HSPT Quantitative Quiz: Compare Quantitative Expressions

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Compare Quantitative ExpressionsQuestion 1 of 20

Compare (a) $37\%$ of 83 with (b) $83\%$ of 37.

(a) is greater than (b)

(b) is greater than (a)

(a) is equal to (b)

Cannot be determined

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HSPT Quantitative Quiz

HSPT Quantitative Quiz: Compare Quantitative Expressions

Practice Compare Quantitative Expressions in HSPT Quantitative with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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.

All questions

Question 1

Compare (a) $37\%$ of 83 with (b) $83\%$ of 37.

(a) is greater than (b)
(b) is greater than (a)
(a) is equal to (b) (correct answer)
Cannot be determined

Explanation: 37% of 83 means 37/100 times 83, and 83% of 37 means 83/100 times 37. Since multiplication is commutative, 37 × 83 equals 83 × 37, so both are the same value, 30.71. The tempting mistake is thinking the order of the percent and the number matters, but it does not.

Question 2

Compare (a) $\sqrt{64}+\sqrt{36}$ with (b) $\sqrt{64+36}$ .

Cannot be determined
(b) is greater than (a)
(a) is equal to (b)
(a) is greater than (b) (correct answer)

Explanation: Compute each side separately. (a) is 8 + 6 = 14. (b) is the square root of 100, which is 10. Since 14 is greater than 10, (a) is greater. The tempting wrong choice is equal, but square roots do not distribute over addition, so sqrt(64) + sqrt(36) is not the same as sqrt(64+36).

Question 3

Compare (a) $8 + 12 \div 4 \times 2$ with (b) $8 + 12 \div (4 \times 2)$ .

(a) is greater than (b) (correct answer)
(b) is greater than (a)
(a) is equal to (b)
Cannot be determined

Explanation: In (a), division and multiplication are done left to right: 12 ÷ 4 = 3, then 3 × 2 = 6, so 8 + 6 = 14. In (b), parentheses come first: 4 × 2 = 8, then 12 ÷ 8 = 1.5, so 8 + 1.5 = 9.5. Therefore (a) is greater. A common mistake is doing 4 × 2 before 12 ÷ 4 in (a), but without parentheses you must work left to right.

Question 4

Compare (a) $6^3 \cdot 6^2$ with (b) $(6^2)^3$ .

(a) is greater than (b)
(b) is greater than (a) (correct answer)
(a) is equal to (b)
Cannot be determined

Explanation: When multiplying powers with the same base, add exponents: 6^3 * 6^2 = 6^5. When raising a power to a power, multiply exponents: (6^2)^3 = 6^6. Since 6^6 is greater than 6^5, (b) is greater. The tempting error is treating (6^2)^3 as 6^(2+3), which would wrongly make them equal.

Question 5

Compare (a) $104 \times 96$ with (b) $10000 - 16$ .

(a) is greater than (b)
(b) is greater than (a)
(a) is equal to (b) (correct answer)
Cannot be determined

Explanation: Rewrite 104 × 96 as (100 + 4)(100 - 4), which equals 100^2 - 4^2 = 10,000 - 16 = 9,984. The expression in (b) is also 10,000 - 16 = 9,984, so they are equal. A tempting error is to multiply 104 and 96 as roughly 10,000 without recognizing the difference of squares, but both sides give the same value.

Question 6

Compare the quantities: (a) $\sqrt{144} + \sqrt{25}$ and (b) $\sqrt{169}$

(a) is greater than (b) (correct answer)
(b) is greater than (a)
(a) and (b) are equal
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 7

Compare the values: (A) $\frac{8}{15} \div \frac{2}{5}$ and (B) $\frac{6}{7} \div \frac{9}{14}$

(A) is greater than (B)
(B) is greater than (A)
(A) and (B) are equal (correct answer)
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 transforms each expression into a more manageable multiplication problem. For expression (A):

\frac{8}{15} \div \frac{2}{5} = \frac{8}{15} \times \frac{5}{2} = \frac{40}{30} = \frac{4}{3}

For expression (B):

\frac{6}{7} \div \frac{9}{14} = \frac{6}{7} \times \frac{14}{9} = \frac{84}{63} = \frac{4}{3}

Both expressions equal

\frac{4}{3}

, so they are equal. Looking at the answer choices: Choice (A) suggests that

\frac{8}{15} \div \frac{2}{5}

is greater than

\frac{6}{7} \div \frac{9}{14}

, but since both equal

\frac{4}{3}

, this is incorrect. Choice (B) makes the opposite incorrect claim that the second expression is larger. Choice (C) correctly identifies that the expressions are equal. Choice (D) suggests we can't determine the relationship, but we clearly can through direct calculation. The key insight here is that even though the original fractions look quite different, the division operations transform them into identical values. Always simplify your final answers when comparing fractions—it makes relationships much clearer. Study tip: When comparing complex fraction expressions, calculate each one completely and reduce to lowest terms before making your comparison. Don't try to estimate or compare the original fractions visually, as the operations can dramatically change the relative values.

