Comparing Rational Forms - SHSAT Math

Which statement correctly compares $\dfrac{11}{25}$ and $43\%$ ?
Answer: $\dfrac{11}{25} > 43\%$ When comparing fractions and percentages, you need to convert them to the same form. The most straightforward approach is to convert the fraction to a percentage. To convert $\dfrac{11}{25}$ to a percentage, divide the numerator by the denominator: $11 ÷ 25 = 0.44$ . Then multiply by 100 to get the percentage: $0.44 × 100 = 44\%$ . Now you can directly compare: $44\% > 43\%$ , which means $\dfrac{11}{25} > 43\%$ . This confirms that answer choice C is correct. Let's examine why the other options are wrong. Choice A states $\dfrac{11}{25} < 43\%$ , but since $\dfrac{11}{25} = 44\%$ , this is clearly false. Choice B claims $\dfrac{11}{25} = 43\%$ , which would require the fraction to equal exactly $0.43$ , but our calculation shows it equals $0.44$ . Choice D suggests the fraction is "approximately $43\%$ but slightly smaller," which is incorrect in two ways: the fraction is larger than $43\%$ , not smaller, and the difference of $1\%$ isn't really "approximate" when dealing with such precise values. Study tip: When comparing fractions to percentages on the SHSAT, always convert to the same form first. Converting fractions to percentages is usually easier than the reverse. Remember that dividing by 25 is the same as multiplying by 4 and moving the decimal point, so $\dfrac{11}{25} = 11 × 4 = 44\%$ .
Which pair of numbers are equal in value?
Answer: $0.45$ and $\dfrac{9}{20}$ When comparing numbers in different forms—fractions, decimals, and percentages—you need to convert them all to the same format to determine which pairs are equal. Let's check each pair by converting to decimals. For choice B, $0.45$ is already a decimal. To convert $\dfrac{9}{20}$ to a decimal, divide: $9 \div 20 = 0.45$ . Since both equal $0.45$ , this pair is equal in value. Now let's verify why the other choices are incorrect. For choice A, convert $\dfrac{9}{40}$ to a decimal: $9 \div 40 = 0.225$ . Convert $22\%$ to a decimal by dividing by 100: $22\% = 0.22$ . Since $0.225 \neq 0.22$ , these aren't equal. For choice C, $0.18$ is given as a decimal. Convert $\dfrac{1}{5}$ : $1 \div 5 = 0.20$ . Since $0.18 \neq 0.20$ , these aren't equal. For choice D, convert $\dfrac{7}{25}$ : $7 \div 25 = 0.28$ . Since $0.28 \neq 0.27$ , these aren't equal. The key strategy for these problems is to convert everything to the same format—decimals are usually easiest. When converting fractions to decimals, simply divide the numerator by the denominator. For percentages, divide by 100. Always double-check your division, especially with fractions that might not result in terminating decimals. This systematic approach prevents careless errors and ensures you catch equivalent values that might not be obvious at first glance.
Three students earned the following scores on a test: Student A earned $\frac{22}{25}$ of the points, Student B earned $87.5\%$ , and Student C earned $0.89$ of the points. Which statement about their relative performance is true?
Answer: Student C performed best, followed by Student A, then Student B performed worst Converting to decimals: Student A: $\frac{22}{25} = 0.88$ , Student B: $87.5\% = 0.875$ , Student C: $0.89$ . Ranking from highest to lowest: $0.89 > 0.88 > 0.875$ , so Student C > Student A > Student B. Choice B incorrectly places Student A first, possibly from converting $\frac{22}{25}$ incorrectly. Choice C incorrectly makes Student B the best performer. Choice D reverses the order of Students B and C.
Marcus claims that $\frac{7}{12}$ is greater than $0.583$ . Sarah argues that $58.4\%$ is the largest of the three numbers. Who is correct?
