ISEE Middle Level Quantitative Reasoning Quiz: Comparing Rational Numbers

What this quiz covers

This quiz focuses on Comparing Rational Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Quantitative Reasoning.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 3/7
2. 11/25
3. 9/20 (correct answer)
4. 8/15

Explanation: To find a number between 4/9 and 52%, first convert both to decimals. 4/9 ≈ 0.444... and 52% = 0.52. We are looking for a number n such that 0.444... < n < 0.52. Let's evaluate the choices: A) 3/7 ≈ 0.428, which is less than 0.444... B) 11/25 = 0.44, which is less than 0.444... C) 9/20 = 0.45, which is between 0.444... and 0.52. D) 8/15 ≈ 0.533, which is greater than 0.52. Thus, 9/20 is the only value in the specified range.

Question 2

1. The quantity in Column A is greater.
2. The quantity in Column B is greater.
3. The two quantities are equal. (correct answer)
4. The relationship cannot be determined from the information given.

Explanation: The phrase 'percent of' implies multiplication. Column A is 0.87 * 50. Column B is 0.50 * 87. Due to the commutative property of multiplication (a * b = b * a), the order of the factors does not change the product. Therefore, 0.87 * 50 = 50 * 0.87. The two quantities are equal. We can calculate both to verify: 0.87 * 50 = 43.5. 0.50 * 87 = 43.5.

Question 3

1. Alex
2. Ben (correct answer)
3. Chandra
4. Alex and Chandra had the same highest score.

Explanation: To determine who had the highest score, we need to compare the three values by converting them to a common format, such as decimals. Alex's score: 4/5 = 0.80. Ben's score: 83% = 0.83. Chandra's score: 0.81. Comparing the decimals, 0.83 is the largest number. Therefore, Ben had the highest score.

Question 4

1. The quantity in Column A is greater. (correct answer)
2. The quantity in Column B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Explanation: Let's choose a value for x that is between -1 and 0. For example, let x = -1/2. Then Column A is -1/2. Column B is 1/(-1/2) = -2. Comparing -1/2 and -2, we see that -1/2 is greater because it is closer to zero on the number line. For any number x between -1 and 0, its reciprocal 1/x will be a number less than -1. A number between -1 and 0 is always greater than a number less than -1. Therefore, the quantity in Column A is always greater.

Question 5

1. Mixture A has a higher juice concentration than Mixture B.
2. Mixture B has a higher juice concentration than Mixture A. (correct answer)
3. Both mixtures have the same juice concentration.
4. The relationship cannot be determined from the information given.

Explanation: The concentration of juice in Mixture A is the amount of juice divided by the total amount of the mixture. The ratio is 2 parts juice to 5 parts water, so the total is 2+5=7 parts. The concentration is 2/7. To compare 2/7 with Mixture B's concentration of 30%, convert both to decimals or percentages. 2/7 ≈ 0.2857, which is 28.57%. Mixture B's concentration is 30%. Since 30% > 28.57%, Mixture B has a higher juice concentration.

Question 6

1. 5/8
2. 0.62
3. 62.8%
4. 13/20 (correct answer)

Explanation: To compare the values, convert all of them to decimals. 5/8 = 0.625. 0.62 remains 0.62. 62.8% = 0.628. 13/20 = 0.65. Comparing the decimals, 0.65 is the largest value. Therefore, 13/20 is the greatest.

Question 7

1. -0.61, -0.606, -3/5, -58% (correct answer)
2. -58%, -3/5, -0.606, -0.61
3. -0.61, -3/5, -58%, -0.606
4. -58%, -0.606, -3/5, -0.61

Explanation: First, convert all numbers to decimals to compare them easily. -3/5 = -0.6. -58% = -0.58. The numbers are -0.6, -0.61, -0.58, and -0.606. For negative numbers, the one with the largest absolute value is the smallest. So, the order from least to greatest is -0.61, -0.606, -0.6, -0.58. This corresponds to the list -0.61, -0.606, -3/5, -58%.

