HSPT Quantitative Quiz: Compare Quantitative Expressions

What this quiz covers

This quiz focuses on Compare Quantitative Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT Quantitative.

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. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When you encounter fraction arithmetic problems asking you to compare quantities, your first step is to simplify the given expressions to their most basic form, then make the comparison. Let's evaluate quantity (a): $\frac{15}{4} - \frac{7}{4}$. Since both fractions have the same denominator, you can subtract the numerators directly: $\frac{15-7}{4} = \frac{8}{4} = 2$. So quantity (a) equals 2, and quantity (b) is also 2. Since both quantities equal 2, they are equal to each other. This makes C the correct answer. Let's examine why the other choices are wrong. Choice A claims that (a) is greater than (b), but since $\frac{8}{4} = 2$, quantity (a) equals 2, not something greater than 2. Choice B suggests that (b) is greater than (a), which would mean 2 > 2, an impossible statement. Choice D states the relationship cannot be determined, but we can clearly calculate that $\frac{15}{4} - \frac{7}{4} = 2$, making the relationship perfectly determinable. The key strategy here is to always simplify expressions completely before making comparisons. Don't let fractions intimidate you—convert them to whole numbers or decimals when possible. Also, watch for the trap of assuming that because an expression looks complicated (like a fraction subtraction), it must have a complicated answer. Sometimes the math works out to clean, simple results.

Question 2

1. (a) is less than (b) by exactly 1
2. (b) is less than (a) by exactly 1 (correct answer)
3. (a) and (b) are equal
4. (a) is greater than (b) by exactly 1

Explanation: When you encounter algebraic expressions that look complex, look for patterns or formulas that can simplify your work. The first expression contains a difference of cubes in the numerator and a specific trinomial in the denominator. For expression (a), recognize that $5^3 - 3^3$ is a difference of cubes, and the denominator $5^2 + 5 \cdot 3 + 3^2$ follows the pattern $a^2 + ab + b^2$. The difference of cubes formula states: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$. Therefore: $\frac{5^3 - 3^3}{5^2 + 5 \cdot 3 + 3^2} = \frac{(5-3)(5^2 + 5 \cdot 3 + 3^2)}{5^2 + 5 \cdot 3 + 3^2} = 5-3 = 2$ For expression (b), notice that $7^2 - 4^2$ is a difference of squares: $a^2 - b^2 = (a+b)(a-b)$. So: $\frac{7^2 - 4^2}{7 + 4} = \frac{(7+4)(7-4)}{7+4} = 7-4 = 3$ Since (a) = 2 and (b) = 3, we have (b) - (a) = 3 - 2 = 1, meaning (b) is greater than (a) by exactly 1, or equivalently, (b) is less than (a) by exactly 1. Choice A incorrectly states (a) is less than (b) by 1, which reverses the relationship. Choice C claims they're equal, but 2 ≠ 3. Choice D states (a) is greater than (b) by 1, which is backwards since 2 < 3. Strategy tip: Always look for factoring opportunities in fraction problems—difference of squares, difference of cubes, and perfect square trinomials often lead to dramatic simplification through cancellation.

Question 3

1. (A) is greater than (B)
2. (B) is greater than (A) (correct answer)
3. (A) and (B) are equal
4. The relationship cannot be determined

Explanation: When comparing expressions with mixed decimals and fractions, you need to convert everything to the same format to make an accurate comparison. Let's convert both expressions to decimals. For expression (A): $0.45 + \frac{7}{12}$, you need to convert $\frac{7}{12}$ to a decimal by dividing 7 by 12, which gives you 0.583... (or 0.58̄). So (A) = 0.45 + 0.583... = 1.033... For expression (B): $\frac{5}{6} + 0.28$, convert $\frac{5}{6}$ to a decimal by dividing 5 by 6, which gives you 0.833... (or 0.83̄). So (B) = 0.833... + 0.28 = 1.113... Since 1.113... > 1.033..., expression (B) is greater than expression (A). Choice A incorrectly suggests (A) is greater, likely from miscalculating one of the fraction-to-decimal conversions. Choice C claims they're equal, which would only happen if both expressions had the same sum—they don't. Choice D suggests the relationship can't be determined, but since we can convert fractions to decimals and perform the arithmetic, we can definitely compare these values. The correct answer is B. Strategy tip: When comparing expressions with mixed fractions and decimals, always convert everything to the same format first. Practice converting common fractions like $\frac{1}{6}$, $\frac{5}{6}$, and $\frac{7}{12}$ to decimals—these appear frequently on standardized tests.