Answer: Both are correct because $58.4\% > \frac{7}{12} > 0.583$ Converting to decimals: $\frac{7}{12} = 0.58\overline{3} \approx 0.5833$ , $0.583$ , and $58.4\% = 0.584$ . Since $0.584 > 0.58\overline{3} > 0.583$ , we have $58.4\% > \frac{7}{12} > 0.583$ . Marcus is correct that $\frac{7}{12} > 0.583$ , and Sarah is correct that $58.4\%$ is the largest. Choice A incorrectly orders $0.583$ and $58.4\%$ . Choice B incorrectly claims Marcus is wrong. Choice D incorrectly makes $0.583$ the largest value.
A store advertises a discount of $\dfrac{2}{15}$ off an item. Which advertised percent is closest to the same discount?
Answer: $13\%$ This question tests your ability to convert fractions to percentages and estimate values—a common skill on standardized math tests. To find what percent $\frac{2}{15}$ represents, you need to convert the fraction to a decimal, then to a percentage. Divide the numerator by the denominator: $2 \div 15 = 0.1333...$ To convert to a percentage, multiply by 100: $0.1333... \times 100 = 13.33...\%$ Since $13.33\%$ is closest to $13\%$ , answer choice C is correct. Let's examine why the other choices don't work. Choice A ( $11\%$ ) is too low—it's more than 2 percentage points away from our calculated value of $13.33\%$ . Choice B ( $12\%$ ) is closer but still more than a full percentage point off. Choice D ( $15\%$ ) might seem tempting since 15 appears in the denominator, but this represents a common trap. The denominator tells you the total number of parts, not the percentage itself. The key insight here is that $\frac{2}{15}$ is slightly more than $\frac{2}{16} = \frac{1}{8} = 12.5\%$ , so you'd expect an answer slightly above $12.5\%$ . Strategy tip: When converting fractions to percentages on multiple choice tests, you can often estimate without doing exact division. Compare to familiar benchmark fractions like $\frac{1}{8} = 12.5\%$ or $\frac{1}{10} = 10\%$ . Since $\frac{2}{15}$ is close to $\frac{2}{16} = \frac{1}{8}$ , you know the answer should be close to $12.5\%$ .
A chemistry student measured three different concentrations of a solution: Sample A has a concentration of $\frac{3}{16}$ , Sample B has $18.75\%$ concentration, and Sample C has $0.1875$ concentration. Which statement correctly describes the relationship between these concentrations?
Answer: All three samples have exactly the same concentration Converting all concentrations to decimals: Sample A: $\frac{3}{16} = 0.1875$ , Sample B: $18.75\% = 0.1875$ , Sample C: $0.1875$ . All three samples have identical concentrations of $0.1875$ . Choices A, B, and C all incorrectly assume that different representations must mean different values, failing to recognize that $\frac{3}{16} = 18.75\% = 0.1875$ .
A recipe calls for ingredients in the following proportions: $\frac{3}{8}$ flour, $37.5\%$ sugar, and $0.375$ butter by weight. If these three ingredients must be listed in descending order by weight on the package label, what is the correct order?
Answer: Flour, sugar, butter in equal proportions since all values represent the same amount Converting all values: $\frac{3}{8} = 0.375 = 37.5\%$ . All three ingredients are in equal proportions, so they represent the same amount by weight. Choice B assumes $37.5\% > \frac{3}{8}$ which would result from treating the percent as $37.5$ rather than $0.375$ . Choice C incorrectly assumes $37.5\% = 0.375$ means $37.5 < 375$ . Choice D makes a similar error by not recognizing the equivalence.
A store offers three different discounts: Discount X removes $\frac{2}{5}$ of the original price, Discount Y removes $42\%$ of the original price, and Discount Z removes $0.38$ of the original price. Which discount saves the customer the most money?
Answer: Discount Y saves the most because $42\% > \frac{2}{5} > 0.38$ Converting to decimals: Discount X: $\frac{2}{5} = 0.4 = 40\%$ , Discount Y: $42\% = 0.42$ , Discount Z: $0.38 = 38\%$ . Since $0.42 > 0.4 > 0.38$ , Discount Y saves the most money. Choice A incorrectly assumes $\frac{2}{5} > 42\%$ , possibly from comparing $2$ to $42$ without proper conversion. Choice C incorrectly makes $0.38$ the largest value. Choice D incorrectly assumes all values are equal.
