Comparing & Ordering Fractions - Accuplacer Arithmetic

1. Three friends are comparing their test scores as fractions of total points possible. Alex scored $\frac{17}{20}$, Beth scored $\frac{21}{25}$, and Carlos scored $\frac{33}{40}$. If they want to rank themselves from lowest to highest score, what is the correct order?
   Answer: Carlos, Alex, Beth (lowest to highest) Convert all fractions to a common denominator of 200: Alex: $\frac{17}{20} = \frac{170}{200}$, Beth: $\frac{21}{25} = \frac{168}{200}$, Carlos: $\frac{33}{40} = \frac{165}{200}$. From lowest to highest: $\frac{165}{200} < \frac{168}{200} < \frac{170}{200}$, so Carlos < Beth < Alex. Choice A incorrectly places Alex as lowest. Choice C reverses Beth and Alex. Choice D completely reverses the order by putting Beth first.
2. A recipe calls for $\frac{2}{3}$ cup of flour, but Maria only has a $\frac{1}{4}$ cup measuring cup. She fills it completely 3 times. What fraction represents how much more flour Maria has compared to what the recipe requires?
   Answer: $\frac{1}{12}$ Maria has $3 \times \frac{1}{4} = \frac{3}{4}$ cup of flour. The recipe needs $\frac{2}{3}$ cup. To compare: $\frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12}$. Maria has $\frac{1}{12}$ cup more than needed. Choice B incorrectly subtracts $\frac{2}{3} - \frac{1}{4}$. Choice C adds the fractions instead of subtracting. Choice D represents the total amount Maria has after converting to twelfths.
3. Which symbol makes the following statement true? $\frac{9}{14} \; \underline{\hspace{2em}} \; \frac{11}{17}$
   Answer: < To compare two fractions like $\frac{9}{14}$ and $\frac{11}{17}$, cross-multiplication is an effective method. Multiply the numerator of the first fraction by the denominator of the second, and vice versa. First product: $9 \times 17 = 153$. Second product: $14 \times 11 = 154$. Since the first product (153) is less than the second product (154), the first fraction is less than the second fraction. Therefore, $\frac{9}{14} < \frac{11}{17}$.
4. Four students are reading the same book. Amy has read $\frac{5}{8}$ of the book, Ben has read $\frac{2}{3}$ of the book, Chloe has read $\frac{3}{5}$ of the book, and David has read $\frac{7}{10}$ of the book. Who has read the most?
   Answer: David To find out who has read the most, we need to find the largest fraction among $\frac{5}{8}$, $\frac{2}{3}$, $\frac{3}{5}$, and $\frac{7}{10}$. Converting to decimals is an efficient way to compare: Amy ($\frac{5}{8}$) = 0.625. Ben ($\frac{2}{3}$) ≈ 0.667. Chloe ($\frac{3}{5}$) = 0.60. David ($\frac{7}{10}$) = 0.70. The largest decimal is 0.70, which corresponds to the fraction read by David. Therefore, David has read the most.
5. Which of the following fractions has the greatest value?
   Answer: $\frac{39}{40}$ Each of these fractions is one part less than a whole (1). We can compare them by seeing how far each is from 1. A) $1 - \frac{7}{8} = \frac{1}{8}$. B) $1 - \frac{15}{16} = \frac{1}{16}$. C) $1 - \frac{19}{20} = \frac{1}{20}$. D) $1 - \frac{39}{40} = \frac{1}{40}$. The fraction that is 'missing' the smallest piece to make a whole is the largest fraction. Since $\frac{1}{40}$ is the smallest of these missing pieces, $\frac{39}{40}$ is the fraction with the greatest value.
6. A length of wire is cut into four pieces with lengths of $\frac{5}{9}$ yard, $\frac{4}{7}$ yard, $\frac{1}{2}$ yard, and $\frac{8}{15}$ yard. What is the length of the shortest piece?