Question 4

1. (A) is greater than (B) (correct answer)
2. (B) is greater than (A)
3. (A) and (B) are equal
4. The relationship cannot be determined

Explanation: When comparing expressions with different forms (decimals vs. fractions), you need to calculate both values to determine their relationship accurately. Let's evaluate expression (A): $1.8 \times 2.5$. You can multiply these decimals directly: $1.8 \times 2.5 = 4.5$. For expression (B): $\frac{9}{2} \times \frac{4}{5}$, multiply the fractions by multiplying numerators together and denominators together: $\frac{9 \times 4}{2 \times 5} = \frac{36}{10} = 3.6$. Now you can compare: 4.5 versus 3.6. Since 4.5 > 3.6, expression (A) is greater than expression (B). Choice (A) correctly states that (A) is greater than (B). Choice (B) incorrectly reverses this relationship, suggesting (B) is larger when 3.6 < 4.5. Choice (C) claims they're equal, but 4.5 ≠ 3.6. Choice (D) suggests the relationship cannot be determined, but since both expressions evaluate to definite numbers, we can always determine their relationship through calculation. Study tip: When comparing mixed decimal and fraction expressions, convert everything to the same form (either all decimals or all fractions) before comparing. Also, double-check your decimal multiplication—a common error is misplacing the decimal point, which could lead you to choose the wrong relationship.

Question 5

1. (A) is greater than (B)
2. (B) is greater than (A) (correct answer)
3. (A) and (B) are equal
4. The relationship cannot be determined

Explanation: When comparing expressions with exponents, you need to calculate each value step by step, following the order of operations carefully. Let's evaluate expression (A): $2^3 + 3^2$. First, calculate the exponents: $2^3 = 8$ and $3^2 = 9$. Then add: $8 + 9 = 17$. Now for expression (B): $5^2 - 2^2$. Calculate the exponents: $5^2 = 25$ and $2^2 = 4$. Then subtract: $25 - 4 = 21$. Since $21 > 17$, expression (B) is greater than expression (A), making choice B correct. Choice A states that (A) is greater than (B), but we found that $17 < 21$, so this is incorrect. Choice C claims the expressions are equal, but $17 \neq 21$, so this is wrong. Choice D suggests the relationship cannot be determined, but since we can calculate exact values for both expressions, we can definitely determine their relationship. The key trap here is rushing through the calculations or mixing up the order of operations. Some students might incorrectly calculate $2^3$ as $6$ instead of $8$, or forget that exponents must be calculated before addition and subtraction. Study tip: Always write out each step when evaluating expressions with exponents. Calculate all exponents first, then perform addition and subtraction from left to right. Double-check your basic exponent calculations—knowing perfect squares and cubes by heart will save you time and prevent errors.