Three investment options show the following annual returns: Option 1 gained $\frac{7}{20}$ of its value, Option 2 gained $34\%$ of its value, and Option 3 gained $0.36$ of its value. If you could only choose one investment for the past year, which option would have provided the highest return?
Answer: Option 3 provided the highest return at $36\%$ Converting all returns to percentages: Option 1: $\frac{7}{20} = 0.35 = 35\%$ , Option 2: $34\%$ , Option 3: $0.36 = 36\%$ . Option 3 has the highest return at $36\%$ . Choice A correctly converts Option 1 but incorrectly claims it's highest. Choice B correctly identifies Option 2's return but incorrectly claims it's highest. Choice D incorrectly states that Options 1 and 3 both equal $36\%$ .
Arrange the following numbers from least to greatest: $-\dfrac{5}{8},\; -0.62,\; -64\%,\; -0.6$ . Which order is correct?
Answer: $-64\%,\; -\dfrac{5}{8},\; -0.62,\; -0.6$ Convert everything to decimals: $-\frac{5}{8} = -0.625$ , $-64\% = -0.64$ , and the others are $-0.62$ and $-0.6$ . For negative numbers, the one farthest from zero is the least. Since $0.64 > 0.625 > 0.62 > 0.6$ , the order from least to greatest is $-0.64 < -0.625 < -0.62 < -0.6$ , that is, $-64\%,\; -\dfrac{5}{8},\; -0.62,\; -0.6$ . The order that puts $-0.62$ before $-\frac{5}{8}$ misreads $0.62$ and $0.625$ ; $-0.62$ is closer to zero, so it is greater. The order starting with $-0.6$ treats the numbers as if they were positive. The order starting with $-\frac{5}{8}$ forgets that $-64\%$ is smaller still.
A basketball player's free throw percentage can be expressed as $\frac{17}{25}$ , $0.68$ , or $68\%$ . If another player has a free throw percentage of $\frac{13}{20}$ and a third player shoots $66.5\%$ from the free throw line, what is the correct ordering of these three players from lowest to highest free throw percentage?
Answer: $\frac{13}{20}$ , $66.5\%$ , $\frac{17}{25}$ Convert all three values to decimals: $\frac{17}{25} = 0.68$ , $\frac{13}{20} = 0.65$ and $66.5\% = 0.665$ . From lowest to highest: $0.65 < 0.665 < 0.68$ , so the order is $\frac{13}{20}$ , $66.5\%$ , $\frac{17}{25}$ . Putting $66.5\%$ first treats it as smaller than 65%, often from comparing 66.5 with 0.65 without converting to the same form. Putting $66.5\%$ last misses that 0.665 is less than 0.68. Putting $\frac{17}{25}$ before $\frac{13}{20}$ reverses those two; 0.68 is greater than 0.65.
Which of the following is the smallest value?
Answer: $-0.1$ When comparing negative numbers, remember that the number furthest from zero (with the largest absolute value) is actually the smallest. To solve this systematically, convert all values to the same format - decimals work best here. Let's convert each option to decimal form. Choice A is already $-0.08$ . For choice B, divide: $-\frac{1}{11} = -0.090909...$ or approximately $-0.091$ . Choice C converts from percentage: $-9\% = -\frac{9}{100} = -0.09$ . Choice D is already $-0.1$ . Now compare: $-0.08$ , $-0.091$ , $-0.09$ , and $-0.1$ . Since these are all negative, the number with the greatest distance from zero is smallest. Think of a number line: $-0.1$ is furthest left, making it the smallest value. Choice A ( $-0.08$ ) is incorrect because $0.08 < 0.1$ , so $-0.08 > -0.1$ . Choice B ( $-\frac{1}{11} \approx -0.091$ ) is wrong because $0.091 < 0.1$ , so this value is greater than $-0.1$ . Choice C ( $-0.09$ ) is incorrect for the same reason - since $0.09 < 0.1$ , we have $-0.09 > -0.1$ . Study tip: When comparing negative numbers, convert everything to decimals first, then remember that "more negative" means smaller. The trap here is thinking like positive numbers - always double-check your inequality direction when working with negatives.

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