   Answer: $\frac{1}{2}$ yard To find the shortest piece, we must compare the four fractions. Converting them to decimals is a straightforward way to compare: $\frac{5}{9} \approx 0.556$, $\frac{4}{7} \approx 0.571$, $\frac{1}{2} = 0.5$, and $\frac{8}{15} \approx 0.533$. Ordering these decimals from least to greatest gives 0.5, 0.533, 0.556, 0.571. The smallest decimal is 0.5, which corresponds to $\frac{1}{2}$ yard.
7. A number $n$ satisfies the inequality $\frac{2}{5} < n < \frac{2}{3}$. Which of the following could be the value of $n$?
   Answer: $\frac{5}{8}$ First, establish the range in decimal form for easier comparison. $\frac{2}{5} = 0.4$ and $\frac{2}{3} \approx 0.667$. We need to find the option that falls between 0.4 and 0.667. A) $\frac{1}{3} \approx 0.333$ (too small). B) $\frac{3}{8} = 0.375$ (too small). C) $\frac{5}{8} = 0.625$ (this is between 0.4 and 0.667). D) $\frac{7}{10} = 0.7$ (too large). Therefore, $\frac{5}{8}$ is a possible value for $n$.
8. In an election, Maria received $\frac{5}{12}$ of the votes and David received $\frac{3}{8}$ of the votes. Which of the following fractions represents a share of votes greater than David's share but less than Maria's share?
   Answer: $\frac{2}{5}$ First, compare the shares of Maria and David. Maria's share is $\frac{5}{12}$ and David's is $\frac{3}{8}$. In decimal form, $\frac{5}{12} \approx 0.4167$ and $\frac{3}{8} = 0.375$. We need to find a fraction that is between 0.375 and 0.4167. Now check the options: A) $\frac{1}{3} \approx 0.333$ (too small). B) $\frac{2}{5} = 0.4$ (this is between 0.375 and 0.4167). C) $\frac{7}{16} = 0.4375$ (too large). D) $\frac{4}{9} \approx 0.444$ (too large). Thus, $\frac{2}{5}$ is the correct answer.
9. A water tank is $\frac{3}{8}$ full. After some water is used, it is $\frac{1}{3}$ full. Later, it is filled until it is $\frac{3}{4}$ full. Which list shows these three fractional amounts of water from least to greatest?
   Answer: $\frac{1}{3}, \frac{3}{8}, \frac{3}{4}$ The problem asks to order the fractions $\frac{3}{8}$, $\frac{1}{3}$, and $\frac{3}{4}$. To compare them, we can find a common denominator, which is 24. Converting the fractions: $\frac{3}{8} = \frac{9}{24}$, $\frac{1}{3} = \frac{8}{24}$, and $\frac{3}{4} = \frac{18}{24}$. By comparing the numerators (8, 9, 18), the order from least to greatest is $\frac{8}{24}, \frac{9}{24}, \frac{18}{24}$, which corresponds to $\frac{1}{3}, \frac{3}{8}, \frac{3}{4}$.
10. A pie is cut into 11 equal slices, and John eats 3 of them. A second, identical pie is cut into 9 equal slices, and Mary eats 3 of them. Which statement is true?
    Answer: Mary ate more pie than John. This problem asks us to compare the fractions of pie eaten. John ate $\frac{3}{11}$ of a pie. Mary ate $\frac{3}{9}$ of a pie. We need to compare $\frac{3}{11}$ and $\frac{3}{9}$. When two fractions have the same numerator, the one with the smaller denominator is the larger fraction. Since 9 is smaller than 11, $\frac{3}{9}$ is greater than $\frac{3}{11}$. Therefore, Mary ate more pie than John. Distractor D is incorrect because the problem states the pies are identical.
11. A project requires a hole that is slightly larger than $\frac{5}{8}$ inch but smaller than $\frac{2}{3}$ inch. Which of the following drill bit sizes could be used?