Question 6

1. (A) is greater than (B)
2. (B) is greater than (A)
3. (A) and (B) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When comparing percentages of different numbers, you need to calculate the actual values rather than just looking at the percentages alone. This type of question tests your ability to work with percentages and recognize when different expressions yield equivalent results. Let's calculate each value: For (A): $15\% \text{ of } 120 = 0.15 \times 120 = 18$ For (B): $12\% \text{ of } 150 = 0.12 \times 150 = 18$ Both expressions equal 18, so (A) and (B) are equal. Looking at the wrong answers: Choice A incorrectly concludes that (A) is greater than (B). This might happen if you mistakenly think that since 15% is greater than 12%, the first expression must be larger—but you can't ignore that 120 is smaller than 150. Choice B makes the opposite error, assuming (B) is greater than (A) because 150 is larger than 120, without accounting for the fact that 12% is smaller than 15%. Choice D suggests the relationship cannot be determined, which would only be true if we had variables or missing information—but here we have all the concrete numbers needed for calculation. Strategy tip: When comparing percentage problems, always calculate the actual values rather than making assumptions based on just the percentages or just the base numbers. The interplay between percentage size and base number size often creates surprising results, and the HSPT frequently tests whether you'll do the math or fall for intuitive but incorrect reasoning.

Question 7

1. (A) is greater than (B)
2. (B) is greater than (A)
3. (A) and (B) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When you encounter square root comparison problems, your first step is to simplify each square root by recognizing perfect squares. Perfect squares are numbers like 4, 9, 16, 25, 36, 49, 64, 81, and 100 that result from squaring whole numbers. Let's evaluate expression (A): $\sqrt{64} + \sqrt{36} - \sqrt{25}$. Since $8^2 = 64$, $6^2 = 36$, and $5^2 = 25$, this becomes $8 + 6 - 5 = 9$. Now for expression (B): $\sqrt{100} - \sqrt{9} + \sqrt{4}$. Since $10^2 = 100$, $3^2 = 9$, and $2^2 = 4$, this becomes $10 - 3 + 2 = 9$. Both expressions equal 9, so (A) and (B) are equal. Looking at the wrong answers: Choice A incorrectly concludes that (A) is greater than (B), likely from computational errors in simplifying the square roots or arithmetic mistakes. Choice B makes the opposite error, suggesting (B) is greater than (A), possibly from the same types of calculation mistakes. Choice D claims the relationship cannot be determined, which would only be true if variables were involved or if the expressions couldn't be simplified—but since all the numbers under the square root signs are perfect squares, we can definitely determine exact values. Study tip: Memorize the first 12 perfect squares (1² through 12²). This will help you quickly recognize and simplify square roots on the HSPT, turning what looks like complex expressions into simple arithmetic problems.

Question 8

1. (A) is greater than (B)
2. (B) is greater than (A)
3. (A) and (B) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When comparing percentages of different numbers, you need to calculate the actual values rather than just looking at the percentages themselves. A smaller percentage of a larger number might equal a larger percentage of a smaller number. Let's calculate each value systematically: For (A): $25\% \text{ of } 84$ Convert the percentage to a decimal: $25\% = 0.25$ Multiply: $0.25 \times 84 = 21$ For (B): $30\% \text{ of } 70$ Convert the percentage: $30\% = 0.30$ Multiply: $0.30 \times 70 = 21$ Both expressions equal 21, so (A) and (B) are equal. Looking at the wrong answers: Choice A incorrectly assumes that since 84 is larger than 70, the result must be larger, ignoring that 25% is smaller than 30%. Choice B makes the opposite error, focusing only on the fact that 30% is larger than 25% while ignoring the different base numbers. Choice D suggests the relationship can't be determined, but with concrete numbers and percentages, we can always calculate exact values. Study tip: Never try to compare percentages mentally by just looking at the numbers. Always convert percentages to decimals and multiply to get the actual values. This type of question frequently appears on standardized tests because it tests whether you'll take shortcuts or do the proper calculations.

