Comparing & Ordering Rational Numbers - Accuplacer Qas

For which of the following integer values of $n$ is the inequality $\frac{2}{5} < \frac{n}{12} < \frac{3}{4}$ true?
Answer: 5 To solve the inequality $\frac{2}{5} < \frac{n}{12} < \frac{3}{4}$ , we can multiply all parts of the inequality by 12 to isolate $n$ . $12 \cdot \frac{2}{5} < 12 \cdot \frac{n}{12} < 12 \cdot \frac{3}{4}$ $\frac{24}{5} < n < 9$ Now, convert $\frac{24}{5}$ to a decimal: $4.8$ . The inequality becomes $4.8 < n < 9$ . The integers $n$ that satisfy this condition are 5, 6, 7, and 8. Of the choices given, only $n=5$ is in this range.
If $a = -\frac{3}{2}$ , $b = \frac{5}{4}$ , and $c = -\frac{7}{5}$ , which of the following is the correct ordering of $a$ , $b$ , and $c$ from least to greatest?
Answer: $a < c < b$ To order the numbers, it is helpful to convert them to decimals. $a = -\frac{3}{2} = -1.5$ $b = \frac{5}{4} = 1.25$ $c = -\frac{7}{5} = -1.4$ Now, order the decimal values from least to greatest. The only positive number, 1.25, is the greatest. Comparing the negative numbers, -1.5 is less than -1.4. So, the order is $-1.5 < -1.4 < 1.25$ . This corresponds to the order $a < c < b$ .
Three shelves are being installed. Shelf A has a length of $3 \frac{3}{8}$ feet. Shelf B has a length of $3.4$ feet. Shelf C has a length of $\frac{10}{3}$ feet. Which of the following correctly orders the shelves from shortest to longest?
Answer: C, A, B To compare the lengths, convert all measurements to the same format, such as decimals. Shelf A: $3 \frac{3}{8} = 3.375$ feet. Shelf B: $3.4$ feet. Shelf C: $\frac{10}{3} = 3.333...$ feet. Now, compare the decimal values: $3.333... < 3.375 < 3.4$ . Therefore, the correct order from shortest to longest is C, A, B.
Let $x = -\frac{5}{4}$ . Which of the following lists the values $x$ , $|x|$ , and $x+2$ in order from least to greatest?
Answer: $x, x+2, |x|$ First, calculate the value of each expression given $x = -\frac{5}{4}$ . $x = -\frac{5}{4} = -1.25$ $|x| = |-\frac{5}{4}| = \frac{5}{4} = 1.25$ $x+2 = -\frac{5}{4} + 2 = -\frac{5}{4} + \frac{8}{4} = \frac{3}{4} = 0.75$ The three values are -1.25, 1.25, and 0.75. Ordering these from least to greatest gives: $-1.25 < 0.75 < 1.25$ . This corresponds to the order $x, x+2, |x|$ .
A student claims that $\frac{-7}{12} > -\frac{3}{5}$ because "7 is greater than 3 and 12 is greater than 5." What is the correct relationship between these fractions?
Answer: The student is incorrect; $\frac{-7}{12} > -\frac{3}{5}$ but not for the stated reason To compare $\frac{-7}{12}$ and $-\frac{3}{5}$ , find a common denominator. LCM of 12 and 5 is 60. $\frac{-7}{12} = \frac{-35}{60}$ and $-\frac{3}{5} = \frac{-36}{60}$ . Since $-35 > -36$ , we have $\frac{-7}{12} > -\frac{3}{5}$ . The student's conclusion is correct, but the reasoning is flawed. You cannot compare negative fractions by simply comparing the absolute values of numerators and denominators. The correct comparison requires finding equivalent fractions with the same denominator or converting to decimals: $\frac{-7}{12} ≈ -0.583$ and $-\frac{3}{5} = -0.6$ . Since $-0.583 > -0.6$ , the inequality $\frac{-7}{12} > -\frac{3}{5}$ is true.
A temperature reading was $-\frac{4}{5}$ °C. An hour later, it was $-\frac{5}{6}$ °C. Which of the following temperatures was recorded between these two readings?
