the garden. It accurately reflects the total length of flower border needed, derived from adding the length and width of the garden and multiplying by two.
C) 25 meters This option misrepresents the perimeter calculation. The value of 25 meters represents the sum of the length and width of the garden (15 m + 10 m), but the total length of the border requires multiplying this sum by 2, resulting in 50 meters, not 25.
D) 100 meters Selecting this option suggests a misunderstanding of the perimeter formula. A perimeter of 100 meters would require a garden of dimensions that are significantly larger than the given 15 meters by 10 meters. This indicates a miscalculation or confusion in the application of the perimeter formula.
Conclusion In summary, the total length of flower border needed for the rectangular garden is 50 meters, correctly calculated from the perimeter formula. Understanding how to compute the perimeter is essential for planning landscaping projects accurately. The other options either miscalculate or misinterpret the dimensions of the garden, leading to incorrect perimeter values.
Which of the following fractions is equivalent to 12/16?
Rationale
{6/8} is equivalent to 12/16.
To determine if a fraction is equivalent to another, both fractions must simplify to the same value. In this case, 12/16 simplifies to 3/4, and {6/8} also simplifies to 3/4, confirming their equivalence.
A) {6/8} This fraction can be simplified by dividing both the numerator and denominator by 2, resulting in {3/4}. This matches the simplified form of 12/16, which also equals 3/4, confirming that {6/8} is indeed equivalent to 12/16.
B) {4/3} This fraction equals approximately 1.33 when expressed as a decimal. In contrast, 12/16 simplifies to 0.75, indicating that {4/3} is not equivalent to 12/16, as their values do not match.
C) {20/32} When simplified by dividing both the numerator and denominator by 4, {20/32} becomes {5/8}. Since 5/8 is approximately 0.625, it does not match the simplified form of 12/16, which is 0.75. Therefore, {20/32} is not equivalent to 12/16.
D) {2/6} This fraction simplifies to {1/3} when both the numerator and denominator are divided by 2. Since 1/3 approximates to about 0.33, it does not equal 0.75, confirming that {2/6} is not equivalent to 12/16.
Conclusion When comparing fractions, equivalence is found when they simplify to the same value. Here, {6/8} simplifies to the same number as 12/16, while the other choices do not. Understanding such equivalences is essential in mathematics, enabling accurate fraction manipulation and comparison.
Based on the table showing healthcare spending per capita in a group of African nations, which country experienced the greatest change in dollars spent per capita from 2013 to 2015?
Rationale
Tanzania experienced the greatest change in dollars spent per capita from 2013 to 2015.
The difference in healthcare spending per capita for Tanzania between 2013 and 2015 is $60 ($68 in 2013 minus $8 in 2015), which is larger than the differences for the other countries listed.
A) Eritrea 2013 ($56), 2015 ($42) The difference in healthcare spending per capita for Eritrea between 2013 and 2015 is $14 ($56 in 2013 minus $42 in 2015). This is smaller than the difference for Tanzania.
B) Nigeria 2013 ($16), 2015 ($21) The difference in healthcare spending per capita for Nigeria between 2013 and 2015 is $5 ($21 in 2015 minus $16 in 2013). This is smaller than the difference for Tanzania.
C) Tanzania 2013 ($68), 2015 ($8) The difference in healthcare spending per capita for Tanzania between 2013 and 2015 is $60 ($68 in 2013 minus $8 in 2015). This is the largest difference among the four countries listed.
D) Kenya 2013 ($81), 2015 ($88) The difference in healthcare spending per capita for Kenya between 2013 and 2015 is $7 ($88 in 2015 minus $81 in 2013). This is smaller than the difference for Tanzania.
Conclusion The change in healthcare spending per capita from 2013 to 2015 is calculated by subtracting the value in 2013 from the value in 2015. The country that experienced the greatest change in healthcare spending per capita during this period is Tanzania, with a difference of $60. The differences for Eritrea, Nigeria, and Kenya are smaller, at $14, $5, and $7, respectively.
A professional football stadium has 36 sections labeled 101-136. Each section has 38 rows of standard stadium seating. Each row of each section on this level averages 22 seats. Estimate the number of seats available in the stadium by rounding each value to the nearest tens place.
Rationale
Approximately 32,000 seats are available in the stadium.
To estimate the total number of seats, we round each component to the nearest ten: 36 sections to 40, 38 rows to 40, and 22 seats per row to 20. Multiplying these rounded values (40 sections × 40 rows × 20 seats) gives us an estimated total of 32,000 seats.
A) 30096 This option is based on a more precise calculation rather than estimation. The actual calculation using the original values (36 sections × 38 rows × 22 seats) results in a figure that does not align with the rounded estimation method required in the question. Thus, it does not reflect the estimated total.
B) 48000 This choice overestimates the number of seats by rounding the sections and rows too high. Rounding 36 sections to 40 and 38 rows to 40, while using 22 seats rounded to 30, leads to an inflated total. Therefore, this option fails to provide a reasonable estimate.