Question 9

1. (A) is greater than (B) (correct answer)
2. (B) is greater than (A)
3. (A) and (B) are equal
4. The relationship cannot be determined

Explanation: When comparing fractions through addition and subtraction, you need to find common denominators and work systematically through each expression. For expression (A): $\frac{5}{8} + \frac{1}{4} - \frac{1}{8}$, convert $\frac{1}{4}$ to eighths: $\frac{1}{4} = \frac{2}{8}$. This gives you $\frac{5}{8} + \frac{2}{8} - \frac{1}{8} = \frac{6}{8} = \frac{3}{4}$. For expression (B): $\frac{7}{12} + \frac{1}{6} - \frac{1}{12}$, convert $\frac{1}{6}$ to twelfths: $\frac{1}{6} = \frac{2}{12}$. This gives you $\frac{7}{12} + \frac{2}{12} - \frac{1}{12} = \frac{8}{12} = \frac{2}{3}$. Now compare $\frac{3}{4}$ and $\frac{2}{3}$ by finding a common denominator of 12: $\frac{3}{4} = \frac{9}{12}$ and $\frac{2}{3} = \frac{8}{12}$. Since $\frac{9}{12} > \frac{8}{12}$, expression (A) is greater. Choice A is correct because $\frac{3}{4} > \frac{2}{3}$. Choice B incorrectly reverses this relationship. Choice C is wrong because the values aren't equal—$\frac{9}{12} \neq \frac{8}{12}$. Choice D is incorrect because these expressions can definitely be determined through straightforward fraction arithmetic. Strategy tip: When comparing fraction expressions, simplify each one completely first, then convert to a common denominator for the final comparison. Don't try to compare the original complex expressions directly—you'll make errors and waste time.

Question 10

1. (a) is greater than (b) (correct answer)
2. (b) is greater than (a)
3. (a) and (b) are equal
4. The relationship cannot be determined

Explanation: When comparing expressions with exponents, you need to calculate each value carefully before making the comparison. Let's work through both quantities step by step. For quantity (a): $3^4 - 2^5$ First, calculate $3^4 = 3 \times 3 \times 3 \times 3 = 81$ Then, calculate $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$ So quantity (a) = $81 - 32 = 49$ For quantity (b): $4^3 - 5^2$ First, calculate $4^3 = 4 \times 4 \times 4 = 64$ Then, calculate $5^2 = 5 \times 5 = 25$ So quantity (b) = $64 - 25 = 39$ Since 49 > 39, quantity (a) is greater than quantity (b), making choice A correct. Choice B is wrong because 39 is not greater than 49. Choice C is incorrect since 49 ≠ 39. Choice D is wrong because we can definitively determine the relationship by calculating both values. The key strategy here is to resist the urge to estimate or compare the expressions without fully calculating them. While $4^3$ is larger than $3^4$, and $5^2$ is smaller than $2^5$, the differences in the subtractions create a different outcome than you might initially expect. Always compute the entire expression before comparing quantities involving multiple operations.

Question 11

1. (a) is greater than (b) (correct answer)
2. (b) is greater than (a)
3. (a) and (b) are equal
4. The relationship cannot be determined

Explanation: When comparing square root expressions, you need to simplify each quantity first before making any comparisons. Don't be tempted to combine terms under a single radical or make assumptions about their relative sizes. Let's evaluate quantity (a): $\sqrt{144} + \sqrt{25}$. Since $12^2 = 144$ and $5^2 = 25$, we have $\sqrt{144} = 12$ and $\sqrt{25} = 5$. Therefore, quantity (a) equals $12 + 5 = 17$. Now for quantity (b): $\sqrt{169}$. Since $13^2 = 169$, we have $\sqrt{169} = 13$. Comparing our results: quantity (a) = 17 and quantity (b) = 13. Since 17 > 13, quantity (a) is greater than quantity (b). Looking at the wrong answers: Choice B incorrectly concludes that (b) is greater than (a), which would happen if you miscalculated one of the square roots or made an arithmetic error. Choice C suggests the quantities are equal, which might occur if you confused $\sqrt{144} + \sqrt{25}$ with $\sqrt{144 + 25} = \sqrt{169}$ – a common error of incorrectly distributing the square root over addition. Choice D claims the relationship cannot be determined, but since we're dealing with specific numerical values, the relationship is clearly determinable. The correct answer is A. Study tip: Always simplify square roots of perfect squares first, then perform arithmetic operations. Remember that $\sqrt{a} + \sqrt{b} \neq \sqrt{a + b}$ – you cannot combine square roots through addition under a single radical.