Question 8

Compare the values: (A) $\frac{3}{8} \times \frac{16}{9}$ and (B) $\frac{2}{5} \times \frac{15}{8}$

(A) is greater than (B)
(B) is greater than (A) (correct answer)
(A) and (B) are equal
The relationship cannot be determined

Explanation: When comparing fractions that involve multiplication, you need to calculate the actual values rather than trying to estimate. Let's work through both expressions step by step. For expression (A):

\frac{3}{8} \times \frac{16}{9}

, multiply the numerators and denominators:

\frac{3 \times 16}{8 \times 9} = \frac{48}{72}

. You can simplify this by dividing both numerator and denominator by their greatest common factor of 24:

\frac{48}{72} = \frac{2}{3}

. For expression (B):

\frac{2}{5} \times \frac{15}{8}

, follow the same process:

\frac{2 \times 15}{5 \times 8} = \frac{30}{40}

. Simplify by dividing both by 10:

\frac{30}{40} = \frac{3}{4}

. Now compare

\frac{2}{3}

and

\frac{3}{4}

. To compare fractions with different denominators, find a common denominator. The LCD of 3 and 4 is 12:

\frac{2}{3} = \frac{8}{12}

and

\frac{3}{4} = \frac{9}{12}

. Since

\frac{9}{12} > \frac{8}{12}

, expression (B) is greater than (A). Choice (A) incorrectly states that (A) is greater than (B). Choice (C) claims they're equal, which they're not. Choice (D) suggests the relationship can't be determined, but with definite numerical values, comparison is always possible. Study tip: When comparing fraction products, always calculate the exact values first. Don't try to estimate or use shortcuts—the HSPT often includes answer choices that trap students who guess without calculating.

Question 9

Compare the values: $(a)$ $\left(\frac{3}{4}\right)^3 \times \frac{64}{27}$ and $(b)$ $\frac{16}{9} \times \left(\frac{3}{4}\right)^2$

(a) exceeds (b) by approximately 0.33
(b) exceeds (a) by approximately 0.44
(a) and (b) differ by less than 0.1 (correct answer)
(b) exceeds (a) by exactly 0.25

Explanation: When comparing expressions with fractions and exponents, you need to calculate each value precisely and then find their difference. This type of problem tests your ability to work with fractional exponents and perform accurate arithmetic. Let's calculate expression (a):

\left(\frac{3}{4}\right)^3 \times \frac{64}{27}

. First,

\left(\frac{3}{4}\right)^3 = \frac{27}{64}

. Then multiply:

\frac{27}{64} \times \frac{64}{27} = \frac{27 \times 64}{64 \times 27} = 1

. Now for expression (b):

\frac{16}{9} \times \left(\frac{3}{4}\right)^2

. We have

\left(\frac{3}{4}\right)^2 = \frac{9}{16}

. Then multiply:

\frac{16}{9} \times \frac{9}{16} = \frac{16 \times 9}{9 \times 16} = 1

. Since both expressions equal 1, their difference is exactly 0, which is certainly less than 0.1. This confirms answer C is correct. Answer A is wrong because (a) doesn't exceed (b) at all—they're equal. Answer B is similarly incorrect since (b) doesn't exceed (a). Answer D is wrong because the difference isn't 0.25; it's 0. The key insight here is recognizing that both expressions involve reciprocal multiplication patterns. In (a), you're essentially multiplying

\frac{27}{64}

\frac{64}{27}

, and in (b), you're multiplying

\frac{16}{9}

\frac{9}{16}

. When you multiply a fraction by its reciprocal, the result is always 1. Watch for these reciprocal relationships—they often appear on standardized tests to test whether you'll do unnecessary decimal conversions instead of recognizing the pattern.

Question 10

Compare the quantities: (a) $2.5^2$ and (b) $6.25$

(a) is greater than (b)
(b) is greater than (a)
(a) and (b) are equal (correct answer)
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 11

Compare the quantities: (a) $0.6^2$ and (b) $0.36$

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

Explanation: When comparing decimal expressions, you need to evaluate each quantity precisely. This question tests your ability to work with decimal exponents and recognize when two different-looking expressions are actually equal. Let's calculate

0.6^2

step by step. When you square 0.6, you multiply it by itself:

0.6 \times 0.6 = 0.36

. So quantity (a) equals 0.36, and quantity (b) is also 0.36. Since both quantities equal exactly the same value, they are equal. Looking at the wrong answers: Choice A claims that

0.6^2

is greater than 0.36, but since

0.6^2 = 0.36

, this is incorrect. Choice B suggests that 0.36 is greater than

0.6^2

, which is also wrong for the same reason—they're identical values. Choice D states the relationship cannot be determined, but since we can calculate

0.6^2

exactly, we can definitely determine the relationship. The correct answer is C because

0.6^2 = 0.36

. Study tip: When you see decimal numbers raised to powers, don't let the exponent intimidate you—calculate it out systematically. Also, remember that squaring a decimal between 0 and 1 always gives you a smaller number, but in this case, the "smaller" result happens to match the comparison value exactly. Practice decimal multiplication to build confidence with these calculations.