Answer: -0.82°C First, convert the fractional temperatures to decimals to establish the range. $-\frac{4}{5} = -0.8$ °C. $-\frac{5}{6} \approx -0.833$ °C. Since $-0.833 < -0.8$ , the temperature dropped. We are looking for a temperature $T$ such that $-\frac{5}{6} < T < -\frac{4}{5}$ , or $-0.833 < T < -0.8$ . A) -0.9 is less than -0.833. B) -0.85 is less than -0.833. C) -0.82 is between -0.833 and -0.8. D) -0.79 is greater than -0.8. Therefore, -0.82°C is a temperature between the two readings.
Let $x = \frac{3/4}{2/5}$ and $y = \frac{4/5}{1/2}$ . Which of the following inequalities is true?
Answer: $x > y$ First, simplify the expressions for $x$ and $y$ . $x = \frac{3/4}{2/5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$ $y = \frac{4/5}{1/2} = \frac{4}{5} \times \frac{2}{1} = \frac{8}{5}$ Now, compare $x = \frac{15}{8}$ and $y = \frac{8}{5}$ . Convert them to decimals: $x = 1.875$ and $y = 1.6$ . Since $1.875 > 1.6$ , we have $x > y$ .
Which of the following numbers is farthest from $-\frac{2}{3}$ on the number line?
Answer: 1 To find which number is farthest, we calculate the absolute distance between each choice and $-\frac{2}{3}$ . A) Distance from -2: $|-2 - (-\frac{2}{3})| = |-\frac{6}{3} + \frac{2}{3}| = |-\frac{4}{3}| = \frac{4}{3}$ B) Distance from $-\frac{1}{2}$ : $|-\frac{1}{2} - (-\frac{2}{3})| = |-\frac{3}{6} + \frac{4}{6}| = |\frac{1}{6}| = \frac{1}{6}$ C) Distance from 0: $|0 - (-\frac{2}{3})| = |\frac{2}{3}| = \frac{2}{3}$ D) Distance from 1: $|1 - (-\frac{2}{3})| = |\frac{3}{3} + \frac{2}{3}| = |\frac{5}{3}| = \frac{5}{3}$ Comparing the distances $\frac{4}{3}, \frac{1}{6}, \frac{2}{3}, \frac{5}{3}$ , the largest value is $\frac{5}{3}$ . Therefore, 1 is the farthest from $-\frac{2}{3}$ .
What rational number is exactly halfway between $-\frac{3}{5}$ and $\frac{7}{10}$ on the number line?
Answer: $\frac{1}{20}$ To find the number exactly halfway between two numbers, we calculate their average. First, express both numbers with a common denominator, which is 10. $-\frac{3}{5} = -\frac{6}{10}$ . The two numbers are $-\frac{6}{10}$ and $\frac{7}{10}$ . Average = $\frac{-\frac{6}{10} + \frac{7}{10}}{2} = \frac{\frac{1}{10}}{2} = \frac{1}{10} \times \frac{1}{2} = \frac{1}{20}$ .
Let $A = \frac{5}{6} - \frac{2}{5}$ and $B = \frac{2}{3} - \frac{1}{4}$ . Which of the following statements is true?
Answer: $A > B$ First, calculate the value of A: $A = \frac{5}{6} - \frac{2}{5}$ . The common denominator is 30. $A = \frac{25}{30} - \frac{12}{30} = \frac{13}{30}$ . Next, calculate the value of B: $B = \frac{2}{3} - \frac{1}{4}$ . The common denominator is 12. $B = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}$ . Now, compare A and B. To compare $\frac{13}{30}$ and $\frac{5}{12}$ , find a common denominator, which is 60. $A = \frac{13}{30} = \frac{26}{60}$ and $B = \frac{5}{12} = \frac{25}{60}$ . Since $\frac{26}{60} > \frac{25}{60}$ , it follows that $A > B$ .
The number $k$ is a rational number such that $k > 1$ . The number $m$ is a rational number such that $0 < m < 1$ . Which of the following inequalities must be true?
Answer: $k > m^2$ Let's analyze the properties of $k$ and $m$ . If $k > 1$ , squaring it results in a larger number. For example, if $k=2$ , $k^2=4$ , so $k^2 > k$ . This makes choice A false. If $0 < m < 1$ , squaring it results in a smaller number. For example, if $m=0.5$ , $m^2=0.25$ , so $m^2 < m$ . This makes choice B false. Now let's compare $k$ and $m^2$ . We know $k > 1$ . We also know that if $0 < m < 1$ , then $0 < m^2 < 1$ . Since $k$ is greater than 1 and $m^2$ is less than 1, it must be true that $k > m^2$ . This makes choice C correct. For choice D, $m > k^2$ is false because $m$ is less than 1 and $k^2$ is greater than 1.