Question 8

1. The quantity in Column A is greater.
2. The relationship cannot be determined from the information given.
3. The two quantities are equal.
4. The quantity in Column B is greater. (correct answer)

Explanation: When comparing exponential expressions with the same positive integer exponent, you need to focus on the bases. Since both expressions have the same exponent $y$ (where $y$ is a positive integer), the key is determining which base is larger. Let's compare the bases: $\frac{3}{4} = 0.75$ and $\frac{4}{5} = 0.8$. Since $0.8 > 0.75$, we know that $\frac{4}{5} > \frac{3}{4}$. For any positive integer exponent, if the base is larger, the result will be larger. This means $\left(\frac{4}{5}\right)^y > \left(\frac{3}{4}\right)^y$ for any positive integer $y$. You can verify this with a simple example: when $y = 2$, $\left(\frac{3}{4}\right)^2 = \frac{9}{16} = 0.5625$ while $\left(\frac{4}{5}\right)^2 = \frac{16}{25} = 0.64$. Choice A incorrectly suggests Column A is greater, likely from miscomparing the fractions or confusing the relationship between bases and results. Choice B is wrong because we can definitively determine the relationship—we have enough information since both bases are positive and different. Choice C is incorrect because the bases are different, so the exponential expressions cannot be equal for any positive integer exponent. Therefore, D is correct: Column B is greater. Study tip: When comparing exponential expressions with the same positive exponent, convert the bases to decimals to quickly see which is larger. The larger base always produces the larger result.

Question 9

1. The quantity in Column A is greater. (correct answer)
2. The quantity in Column B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Explanation: When comparing two positive fractions, taking the reciprocal reverses the inequality. Since it is given that a < b, it follows that 1/a > 1/b. We can test this with numbers. Let a = 1/3 and b = 1/2. Both are between 0 and 1, and 1/3 < 1/2. Column A is 1/(1/3) = 3. Column B is 1/(1/2) = 2. Since 3 > 2, the quantity in Column A is greater. This holds for any values of a and b that fit the given conditions.

Question 10

1. The quantity in Column A is greater. (correct answer)
2. The quantity in Column B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Explanation: Calculate the sale price for each column. Column A: The discount is (1/5) * $90 =$18. The sale price is $90 -$18 = $72. Column B: The discount is 15$85, which is 0.15 * $85 =$12.75. The sale price is $85 -$12.75 = $72.25. Comparing the two prices,$72.25 is greater than $72. Therefore, the quantity in Column B is greater.

Question 11

1. The bike that is 1/3 off. (correct answer)
2. The bike that is 30% off.
3. Both bikes have the same sale price.
4. The relationship cannot be determined without knowing the final prices.

Explanation: To find the lower sale price, we must determine which discount is larger. We need to compare 1/3 and 30%. Let's convert 1/3 to a percentage. 1/3 is approximately 33.3...%. Since 33.3...% is greater than 30%, the 1/3 off discount is larger. A larger discount results in a lower sale price. Therefore, the bike that is 1/3 off has a lower sale price. We can also calculate the prices: Price A = $210 * (1 - 1/3) =$210 * (2/3) = $140. Price B =$210 * (1 - 0.30) = $210 * 0.70 =$147. $140 is lower than$147.

Question 12

1. Blueberry
2. Banana Nut (correct answer)
3. Chocolate Chip
4. Poppy Seed

Explanation: To compare the amounts, convert each to a decimal. Blueberry: 1/4 = 0.25. Banana Nut: 35% = 0.35. Chocolate Chip: 0.3. Poppy Seed: 1 - (0.25 + 0.35 + 0.3) = 1 - 0.9 = 0.1. Comparing the decimals: 0.35 > 0.3 > 0.25 > 0.1. Therefore, Banana Nut muffins were sold the most.

Question 13

1. The quantity in Column A is greater. (correct answer)
2. The quantity in Column B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Explanation: To compare the two quantities, we should express them in the same format. Let's convert 3/7 to a percentage. 3/7 ≈ 0.42857... As a percentage, this is approximately 42.86%. Since 42.86% is greater than 42%, the quantity in Column A is greater.

Question 14

1. Pizza
2. Burgers (correct answer)
3. Tacos
4. No preference was the most popular response.

Explanation: To find the most popular food, we need to compare the proportions for each choice. Let's convert them to decimals. Pizza: 25% = 0.25. Burgers: 1/3 ≈ 0.333... Tacos: 0.3. Comparing these values, 0.333... is the largest. Therefore, burgers were the most popular choice. The number of students (60) is extra information not needed to determine the most popular choice based on proportion.

Question 15

1. The quantity in Column A is greater.
2. The quantity in Column B is greater. (correct answer)
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Explanation: To compare the two negative numbers, first convert the fraction in Column A to a decimal. 5/6 = 0.8333... So, Column A is -0.8333... and Column B is -0.83. On a number line, a number to the right is greater. Since -0.83 is to the right of -0.8333..., -0.83 is the greater number. Therefore, the quantity in Column B is greater.