Question 12

Compare the quantities: (a) $\sqrt{36} \times \sqrt{4}$ and (b) $\sqrt{144}$

(a) is greater than (b)
(b) is greater than (a)
(a) and (b) are equal (correct answer)
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 13

Compare the values: (A) $\frac{2}{3} \text{ of } 45$ and (B) $\frac{3}{5} \text{ of } 50$

(A) is greater than (B)
(B) is greater than (A)
(A) and (B) are equal (correct answer)
The relationship cannot be determined

Explanation: When you encounter fraction comparison problems like this, you need to calculate each value precisely before making any judgments about their relationship. Let's evaluate each expression step by step. For (A),

\frac{2}{3} \text{ of } 45

means

\frac{2}{3} \times 45

. To calculate this, multiply:

\frac{2 \times 45}{3} = \frac{90}{3} = 30

. For (B),

\frac{3}{5} \text{ of } 50

means

\frac{3}{5} \times 50

. Calculate:

\frac{3 \times 50}{5} = \frac{150}{5} = 30

. Since both expressions equal 30, they are equal. Choice A states that (A) is greater than (B), but since 30 = 30, this is incorrect. The values are identical, not different with one being larger. Choice B claims that (B) is greater than (A), which is also wrong for the same reason—both values equal 30, so neither is greater than the other. Choice C correctly identifies that (A) and (B) are equal, since both equal 30. Choice D suggests the relationship cannot be determined, but this is false because we can calculate exact values for both expressions. There's no missing information or variables that would prevent us from determining the relationship. Study tip: Don't try to compare fractions and whole numbers mentally—always calculate the actual values first. Many students make errors by attempting shortcuts or estimating when precise calculation is both possible and necessary. Convert "of" problems to multiplication and solve completely before comparing.

Question 14

Compare the values: (A) $3^4 \div 9$ and (B) $2^6 \div 8$

(A) is greater than (B) (correct answer)
(B) is greater than (A)
(A) and (B) are equal
The relationship cannot be determined

Explanation: When comparing expressions with exponents and division, you need to evaluate each expression step by step, being careful to apply the order of operations correctly. Let's calculate expression (A):

3^4 \div 9

. First, evaluate the exponent:

3^4 = 3 \times 3 \times 3 \times 3 = 81

. Then divide:

81 \div 9 = 9

. Now for expression (B):

2^6 \div 8

. Calculate the exponent:

2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64

. Then divide:

64 \div 8 = 8

. Since 9 > 8, expression (A) is greater than expression (B). Choice A is correct because our calculations show that 9 > 8. Choice B is wrong because it claims the opposite relationship. Choice C incorrectly states the values are equal when 9 ≠ 8. Choice D suggests we can't determine the relationship, but since both expressions contain only constants (no variables), we can always calculate exact values and make the comparison. Study tip: When comparing numerical expressions, always evaluate completely rather than trying shortcuts. Calculate each expression to its final numerical value first, then compare. Also, remember that expressions with larger bases or exponents don't automatically yield larger results—the division or other operations can change the final comparison, so you must work through each step methodically.

Question 15

Compare the quantities: (a) $6^2 \div 3^2$ and (b) $4$

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

Explanation: When you encounter comparison problems involving exponents and division, work systematically by evaluating each expression using the order of operations. Let's evaluate quantity (a):

6^2 \div 3^2

. First, calculate the exponents:

6^2 = 36

and

3^2 = 9

. Then perform the division:

36 \div 9 = 4

. So quantity (a) equals 4. Quantity (b) is simply 4. Since both quantities equal 4, they are equal to each other. Looking at the wrong answers: Choice A claims (a) is greater than (b), but since

6^2 \div 3^2 = 4

and quantity (b) is also 4, this is incorrect. Choice B suggests (b) is greater than (a), which fails for the same reason—both expressions have the same value. Choice D states the relationship cannot be determined, but this is wrong because we can clearly calculate both quantities and compare them definitively. A common mistake students make is trying to simplify

6^2 \div 3^2

by writing it as

(6 \div 3)^2 = 2^2 = 4

. While this happens to give the correct answer here, this approach is mathematically incorrect. The proper rule is

\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n

, but you should stick to the order of operations: calculate exponents first, then divide. Study tip: Always use the standard order of operations (PEMDAS) when evaluating expressions. Don't look for shortcuts with exponents unless you're certain about the underlying rules—it's safer and more reliable to calculate step by step.

