Comparing Fractions, Decimals & Percents - Accuplacer Arithmetic

1. A value is calculated by taking 25% of 3. Which of the following is greater than this value?
   Answer: $\frac{4}{5}$ First, calculate the value: 25% of 3 is $0.25 \times 3 = 0.75$. Next, compare this value to the answer choices by converting them to decimals. A: $\frac{3}{5} = 0.6$. B: $0.7$. C: $72\% = 0.72$. D: $\frac{4}{5} = 0.8$. We are looking for a value greater than $0.75$. Only $0.8$ is greater than $0.75$.
2. The length of a part must be between 0.45 inches and $\frac{1}{2}$ inch. Which of the following lengths is NOT an acceptable length?
   Answer: $\frac{11}{25}$ inches First, define the acceptable range in a consistent format, like decimals. The range is between 0.45 inches and $\frac{1}{2} = 0.5$ inches. Now check each option. A: $\frac{11}{25} = 0.44$ inches. This is less than 0.45, so it is outside the acceptable range. B: 0.46 is between 0.45 and 0.5. C: $\frac{19}{40} = 0.475$ is between 0.45 and 0.5. D: 0.49 is between 0.45 and 0.5. Therefore, $\frac{11}{25}$ inches is not an acceptable length.
3. Which of the following numbers has the greatest value?
   Answer: $\frac{5}{8}$ To find the greatest value, convert all numbers to decimals. A: $\frac{3}{5} = 0.60$. B: $60.5\% = 0.605$. C: $0.62$. D: $\frac{5}{8} = 0.625$. Comparing the decimals $0.60$, $0.605$, $0.62$, and $0.625$, the greatest value is $0.625$, which corresponds to $\frac{5}{8}$.
4. A certain length is 20% of $\frac{3}{2}$ meters. Which of the following lengths is shorter than this length?
   Answer: $\frac{1}{4}$ meter First, calculate the length. $\frac{3}{2}$ meters is 1.5 meters. 20% of 1.5 meters is $0.20 \times 1.5 = 0.3$ meters. The question asks which option is shorter than 0.3 meters. Convert the options to decimals: A: $\frac{1}{4} = 0.25$. B: $\frac{1}{3} \approx 0.333...$. C: $0.35$. D: $40\%$ of a meter is $0.4$ meters. Comparing these, only $0.25$ is shorter than $0.3$.
5. In one hour, Cyclist A completes $\frac{5}{8}$ of a race, and Cyclist B completes 65% of the same race. Which statement correctly compares their progress?
   Answer: Cyclist B has completed more of the race than Cyclist A. To compare their progress, convert the fraction and percent to a common format. Cyclist B completed 65% of the race. For Cyclist A, convert $\frac{5}{8}$ to a percent. $\frac{5}{8} = 5 \div 8 = 0.625$, which is 62.5%. Since $65\% > 62.5\%$, Cyclist B has completed more of the race.
6. A recipe calls for ingredients in the following proportions: sugar $\frac{2}{5}$, flour $0.38$, and butter $39\%$ of the total weight. Which ingredient makes up the largest proportion?
   Answer: Sugar, because $\frac{2}{5}$ converts to the largest decimal value Convert all to decimals: $\frac{2}{5} = 0.40$, $0.38 = 0.38$, and $39\% = 0.39$. Sugar at $0.40$ is largest. Choice A shows a fundamental misunderstanding. Choice B incorrectly identifies the middle value as largest. Choice C incorrectly claims 39% is largest when $0.39 < 0.40$.
7. A quality control inspector found that $\frac{11}{25}$ of widgets from Machine A, $0.445$ of widgets from Machine B, and $44\%$ of widgets from Machine C met specifications. Which machine produced the highest proportion of acceptable widgets?
   Answer: Machine B, because $0.445$ is the highest rate Convert to decimals: Machine A: $\frac{11}{25} = 0.44$, Machine B: $0.445$, Machine C: $44\% = 0.44$. Machine B has the highest rate at $0.445$. Choice A incorrectly identifies Machine A as highest. Choice C incorrectly identifies Machine C as highest. Choice D incorrectly claims A and C tie, ignoring that B is higher.