C) 32000 This option accurately represents the estimated total based on rounding the components to the nearest ten: 40 sections, 40 rows, and 20 seats. The multiplication of these rounded values yields the correct estimated total of 32,000 seats available in the stadium.
D) 18000 This answer significantly underestimates the total number of seats. By not rounding the values to the nearest ten correctly or by miscalculating, this figure does not reflect the capacity of the stadium as per the provided data.
Conclusion Estimating the number of seats in the stadium involves rounding each value to the nearest ten and calculating the total. With 40 sections, 40 rows, and 20 seats per row, we arrive at an estimated total of approximately 32,000 seats. This estimation method highlights the importance of rounding in simplifying calculations while still achieving a reasonable approximation of the total seating capacity.
Which of the following values is the greatest?
Rationale
9/2 is the greatest value among the choices.
When converted to decimal form, 9/2 equals 4.5, which is greater than all other provided values. This makes it the highest value in the given set of options.
A) 4.25 4.25 is a decimal number that equals 4.25. When compared to 9/2 (4.5), it is clear that 4.25 is less than 4.5, making it not the greatest value.
B) 4.4 4.4 is another decimal number, which is equivalent to 4.4. Although it is close to 4.5, it is still less than 9/2. Therefore, it cannot be considered the greatest value.
C) {10/3} 10/3 can be calculated as approximately 3.33 in decimal form. This value is significantly less than 4.5, thus it is not the highest among the options presented.
D) {9/2} 9/2, which equals 4.5, is indeed the greatest value in this set. This fraction represents a value higher than all the other options provided.
Conclusion In comparing the values 4.25, 4.4, 10/3, and 9/2, 9/2 stands out as the largest value at 4.5. The other options fall short in comparison, clearly establishing 9/2 as the greatest. Understanding such comparisons is essential in mathematical evaluations to determine the highest numerical value accurately.
An employer wants to purchase one cup of coffee for each of their 214 employees. A cup of coffee costs $3.95. To estimate the total cost, the employer plans to round the number of employees to the nearest ten and round the cost of the coffee to the nearest dollar. Which of the following prices represents the estimated total cost for the coffee?
Rationale
The estimated total cost for the coffee is $840.00.
The employer plans to round the number of employees to the nearest ten and the cost of the coffee to the nearest dollar for estimation. Thus, 214 employees round to 210 and $3.95 rounds to $4.00. Multiplying these rounded values gives an estimated total cost of $840.00.
A) $845.30 This choice might be the result of rounding only the number of employees (214 to 210) but not the cost of coffee. Multiplying 210 employees with the exact cost of coffee ($3.95) results in $829.50, not $845.30. Therefore, this option is incorrect.
B) $840.00 This choice correctly represents the product of the rounded number of employees (210) and the rounded cost of coffee ($4.00). Hence, this is the correct answer.
C) $829.50 This choice appears to be the result of an error in rounding. It seems the number of employees was rounded correctly to 210 but the cost of the coffee was not rounded. Multiplying 210 employees with the exact cost of coffee ($3.95) results in $829.50. However, the employer wanted to round both values for estimation, so this option is incorrect.
D) $850.00 This choice seems to be the result of an overestimate. Neither the rounded number of employees (210) nor the rounded cost of coffee ($4.00) would result in this total. Therefore, this option is incorrect.
Conclusion In this situation, the employer wants to estimate the total cost by rounding both the number of employees and the cost of coffee. By rounding 214 employees to 210 and the cost of coffee from $3.95 to $4.00, the estimated total comes to $840.00. All other choices appear to be based on errors in rounding or calculation.
Which of the following is the correct simplification of the expression below?
Rationale
16
To simplify the expression correctly, we perform the arithmetic operations as indicated. The final result of the calculations yields the value of 16, making it the correct simplification of the expression.
A) -8 This result is incorrect because it suggests that the operations performed led to a negative outcome, which does not align with the arithmetic calculations involved in the expression. The operations are designed to yield a positive result, and a final value of -8 indicates a miscalculation in either addition or subtraction.
B) 16 This is the correct answer as it accurately reflects the outcome of the arithmetic operations applied in the expression. Upon simplification, the terms combine to yield a total of 16, confirming that this is the right choice.
C) -6 This choice is incorrect because a result of -6 implies that the calculations involved either an incorrect addition or subtraction of positive and negative numbers. The expression does not lead to a total that could result in a negative number, thus making this option invalid.
D) 18 Choosing 18 suggests an overestimation of the total derived from the expression. This indicates that either too many positive values were counted or a mistake was made in the arithmetic operations. The actual simplification does not support this outcome.
Conclusion The correct simplification of the expression yields the answer 16, which accurately reflects the proper execution of arithmetic operations. Other options, such as -8, -6, and 18, demonstrate either miscalculations or misunderstandings of the original expression's structure. Thus, 16 stands as the only correct and logical outcome.
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