Ordering Rational Numbers - SSAT Upper Level Quantitative

Which value is between 0.44 and 4/9?
Answer: 0.4425 Rewrite 4/9 as 0.4444..., so you need a value greater than 0.44 but less than 0.4444. Since 0.4400 < 0.4425 < 0.4444, 0.4425 fits. The tempting 0.4448 is wrong because it is greater than 0.4444..., not between them.
Which of the following is true?
Answer: -2/3 < -0.5 Compare the values on a number line. Since -2/3 is about -0.666 and -0.5 is -0.5, -2/3 is farther left, so it is less than -0.5. The tempting wrong comparison is -0.7 > -1/4, but -0.7 is actually less than -0.25 because the farther a negative number is from zero, the smaller it is.
Which of the following lists these interest rates in ascending order: $12\%$ , $0.09$ , $\frac{1}{10}$ , $8\%$ , $0.11$ ?
Answer: $8\%,\ 0.09,\ \frac{1}{10},\ 0.11,\ 12\%$ This question tests SSAT Upper Level quantitative skills: comparing and ordering integers and rational numbers. Students need to understand how to accurately compare and order numbers presented in various formats such as fractions, decimals, and percentages. For this question, the interest rates must be arranged in ascending order by converting all to decimals: 12% = 0.12, 0.09 = 0.09, 1/10 = 0.10, 8% = 0.08, and 0.11 = 0.11. The correct answer is choice A because it accurately represents the numbers in ascending order: 8% (0.08), 0.09, 1/10 (0.10), 0.11, 12% (0.12). A common mistake is shown in choice B, where students might incorrectly place 0.09 before 8%, not recognizing that 8% = 0.08 is smaller than 0.09. To assist students: Encourage practice with converting percentages to decimals by dividing by 100, and fractions to decimals through division. Provide exercises that mix all three formats to build confidence in ordering rational numbers.
Three real numbers $r$ , $s$ , and $t$ satisfy the following conditions: $r < s < t$ , $r + s + t = 0$ , and $rst = \frac{1}{8}$ . If $r = -\frac{1}{2}$ , which of the following represents the correct ordering of $|r|$ , $|s|$ , and $|t|$ from least to greatest?
Answer: $|s|, |r|, |t|$ Since $r + s + t = 0$ and $r = -\frac{1}{2}$ , we get $s + t = \frac{1}{2}$ . Since $rst = \frac{1}{8}$ , we get $-\frac{1}{2}st = \frac{1}{8}$ , so $st = -\frac{1}{4}$ . So $s$ and $t$ are the roots of $x^2 - \frac{1}{2}x - \frac{1}{4} = 0$ , or $4x^2 - 2x - 1 = 0$ . The quadratic formula gives $x = \frac{2 \pm \sqrt{4 + 16}}{8} = \frac{1 \pm \sqrt{5}}{4}$ . Because $s < t$ , $s = \frac{1 - \sqrt{5}}{4} \approx -0.309$ and $t = \frac{1 + \sqrt{5}}{4} \approx 0.809$ , and both exceed $r = -0.5$ . The absolute values are $|s| \approx 0.309$ , $|r| = 0.5$ , $|t| \approx 0.809$ , so the order is $|s|, |r|, |t|$ . The order $|r|, |s|, |t|$ copies the order of the numbers themselves, forgetting that taking absolute values makes the negatives positive. The orders ending in $|r|$ assume the most negative number has the largest size, but $t$ is farther from zero than $r$ .
A science experiment lists solution concentrations: $0.125$ , $\frac{1}{10}$ , $13\%$ , $0.12$ , $\frac{3}{25}$ . Which is greatest?
Answer: $13\%$ This question tests SSAT Upper Level quantitative skills: comparing and ordering integers and rational numbers. Students need to understand how to accurately compare and order numbers presented in various formats such as fractions, decimals, and percentages. For this question, students must identify the greatest concentration by converting all to decimals: 0.125 = 0.125, 1/10 = 0.10, 13% = 0.13, 0.12 = 0.12, and 3/25 = 0.12. The correct answer is choice C because 13% = 0.13 is the largest value among all the options. A common mistake might be choosing choice B (0.125), where students might think the number with more decimal places is larger, not recognizing that 0.13 > 0.125. To assist students: Emphasize place value understanding and that 0.13 = 0.130 when comparing decimals. Provide exercises focusing on decimal comparison with different numbers of decimal places.
Scores: 7/8, 0.82, 83%. Which shows least-to-greatest order?
Answer: 0.82 < 83% < 7/8 Convert each to decimals: 7/8 = 0.875, 83% = 0.83, and 0.82 stays 0.82. So from least to greatest: 0.82, 0.83, 0.875. The tempting error is putting 7/8 before 83%, but 7/8 is actually 0.875, the largest value.
What is the least integer greater than -7/4?
Answer: -1 Since -7/4 equals -1.75, the integers greater than it are -1, 0, 1, and so on. The least of those is -1. A common trap is choosing -2, but -2 is less than -1.75, so it is not greater than -7/4.