8. Which of the following numbers is greater than $\frac{1}{3}$ and less than 40%?
   Answer: $\frac{3}{8}$ First, convert the given boundaries to a common format, like decimals. $\frac{1}{3}$ is approximately $0.333...$. $40\%$ is $0.40$. The question asks for a number between $0.333...$ and $0.40$. Now, convert the answer choices: A is $0.3$, B is $\frac{3}{8} = 0.375$, C is $\frac{2}{5} = 0.40$, and D is $41\% = 0.41$. The only value that falls between $0.333...$ and $0.40$ is $0.375$.
9. A store offers two discounts on an appliance. The first offer is 30% off the original price. The second offer is $\frac{1}{4}$ off the original price. Which statement accurately compares the two offers?
   Answer: The 30% off offer is a better discount. To compare the discounts, convert them to the same format. The first discount is 30%. The second discount is $\frac{1}{4}$, which is equivalent to $\frac{25}{100}$ or 25%. Since 30% is a larger value than 25%, the 30% off offer is a better discount for the customer. The actual price of the appliance is not needed to compare the rates of discount.
10. A student's score on a test was $\frac{21}{25}$. On a second test, the score was 82%. On which test was the score higher?
    Answer: The score on the first test was higher. To compare the scores, they must be in the same format. Let's convert the fraction $\frac{21}{25}$ to a percent. To do this, find an equivalent fraction with a denominator of 100: $\frac{21 \times 4}{25 \times 4} = \frac{84}{100}$, which is 84%. The score on the first test was 84%, and the score on the second test was 82%. Since 84% > 82%, the score on the first test was higher.
11. Three of the following values are equivalent. Which value is NOT equivalent to the others?
    Answer: $\frac{6}{10}$ To find the value that is not equivalent, convert all options to a common format, such as a decimal. $0.625$ is already a decimal. $\frac{5}{8} = 5 \div 8 = 0.625$. $62.5\% = 62.5 \div 100 = 0.625$. $\frac{6}{10} = 0.6$. The values $0.625$, $\frac{5}{8}$, and $62.5\%$ are all equivalent. The value $\frac{6}{10}$ is different.
12. Which list shows the numbers $\frac{7}{10}$, 0.72, and 68% in order from greatest to least?
    Answer: 0.72, $\frac{7}{10}$, 68% Convert all numbers to decimals to compare them. $\frac{7}{10} = 0.7$. $0.72$ is already a decimal. $68\% = 0.68$. The three numbers are $0.7$, $0.72$, and $0.68$. Ordering them from greatest to least gives $0.72$, $0.7$, $0.68$. This corresponds to the original list: 0.72, $\frac{7}{10}$, 68%.
13. Which of the following comparisons is true?
    Answer: $0.55 = \frac{11}{20}$ Evaluate each statement by converting to a common format. A: $\frac{5}{8} = 0.625$, so $0.625 < 0.6$ is false. B: $\frac{3}{4} = 0.75 = 75\%$, so $70\% > 75\%$ is false. C: $\frac{11}{20} = \frac{11 \times 5}{20 \times 5} = \frac{55}{100} = 0.55$, so $0.55 = 0.55$ is true. D: $\frac{1}{3} \approx 33.3\%$, so $33.3\% > 35\%$ is false.
14. Which of the following inequalities is true?
    Answer: $-0.7 > -\frac{4}{5}$ Convert fractions to decimals to compare. A: $-\frac{3}{4} = -0.75$. Is $-0.75 < -0.8$? False. B: $-\frac{2}{3} \approx -0.667$. Is $-0.667 < -0.7$? False. C: $-\frac{1}{2} = -0.5$. Is $-0.6 > -0.5$? False. For negative numbers, the number with the smaller absolute value is greater. D: $-\frac{4}{5} = -0.8$. Is $-0.7 > -0.8$? True. On a number line, -0.7 is to the right of -0.8.