If $x = -\frac{2}{3}$ , which of the following expressions has the greatest value?
Answer: $-x$ We evaluate each expression with $x = -\frac{2}{3}$ . A) $x = -\frac{2}{3}$ B) $x^2 = (-\frac{2}{3})^2 = \frac{4}{9}$ C) $\frac{1}{x} = \frac{1}{-2/3} = -\frac{3}{2}$ D) $-x = -(-\frac{2}{3}) = \frac{2}{3}$ Now, we compare the values: $-\frac{2}{3} \approx -0.667$ , $\frac{4}{9} \approx 0.444$ , $-\frac{3}{2} = -1.5$ , and $\frac{2}{3} \approx 0.667$ . The greatest value among these is $\frac{2}{3}$ , which corresponds to the expression $-x$ .
Three friends are running a race. After 10 minutes, Alex has completed $\frac{2}{5}$ of the race, Ben has completed $\frac{3}{8}$ of the race, and Carlos has completed 0.41 of the race. Who is in the lead, and who is in last place?
Answer: Carlos is in the lead; Ben is in last place. To determine the order, we need to compare the fractions of the race each person has completed. It's easiest to convert all numbers to decimals. Alex: $\frac{2}{5} = 0.40$ Ben: $\frac{3}{8} = 0.375$ Carlos: $0.41$ Comparing the decimals: $0.41 > 0.40 > 0.375$ . This means Carlos has completed the most of the race and is in the lead. Ben has completed the least and is in last place.
Which of the following rational numbers is located between $\frac{4}{7}$ and $\frac{5}{8}$ on a number line?
Answer: $\frac{3}{5}$ To find a number between $\frac{4}{7}$ and $\frac{5}{8}$ , we can convert the fractions to decimals. $\frac{4}{7} \approx 0.5714$ and $\frac{5}{8} = 0.625$ . We need to find the answer choice that falls between these two values. A) $\frac{9}{16} = 0.5625$ , which is less than 0.5714. B) $\frac{3}{5} = 0.6$ , which is between 0.5714 and 0.625. C) $\frac{5}{7} \approx 0.714$ , which is greater than 0.625. D) $\frac{7}{8} = 0.875$ , which is greater than 0.625.
Which of the following inequalities is a true statement?
Answer: $-\frac{3}{4} < -\frac{8}{11}$ To compare the negative fractions, we can convert them to decimals or use a common denominator. For negative numbers, the number with the smaller absolute value is greater. A) $-\frac{5}{8} = -0.625$ and $-\frac{3}{5} = -0.6$ . Since $-0.625 < -0.6$ , this statement is false. B) $-\frac{7}{9} \approx -0.778$ and $-\frac{6}{7} \approx -0.857$ . Since $-0.778 > -0.857$ , this statement is false. C) $-\frac{2}{3} \approx -0.667$ and $-\frac{5}{8} = -0.625$ . Since $-0.667 < -0.625$ , this statement is false. D) $-\frac{3}{4} = -0.75$ and $-\frac{8}{11} \approx -0.727$ . Since $-0.75 < -0.727$ , this statement is true.
A recipe requires $\frac{3}{4}$ cup of flour. A baker has a scoop that holds $\frac{1}{3}$ cup and another that holds $\frac{1}{2}$ cup. The baker uses two scoops of the smaller size and one scoop of the larger size. Which statement accurately compares the amount of flour used to the amount required?
Answer: The amount used is $\frac{5}{12}$ cup more than required. First, calculate the total amount of flour used. Two scoops of $\frac{1}{3}$ cup is $2 \times \frac{1}{3} = \frac{2}{3}$ cup. One scoop of $\frac{1}{2}$ cup is $\frac{1}{2}$ cup. The total amount used is $\frac{2}{3} + \frac{1}{2}$ . To add these, find a common denominator, which is 6: $\frac{4}{6} + \frac{3}{6} = \frac{7}{6}$ cup. Next, compare the amount used ( $\frac{7}{6}$ cup) to the amount required ( $\frac{3}{4}$ cup). Find a common denominator, which is 12. Amount used: $\frac{7}{6} = \frac{14}{12}$ . Amount required: $\frac{3}{4} = \frac{9}{12}$ . The difference is $\frac{14}{12} - \frac{9}{12} = \frac{5}{12}$ . Since the amount used is greater than the amount required, the baker used $\frac{5}{12}$ cup more than required.