    Answer: $\frac{13}{20}$ inch We need to find a fraction that is between $\frac{5}{8}$ and $\frac{2}{3}$. Converting the given fractions to decimals gives a range: $\frac{5}{8} = 0.625$ and $\frac{2}{3} \approx 0.667$. Now we check the answer choices: A) $\frac{3}{5} = 0.6$, which is too small. B) $\frac{5}{9} \approx 0.556$, which is too small. C) $\frac{11}{16} = 0.6875$, which is too large. D) $\frac{13}{20} = 0.65$, which is between 0.625 and 0.667. Thus, $\frac{13}{20}$ inch is the correct size.
12. Three of the following fractions are greater than $\frac{1}{2}$. Which fraction is NOT greater than $\frac{1}{2}$?
    Answer: $\frac{5}{11}$ To compare each fraction to $\frac{1}{2}$, we can find a common denominator or see if the numerator is more than half the denominator. A) For $\frac{4}{7}$, half of 7 is 3.5. Since 4 > 3.5, $\frac{4}{7} > \frac{1}{2}$. B) For $\frac{5}{11}$, half of 11 is 5.5. Since 5 < 5.5, $\frac{5}{11} < \frac{1}{2}$. C) For $\frac{7}{12}$, half of 12 is 6. Since 7 > 6, $\frac{7}{12} > \frac{1}{2}$. D) For $\frac{9}{16}$, half of 16 is 8. Since 9 > 8, $\frac{9}{16} > \frac{1}{2}$. The fraction that is not greater than $\frac{1}{2}$ is $\frac{5}{11}$.
13. Which of the following lists of fractions is ordered from greatest to least?
    Answer: $\frac{4}{5}, \frac{4}{7}, \frac{4}{9}, \frac{4}{11}$ When comparing fractions that have the same numerator, the fraction with the smallest denominator has the greatest value. The numerators are all 4. The denominators are 5, 7, 9, and 11. Ordering the denominators from least to greatest gives 5, 7, 9, 11. Therefore, the corresponding fractions ordered from greatest to least are $\frac{4}{5}, \frac{4}{7}, \frac{4}{9}, \frac{4}{11}$. Distractor A incorrectly lists them from least to greatest.
14. Which of the following inequalities is true?
    Answer: $-\frac{5}{12} < -\frac{3}{8}$ To compare negative fractions, remember that the fraction with the smaller absolute value is the greater number (closer to zero). Let's convert to decimals or use a common denominator. For D: $-\frac{5}{12}$ and $-\frac{3}{8}$. The common denominator is 24. $-\frac{5}{12} = -\frac{10}{24}$ and $-\frac{3}{8} = -\frac{9}{24}$. Since -10 is less than -9, $-\frac{10}{24} < -\frac{9}{24}$, which means $-\frac{5}{12} < -\frac{3}{8}$ is true. For A: $-\frac{2}{3} \approx -0.667$ and $-\frac{3}{5} = -0.6$; $-0.667 < -0.6$. For B: $\frac{7}{8} < \frac{8}{9}$, so $-\frac{7}{8} > -\frac{8}{9}$. For C: $\frac{1}{4} > \frac{1}{5}$, so $-\frac{1}{4} < -\frac{1}{5}$.
15. Which of the following fractions is the largest?
    Answer: $\frac{18}{27}$ To find the largest fraction, we can compare them. Notice that the fraction $\frac{18}{27}$ can be simplified. Both 18 and 27 are divisible by 9, so $\frac{18}{27} = \frac{2}{3}$. Now compare $\frac{2}{3}$ with the other fractions by converting to decimals: A) $\frac{11}{20} = 0.55$. B) $\frac{4}{7} \approx 0.571$. C) $\frac{5}{8} = 0.625$. D) $\frac{18}{27} = \frac{2}{3} \approx 0.667$. Since 0.667 is the largest decimal value, $\frac{18}{27}$ is the largest fraction.