If $a = -\frac{5}{8}$ , $b = -0.62$ , and $c = -\frac{3}{5}$ , which of the following correctly orders these three numbers from least to greatest?
Answer: $a, b, c$ First convert all numbers to decimals: $a = -\frac{5}{8} = -0.625$ , $b = -0.62$ , and $c = -\frac{3}{5} = -0.6$ . Since these are all negative numbers, the number closest to zero is the greatest. Comparing: $-0.625 < -0.62 < -0.6$ , so $a < b < c$ . Therefore, from least to greatest: $a, b, c$ .
In a science lab, which recorded value is the greatest: $0.62$ , $\frac{5}{8}$ , $61\%$ , $0.59$ , or $\frac{3}{5}$ ?
Answer: $\frac{5}{8}$ This question tests SSAT Upper Level quantitative skills: comparing and ordering integers and rational numbers. Students need to understand how to accurately compare and order numbers presented in various formats such as fractions, decimals, and percentages. For this question, students must identify the greatest value by converting all to decimals: 0.62 = 0.62, 5/8 = 0.625, 61% = 0.61, 0.59 = 0.59, and 3/5 = 0.60. The correct answer is choice B because 5/8 = 0.625 is the largest value among all the options. A common mistake might be choosing choice A (0.62), where students fail to recognize that 5/8 = 0.625 is actually greater than 0.62. To assist students: Encourage careful conversion of fractions to decimals and remind them that 5/8 means 5 ÷ 8. Provide practice problems that require finding the maximum or minimum value among mixed formats.
Which list is ordered from least to greatest?
Answer: -7/8, -3/4, -0.6, -1/2 On the negative side, the smaller number has the larger absolute value. Since -7/8 = -0.875, -3/4 = -0.75, -0.6, and -1/2 = -0.5, the order from least to greatest is -7/8, -3/4, -0.6, -1/2. A tempting wrong answer is -1/2, -0.6, -3/4, -7/8, because that lists them from greatest to least instead.
A sequence of rational numbers is defined by the pattern: $-\frac{3}{2}, -\frac{5}{4}, -\frac{7}{8}, -\frac{9}{16}, ...$ If this pattern continues, which of the following represents the correct ordering of the 6th, 7th, and 8th terms from least to greatest?
Answer: 6th, 7th, 8th First, identify the pattern. The numerators are $-3, -5, -7, -9, ...$ which follows $-(2n+1)$ for $n = 1, 2, 3, 4, ...$ The denominators are $2, 4, 8, 16, ...$ which follows $2^n$ . So the $n$ th term is $-\frac{2n+1}{2^n}$ . The 6th term: $-\frac{13}{64} ≈ -0.203$ . The 7th term: $-\frac{15}{128} ≈ -0.117$ . The 8th term: $-\frac{17}{256} ≈ -0.066$ . Since $-0.203 < -0.117 < -0.066$ , we have 6th < 7th < 8th. Therefore, from least to greatest: 6th, 7th, 8th.
If $p$ and $q$ are integers such that $\frac{p}{q} = 0.\overline{36}$ , what is the value of $p + q$ when the fraction is in lowest terms?
Answer: 15 Let $x = 0.\overline{36} = 0.363636...$ Then $100x = 36.363636...$ Subtracting: $100x - x = 36.363636... - 0.363636...$ gives $99x = 36$ , so $x = \frac{36}{99}$ . To reduce to lowest terms, find GCD(36, 99): $36 = 2^2 \times 3^2$ and $99 = 3^2 \times 11$ , so GCD = 9. Therefore $\frac{36}{99} = \frac{4}{11}$ . Thus $p = 4$ and $q = 11$ , so $p + q = 15$ .
At a store, rank these discounts from smallest to largest value: $\frac{1}{5}$ , $0.18$ , $15\%$ , $\frac{3}{16}$ , $0.22$ .
Answer: $15\%,\ 0.18,\ \frac{3}{16},\ \frac{1}{5},\ 0.22$ This question tests SSAT Upper Level quantitative skills: comparing and ordering integers and rational numbers. Students need to understand how to accurately compare and order numbers presented in various formats such as fractions, decimals, and percentages. For this question, the discounts must be arranged from smallest to largest by converting all values to decimals: 15% = 0.15, 1/5 = 0.20, 0.18 = 0.18, 3/16 = 0.1875, and 0.22 = 0.22. The correct answer is choice B because it accurately represents the numbers in the correct order: 15% (0.15), 0.18, 3/16 (0.1875), 1/5 (0.20), 0.22. A common mistake is shown in choice A, where students might place 1/5 before 3/16, not realizing that 1/5 = 0.20 is greater than 3/16 = 0.1875. To assist students: Encourage practice with converting between formats and stress the importance of attention to detail with decimal placement. Provide exercises that involve ordering mixed formats to develop flexibility in working with percentages, fractions, and decimals.

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