Three pumps remove water from a tank at rates of $\frac{3}{8}$ tank per hour, $0.4$ tank per hour, and $\frac{7}{20}$ tank per hour, respectively. If only one pump can operate at a time, which pump should be used to empty the tank fastest?
Answer: The second pump should be used because $0.4$ tank per hour is the fastest rate To find the fastest pump, convert all rates to the same form and compare. $\frac{3}{8} = 0.375$ , $0.4 = 0.4$ , and $\frac{7}{20} = 0.35$ . Comparing: $0.4 > 0.375 > 0.35$ . Therefore, the second pump with rate $0.4$ tank per hour is fastest. Choice A is incorrect because having the largest numerator doesn't guarantee the largest fraction value. Choice C is incorrect because $\frac{7}{20} = 0.35$ is actually the slowest rate. Choice D is incorrect because $\frac{3}{8} = 0.375$ is not the fastest rate.
Which of the following fractions has the greatest value?
Answer: $\frac{17}{19}$ One way to compare these fractions is to consider how far each is from 1. A) $1 - \frac{11}{13} = \frac{2}{13}$ B) $1 - \frac{13}{15} = \frac{2}{15}$ C) $1 - \frac{15}{17} = \frac{2}{17}$ D) $1 - \frac{17}{19} = \frac{2}{19}$ The fraction with the greatest value will be the one that is closest to 1, meaning its difference from 1 is the smallest. We need to find the smallest fraction among $\frac{2}{13}, \frac{2}{15}, \frac{2}{17}, \frac{2}{19}$ . Since all these fractions have the same numerator (2), the one with the largest denominator will be the smallest. The largest denominator is 19, so $\frac{2}{19}$ is the smallest difference. Therefore, $\frac{17}{19}$ is the fraction closest to 1 and has the greatest value.
A point $P$ on a number line is located at $-\frac{13}{8}$ . Which of the following points is closest to $P$ ?
Answer: -1.6 First, convert $-\frac{13}{8}$ to a decimal: $-13 \div 8 = -1.625$ . Now, find the distance from -1.625 to each of the given points. A) Distance to -1.75: $|-1.75 - (-1.625)| = |-0.125| = 0.125$ B) Distance to -1.6: $|-1.6 - (-1.625)| = |0.025| = 0.025$ C) Distance to $-\frac{11}{7} \approx -1.571$ : $|-1.571 - (-1.625)| = |0.054| = 0.054$ D) Distance to $-\frac{5}{3} \approx -1.667$ : $|-1.667 - (-1.625)| = |-0.042| = 0.042$ Comparing the distances (0.125, 0.025, 0.054, 0.042), the smallest distance is 0.025. Therefore, -1.6 is the closest point to $P$ .
The ideal weight for a manufactured part is $\frac{7}{8}$ ounces. A part is acceptable if its weight is within $\frac{1}{32}$ ounce of the ideal weight. Which of the following weights would cause a part to be rejected?
Answer: $\frac{13}{16}$ ounces First, find the acceptable range of weights. The ideal weight is $\frac{7}{8}$ oz. Convert this to a fraction with a denominator of 32: $\frac{7}{8} = \frac{28}{32}$ oz. The acceptable range is $\frac{28}{32} \pm \frac{1}{32}$ , which is from $\frac{27}{32}$ oz to $\frac{29}{32}$ oz, inclusive. A part is rejected if its weight is outside this range. A) $\frac{13}{16} = \frac{26}{32}$ . This weight is less than $\frac{27}{32}$ , so it is outside the acceptable range and would be rejected. B) $\frac{55}{64} = \frac{27.5}{32}$ . This is within the range $[\frac{27}{32}, \frac{29}{32}]$ , so it is acceptable. C) $\frac{7}{8} = \frac{28}{32}$ . This is the ideal weight, so it is acceptable. D) $\frac{29}{32}$ . This is the upper limit of the acceptable range, so it is acceptable.

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