Question 16

Compare the quantities: (a) $0.4 \times 0.75$ and (b) $\frac{3}{10}$

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

Explanation: When comparing quantities involving decimals and fractions, you need to convert both expressions to the same form to make an accurate comparison. First, let's calculate quantity (a):

0.4 \times 0.75

. To multiply decimals, you can multiply the numbers as if they were whole numbers, then place the decimal point correctly.

4 \times 75 = 300

, and since we have a total of three decimal places (one in 0.4 and two in 0.75), we get

0.300 = 0.3

. Now let's convert quantity (b) to decimal form:

\frac{3}{10} = 0.3

. Since both quantities equal 0.3, they are equal. Looking at the wrong answers: Choice A states that (a) is greater than (b), but we've shown both equal 0.3. 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 with concrete numerical values like these, we can always determine the relationship through calculation. The key insight here is recognizing that

0.4 \times 0.75

simplifies to a clean decimal that matches the fraction

\frac{3}{10}

exactly. Study tip: When comparing quantities with mixed decimals and fractions, always convert everything to the same form first—either all decimals or all fractions. Also, remember that multiplying by 0.75 is the same as multiplying by

\frac{3}{4}

, which can sometimes make mental math easier:

0.4 \times \frac{3}{4} = \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}

Question 17

Compare the values: (A) $\frac{7}{9} \times \frac{18}{14}$ and (B) $\frac{4}{5} \times \frac{15}{12}$

(A) is greater than (B)
(B) is greater than (A)
(A) and (B) are equal (correct answer)
The relationship cannot be determined

Explanation: When comparing fractions involving multiplication, you need to calculate each expression's value rather than trying to estimate. Let's work through both products systematically. For expression (A):

\frac{7}{9} \times \frac{18}{14}

, multiply the numerators and denominators:

\frac{7 \times 18}{9 \times 14} = \frac{126}{126} = 1

. Notice that you can also simplify before multiplying by canceling common factors: the 7 in the numerator cancels with 14 in the denominator (since 14 = 7 × 2), and the 18 in the numerator cancels with 9 in the denominator (since 18 = 9 × 2), giving you

\frac{1 \times 2}{1 \times 2} = 1

. For expression (B):

\frac{4}{5} \times \frac{15}{12}

, again multiply:

\frac{4 \times 15}{5 \times 12} = \frac{60}{60} = 1

. You can also simplify first: 4 and 12 share a factor of 4, and 5 and 15 share a factor of 5, leaving you with

\frac{1 \times 3}{1 \times 3} = 1

. Since both expressions equal 1, choice (C) is correct—they are equal. Choice (A) incorrectly assumes the first expression is larger, while choice (B) incorrectly assumes the second is larger. Choice (D) suggests the relationship can't be determined, but both values are clearly calculable. Study tip: When comparing fraction products, always simplify by canceling common factors before multiplying—it makes calculations faster and reduces errors. Look for numbers that divide evenly across numerators and denominators.

Question 18

Compare the quantities: (a) $\frac{3}{7} + \frac{4}{7}$ and (b) $1$

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

Explanation: When you encounter fraction addition problems on the HSPT, you're testing your ability to perform basic arithmetic operations and compare quantities accurately. To solve

\frac{3}{7} + \frac{4}{7}

, notice that both fractions have the same denominator (7). When adding fractions with identical denominators, you simply add the numerators and keep the denominator the same:

\frac{3}{7} + \frac{4}{7} = \frac{3+4}{7} = \frac{7}{7}

. Since

\frac{7}{7} = 1

, quantity (a) equals 1, which means (a) and (b) are equal. Looking at the wrong answers: Choice A claims (a) is greater than (b), but since

\frac{7}{7} = 1

, they're actually equal, not with (a) being larger. Choice B suggests (b) is greater than (a), which falls into the same error—assuming inequality where equality exists. Choice D states the relationship cannot be determined, but this is incorrect because we can definitively calculate

\frac{3}{7} + \frac{4}{7}

and compare it to 1. The correct answer is C because both quantities equal 1. Study tip: When adding fractions with the same denominator, focus only on adding the numerators. Also, always check if your fraction answer can be simplified—here,

\frac{7}{7}

simplifies to 1. On comparison questions, complete all calculations before determining the relationship between quantities.

Question 19

Compare the quantities: (a) $\frac{15}{4} - \frac{7}{4}$ and (b) $2$

(a) is greater than (b)
(b) is greater than (a)
(a) and (b) are equal (correct answer)
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 20

Compare the values: $(a)$ $\frac{5^3 - 3^3}{5^2 + 5 \cdot 3 + 3^2}$ and $(b)$ $\frac{7^2 - 4^2}{7 + 4}$

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