Question 12

1. (a) is greater than (b)
2. (b) is greater than (a) (correct answer)
3. (a) and (b) are equal
4. The relationship cannot be determined

Explanation: When you encounter problems comparing expressions with exponents, your first step is always to calculate each quantity separately before making any comparisons. Let's evaluate quantity (a): $2^3 + 3^2$. Remember that $2^3 = 2 \times 2 \times 2 = 8$ and $3^2 = 3 \times 3 = 9$. So quantity (a) equals $8 + 9 = 17$. Now for quantity (b): $5^2 - 2^2$. We have $5^2 = 5 \times 5 = 25$ and $2^2 = 2 \times 2 = 4$. Therefore, quantity (b) equals $25 - 4 = 21$. Comparing our results: quantity (a) = 17 and quantity (b) = 21. Since 21 > 17, quantity (b) is greater than quantity (a). Choice A incorrectly states that (a) is greater than (b). This would be wrong since 17 < 21. Choice C claims the quantities are equal, but 17 ≠ 21. Choice D suggests the relationship cannot be determined, but since we can calculate definite numerical values for both expressions, we can always determine their relationship. Choice B correctly identifies that (b) is greater than (a). Strategy tip: Never try to compare expressions with exponents without calculating them first. Even if the expressions look similar or you think you can eyeball the comparison, always work out the actual numerical values. Exponents can create surprising results that don't match intuitive expectations, especially when addition and subtraction are involved.

Question 13

1. (a) is greater than (b)
2. (b) is greater than (a) (correct answer)
3. (a) and (b) are equal
4. The relationship cannot be determined

Explanation: When you encounter problems comparing expressions with negative numbers raised to powers, the key is carefully evaluating each exponent and remembering how negative numbers behave under different powers. Let's calculate quantity (a): $(-3)^2 + (-2)^3$. Since $(-3)^2 = 9$ (negative times negative equals positive) and $(-2)^3 = -8$ (odd powers preserve the negative sign), we get $9 + (-8) = 1$. For quantity (b): $(-1)^4 + (-5)^2$. Here, $(-1)^4 = 1$ (even power makes it positive) and $(-5)^2 = 25$ (even power makes it positive), giving us $1 + 25 = 26$. Since quantity (a) equals 1 and quantity (b) equals 26, quantity (b) is greater than quantity (a), making B correct. Choice A incorrectly states that (a) is greater than (b), which would mean 1 > 26. Choice C claims they're equal, suggesting both expressions equal the same value, but 1 ≠ 26. Choice D suggests the relationship cannot be determined, but since both expressions contain only constants (no variables), we can definitively calculate and compare them. The most common error here is mishandling negative bases with exponents. Remember: even exponents always produce positive results regardless of the base's sign, while odd exponents preserve the original sign of negative bases. Always work through each term systematically before comparing quantities.

Question 14

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When comparing fractions, you need to either convert them to the same denominator or calculate their decimal values. In this case, since one quantity involves multiplication, let's first simplify $\frac{3}{4} \times \frac{8}{9}$. To multiply fractions, multiply the numerators together and the denominators together: $\frac{3}{4} \times \frac{8}{9} = \frac{3 \times 8}{4 \times 9} = \frac{24}{36}$. Now simplify by finding the greatest common factor of 24 and 36, which is 12: $\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}$. So quantity (a) equals $\frac{2}{3}$, and quantity (b) is $\frac{2}{3}$. Therefore, the quantities are equal. Looking at the wrong answers: Choice A claims (a) is greater than (b), but since both equal $\frac{2}{3}$, this is incorrect. Choice B claims (b) is greater than (a), which is also wrong for the same reason. Choice D suggests the relationship cannot be determined, but we can clearly calculate and compare these definite values. The key strategy here is recognizing that fraction multiplication problems often simplify to common fractions that appear elsewhere in the question. Always reduce your fractions to lowest terms before comparing, and look for patterns—the HSPT frequently creates problems where different expressions yield the same simplified result. This tests whether you can perform operations accurately rather than just recognize obvious relationships.