Question 16

1. 0.60 cup
2. 0.63 cup
3. 0.66 cup (correct answer)
4. 0.67 cup

Explanation: First, convert 2/3 to a decimal: 2 ÷ 3 = 0.666... The question asks for the closest value that is not more than 2/3 cup. We must eliminate 0.67 cup because it is greater than 0.666... Now we compare the remaining options to 0.666...: |0.666... - 0.60| = 0.066...; |0.666... - 0.63| = 0.036...; |0.666... - 0.66| = 0.006... The value closest to 0.666... is 0.66.

Question 17

1. Jerome
2. Ken
3. Larry (correct answer)
4. Jerome and Ken have the same highest average.

Explanation: To compare the batting averages, convert all values to decimals. Jerome's average is 0.325. Ken's average is 8/25. To convert 8/25 to a decimal, divide 8 by 25, or multiply the numerator and denominator by 4 to get 32/100, which is 0.32. Larry's average is 33%, which is 0.33. Comparing the decimals 0.325, 0.320, and 0.330, the highest value is 0.33. Therefore, Larry has the highest batting average.

Question 18

1. The quantity in Column A is greater.
2. The relationship cannot be determined from the information given.
3. The two quantities are equal.
4. The quantity in Column B is greater. (correct answer)

Explanation: When comparing expressions involving fractions, you need to carefully evaluate each quantity by performing the operations in the correct order and comparing the final results. Let's calculate Column A first: $1 - (\frac{1}{5} + \frac{1}{6})$. To add the fractions, find a common denominator. The least common multiple of 5 and 6 is 30, so $\frac{1}{5} + \frac{1}{6} = \frac{6}{30} + \frac{5}{30} = \frac{11}{30}$. Therefore, Column A equals $1 - \frac{11}{30} = \frac{30}{30} - \frac{11}{30} = \frac{19}{30}$. For Column B: $1 - (\frac{1}{5} \times \frac{1}{6})$. Multiplying fractions is straightforward: $\frac{1}{5} \times \frac{1}{6} = \frac{1}{30}$. So Column B equals $1 - \frac{1}{30} = \frac{30}{30} - \frac{1}{30} = \frac{29}{30}$. Comparing $\frac{19}{30}$ and $\frac{29}{30}$, we see that $\frac{29}{30} > \frac{19}{30}$, making Column B greater. Choice A is incorrect because $\frac{19}{30} < \frac{29}{30}$. Choice B is wrong since we can definitively calculate both quantities. Choice C fails because the two fractions are clearly unequal. Choice D is correct. Remember that addition of fractions typically yields larger results than multiplication of the same fractions (when the fractions are positive and less than 1). This happens because adding fractions combines their values, while multiplying them makes the result smaller than either original fraction.

Question 19

1. The quantity in Column A is greater.
2. The relationship cannot be determined from the information given.
3. The two quantities are equal.
4. The quantity in Column B is greater. (correct answer)

Explanation: When comparing algebraic expressions involving fractions and square roots, you need to carefully consider how these operations behave differently for numbers between 0 and 1 versus numbers greater than 1. Since $0 < x < 1$, let's test with a specific value to see the pattern. If $x = 0.25$, then $\sqrt{x} = \sqrt{0.25} = 0.5$. Notice that $0.5 > 0.25$, so $\sqrt{x} > x$. This happens because taking the square root of a number between 0 and 1 actually makes it larger, not smaller. Here's why: when $0 < x < 1$, we have $x = x \cdot x < x \cdot 1 = x$, which means $x^2 < x$. Therefore, $x < \sqrt{x}$. Looking at the wrong answers: Choice A incorrectly assumes that $x > \sqrt{x}$, which would be true if $x > 1$, but not when $0 < x < 1$. This is a common misconception—students often think square roots always make numbers smaller. Choice B suggests we can't determine the relationship, but the constraint $0 < x < 1$ gives us enough information to establish that $\sqrt{x} > x$ for all values in this range. Choice C claims the quantities are equal, but $x = \sqrt{x}$ only when $x = 0$ or $x = 1$, neither of which are included in our given range. The correct answer is D because $\sqrt{x} > x$ whenever $0 < x < 1$. Remember: Square roots of numbers between 0 and 1 are always larger than the original number, while square roots of numbers greater than 1 are always smaller.