15. If $\frac{5}{8} = 0.625$ and $63\% = 0.63$, which statement about their relationship is correct?
    Answer: $63\%$ is greater than $\frac{5}{8}$ by exactly $0.005$ When comparing fractions and percentages, you need to express both values in the same format to find their exact difference. Here, you're given that $\frac{5}{8} = 0.625$ and $63\% = 0.63$, so you can work directly with these decimal forms. To find which is larger and by how much, subtract the smaller value from the larger: $0.63 - 0.625 = 0.005$. This tells us that $63\%$ is greater than $\frac{5}{8}$ by exactly $0.005$. Let's examine why the other answers are incorrect. Answer A claims $\frac{5}{8}$ is greater than $63\%$, but $0.625 < 0.63$, so this reverses the relationship. Answer B states that $63\%$ is greater by $0.5\%$, but $0.5\% = 0.005$ as a decimal, not as a percentage difference. The actual percentage difference would be $\frac{0.005}{0.625} \times 100\% \approx 0.8\%$. Answer D again reverses the relationship, incorrectly claiming $\frac{5}{8}$ is the larger value. The key insight is distinguishing between a difference expressed as a raw decimal ($0.005$) versus a percentage point difference ($0.5\%$). Since $0.63 - 0.625 = 0.005$, answer C correctly identifies both the direction and magnitude of the difference. Remember: when comparing decimals, line up the decimal places and subtract carefully. Always double-check whether the question asks for the difference as a decimal value or as a percentage.
16. Which of the following numbers is the smallest?
    Answer: $\frac{13}{20}$ To find the smallest number, convert all options to decimals. A: $\frac{13}{20} = 13 \div 20 = 0.65$. B: $66\% = 0.66$. C: $\frac{2}{3} = 2 \div 3 = 0.666...$. D: $0.67$. Comparing the decimals $0.65$, $0.66$, $0.666...$, and $0.67$, the smallest value is $0.65$.
17. Over one year, the value of Stock X increased by $\frac{1}{10}$. The value of Stock Y increased by 9.5%. The value of Stock Z increased by 0.105. Which stock had the greatest increase in value?
    Answer: Stock Z To compare the increases, convert them all to the same format, such as a decimal or a percent. Stock X: $\frac{1}{10} = 0.10 = 10\%$. Stock Y: $9.5\% = 0.095$. Stock Z: $0.105 = 10.5\%$. Comparing the percentages $10\%$, $9.5\%$, and $10.5\%$, the greatest value is $10.5\%$, which corresponds to Stock Z.
18. Which of the following lists of numbers is in order from least to greatest?
    Answer: $0.3, 35\%, \frac{2}{5}$ To compare the numbers, convert them to the same format, such as decimals. $0.3$ is already a decimal. $35\%$ is equal to $0.35$. $\frac{2}{5}$ is equal to $0.4$. The numbers in decimal form are $0.3$, $0.35$, and $0.4$. Arranging these from least to greatest gives $0.3$, $0.35$, $0.4$, which corresponds to the original list $0.3, 35\%, \frac{2}{5}$.
19. A calculation is performed: $0.5 - \frac{1}{8}$. Which of the following is equivalent to the result?
    Answer: $\frac{3}{8}$ First, perform the calculation. It's easiest to convert both numbers to decimals. $\frac{1}{8} = 0.125$. So, the calculation is $0.5 - 0.125 = 0.375$. Now, find which answer choice is equivalent to $0.375$. A: $3.75\% = 0.0375$. B: $\frac{1}{4} = 0.25$. C: $37\% = 0.37$. D: $\frac{3}{8} = 3 \div 8 = 0.375$. The correct equivalent value is $\frac{3}{8}$.
20. A recipe requires $\frac{3}{4}$ cup of flour. A baker has a measuring cup that holds 0.8 cups of flour when full. If the baker fills this cup, how does the amount of flour compare to the recipe's requirement?
    Answer: The baker has more than the required amount. To compare the amounts, convert the fraction to a decimal. $\frac{3}{4} = 3 \div 4 = 0.75$. The recipe requires 0.75 cups of flour. The baker has 0.8 cups of flour. Since $0.8 > 0.75$, the baker has more than the required amount.

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