Question 15

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When comparing fractions, you need to evaluate each expression first, then determine their relationship. Let's calculate quantity (a): $\frac{7}{8} - \frac{3}{8}$. Since both fractions have the same denominator, you can subtract the numerators directly: $\frac{7-3}{8} = \frac{4}{8}$. Now simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 4: $\frac{4}{8} = \frac{1}{2}$. Quantity (b) is already given as $\frac{1}{2}$. Since both quantities equal $\frac{1}{2}$, they are equal. Looking at the wrong answers: Choice A claims (a) is greater than (b), but since $\frac{1}{2} = \frac{1}{2}$, this is false. Choice B claims (b) is greater than (a), which is also incorrect for the same reason. Choice D suggests the relationship cannot be determined, but we can clearly calculate both values and compare them definitively. The trap here is failing to simplify $\frac{4}{8}$ to $\frac{1}{2}$. If you left the answer as $\frac{4}{8}$ and compared it to $\frac{1}{2}$ without recognizing they're equivalent, you might incorrectly think they're different. Study tip: Always simplify fractions to lowest terms before making comparisons. When subtracting fractions with the same denominator, subtract the numerators and keep the denominator the same, then reduce the result.

Question 16

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When you encounter problems comparing square roots, you can either calculate each expression separately or use the multiplication property of square roots to simplify your work. Let's evaluate both quantities. For quantity (a): $\sqrt{36} \times \sqrt{4}$. First, find each square root: $\sqrt{36} = 6$ and $\sqrt{4} = 2$. Therefore, $\sqrt{36} \times \sqrt{4} = 6 \times 2 = 12$. For quantity (b): $\sqrt{144} = 12$ since $12^2 = 144$. Since both quantities equal 12, they are equal. Alternatively, you could use the property that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$ to transform quantity (a): $\sqrt{36} \times \sqrt{4} = \sqrt{36 \times 4} = \sqrt{144}$, which immediately shows the quantities are identical. Looking at the wrong answers: Choice A suggests quantity (a) is greater, but $12 = 12$, not $12 > 12$. Choice B suggests quantity (b) is greater, but again, both equal 12. Choice D claims the relationship cannot be determined, but since we can calculate exact values for both square roots involving perfect squares, the relationship is clearly determinable. Remember that when comparing expressions with square roots of perfect squares, calculate the actual values rather than leaving them under the radical. Also, keep the multiplication property of square roots handy—it often reveals that seemingly different expressions are actually equivalent.

Question 17

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When you encounter division of fractions, remember that dividing by a fraction is the same as multiplying by its reciprocal. This fundamental rule will help you solve quantity (a) accurately. To find $\frac{5}{6} \div \frac{5}{12}$, you multiply $\frac{5}{6}$ by the reciprocal of $\frac{5}{12}$, which is $\frac{12}{5}$: $\frac{5}{6} \times \frac{12}{5} = \frac{5 \times 12}{6 \times 5} = \frac{60}{30} = 2$ So quantity (a) equals 2, and quantity (b) is also 2. Choice A states that (a) is greater than (b), but since both quantities equal 2, this is incorrect. Choice B claims that (b) is greater than (a), which is also wrong for the same reason. Choice D suggests the relationship cannot be determined, but we can clearly calculate that both quantities equal 2, making this incorrect as well. Choice C correctly identifies that (a) and (b) are equal. The most common error here is incorrectly handling fraction division. Some students might try to divide numerators and denominators separately or forget to flip the second fraction when converting division to multiplication. Others might make arithmetic errors during the multiplication step. Study tip: Always remember "keep, change, flip" for fraction division: keep the first fraction unchanged, change division to multiplication, and flip the second fraction to its reciprocal. Practice this method until it becomes automatic, as fraction operations appear frequently on standardized tests.

Question 18

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When you encounter comparison problems involving exponents and decimals, your goal is to evaluate each quantity and determine their relationship. Let's calculate quantity (a): $2.5^2$. This means $2.5 \times 2.5$. You can work this out step by step: $2.5 \times 2.5 = 6.25$. Alternatively, you might recognize that $2.5 = \frac{5}{2}$, so $2.5^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6.25$. Quantity (b) is simply $6.25$. Since $2.5^2 = 6.25$ and quantity (b) equals $6.25$, the two quantities are identical. Looking at the wrong answers: Choice A claims that $2.5^2 > 6.25$, which would mean $6.25 > 6.25$—clearly impossible. Choice B suggests that $6.25 > 2.5^2$, or $6.25 > 6.25$—equally impossible. Choice D states the relationship cannot be determined, but since we can calculate exact values for both quantities, we can definitively compare them. The correct answer is C because the quantities are equal. Study tip: When comparing quantities involving exponents, always calculate the exact values rather than estimating. Small numbers raised to powers can sometimes yield surprising results, so precise computation is essential. Also, remember that perfect squares like $2.5^2$ often appear as "disguised" equal comparisons on standardized tests—if one quantity looks suspiciously close to a simple calculation of the other, check if they're actually the same value.

Question 19

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: When comparing numerical expressions, you need to calculate each quantity first, then determine their relationship. Let's evaluate both expressions step by step. For quantity (a): $5^2 - 4^2 = 25 - 16 = 9$. For quantity (b): $3^2 = 9$. Since both quantities equal 9, they are equal to each other. Now let's examine why each answer choice is correct or incorrect. Choice A claims that (a) is greater than (b), but since $9 = 9$, this is false. Choice B states that (b) is greater than (a), which is also false for the same reason. Choice C correctly identifies that (a) and (b) are equal, since both expressions equal 9. Choice D suggests the relationship cannot be determined, but since we can calculate definite values for both expressions, we can definitely determine their relationship. You might also recognize a shortcut here using the difference of squares formula: $5^2 - 4^2 = (5+4)(5-4) = 9 \times 1 = 9$. This confirms our answer without having to calculate $25 - 16$ separately. Study tip: On HSPT quantitative questions, always calculate the actual values when comparing expressions with exponents. Don't try to guess based on the numbers you see – small differences in bases or exponents can dramatically change the results. Also, watch for opportunities to use algebraic shortcuts like the difference of squares formula to save time.

Question 20

1. (a) is greater than (b)
2. (b) is greater than (a)
3. (a) and (b) are equal (correct answer)
4. The relationship cannot be determined

Explanation: This question tests your ability to simplify square roots and compare numerical values. When you see square roots in a comparison problem, your first step should be to evaluate each square root to get concrete numbers you can work with. Let's calculate quantity (a): $\sqrt{81} - \sqrt{16}$. You need to find the principal square roots of both numbers. Since $9 \times 9 = 81$, we have $\sqrt{81} = 9$. Since $4 \times 4 = 16$, we have $\sqrt{16} = 4$. Therefore, quantity (a) equals $9 - 4 = 5$. Quantity (b) is simply $5$. Since both quantities equal 5, they are equal, making answer choice C correct. Let's examine why the other answers are wrong. Answer choice A claims that (a) is greater than (b), but since both equal 5, this is false. Answer choice B claims that (b) is greater than (a), which is also false for the same reason. Answer choice D suggests the relationship cannot be determined, but this is incorrect because both quantities have definite, calculable values. The key strategy here is recognizing perfect squares. Memorize the perfect squares from 1 to 144 (1², 2², 3², ..., 12²) so you can quickly evaluate square roots like $\sqrt{81}$ and $\sqrt{16}$ without hesitation. This foundational knowledge will save you time on quantitative comparison questions and help you avoid calculation errors under test pressure.