16. A recipe for soup requires $\frac{2}{3}$ cup of stock, and a recipe for a sauce requires $\frac{3}{4}$ cup of stock. If you have two separate 1-cup containers of stock and use one for each recipe, which recipe leaves a greater amount of stock remaining in its container?
    Answer: The soup recipe leaves a greater amount. First, calculate the amount of stock remaining for each recipe. For the soup: $1 - \frac{2}{3} = \frac{1}{3}$ cup remains. For the sauce: $1 - \frac{3}{4} = \frac{1}{4}$ cup remains. Now, compare the remaining amounts, $\frac{1}{3}$ and $\frac{1}{4}$. Since $\frac{1}{3}$ is greater than $\frac{1}{4}$, the soup recipe leaves a greater amount of stock remaining. A common error is to compare the initial amounts used ($\frac{2}{3}$ and $\frac{3}{4}$) instead of the remainders.
17. Which of the following lists the numbers $1\frac{1}{4}$, $\frac{6}{5}$, and $\frac{9}{7}$ in order from least to greatest?
    Answer: $\frac{6}{5}, 1\frac{1}{4}, \frac{9}{7}$ To order the numbers, it's helpful to convert them to the same format, such as decimals. $1\frac{1}{4} = 1.25$. $\frac{6}{5} = 1.2$. $\frac{9}{7} \approx 1.286$. Now, order these decimal values from least to greatest: 1.2, 1.25, 1.286. This corresponds to the original numbers in the order $\frac{6}{5}, 1\frac{1}{4}, \frac{9}{7}$.
18. Which of the following fractions is greater than $\frac{3}{4}$ but less than $\frac{4}{5}$?
    Answer: $\frac{31}{40}$ To solve this, find a common denominator or convert the fractions to decimals. As decimals, $\frac{3}{4} = 0.75$ and $\frac{4}{5} = 0.80$. We need to find a fraction between 0.75 and 0.80. Let's convert the options to decimals: A) $\frac{7}{10} = 0.70$ (too small). B) $\frac{11}{15} \approx 0.733$ (too small). C) $\frac{13}{16} = 0.8125$ (too large). D) $\frac{31}{40} = 0.775$ (which is between 0.75 and 0.80). Therefore, $\frac{31}{40}$ is the correct answer.
19. Three runners are practicing on a track. Ali ran $\frac{7}{8}$ of a mile, Ben ran $\frac{5}{6}$ of a mile, and Carmen ran $\frac{13}{15}$ of a mile. Which statement correctly compares the distances they ran?
    Answer: Ali ran the farthest distance. To compare the distances, we compare the fractions $\frac{7}{8}$, $\frac{5}{6}$, and $\frac{13}{15}$. Converting to decimals is an efficient method: Ali: $\frac{7}{8} = 0.875$. Ben: $\frac{5}{6} \approx 0.833$. Carmen: $\frac{13}{15} \approx 0.867$. Comparing these decimals, we see that $0.875 > 0.867 > 0.833$. Therefore, Ali ran the farthest distance. Distractor A is wrong because Ben ran the shortest distance. Distractor B is wrong because Carmen (0.867) ran farther than Ben (0.833).
20. A recipe is scaled down so that $\frac{2}{3}$ of the original amount of each ingredient is used. If the original recipe called for $\frac{3}{4}$ cup of milk and $\frac{5}{6}$ cup of flour, which ingredient quantity is larger in the scaled recipe?
    Answer: Flour is larger by $\frac{1}{18}$ cup Scale each ingredient by $\frac{2}{3}$: Milk: $\frac{2}{3} × \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ cup. Flour: $\frac{2}{3} × \frac{5}{6} = \frac{10}{18} = \frac{5}{9}$ cup. To compare $\frac{1}{2}$ and $\frac{5}{9}$, convert to common denominator 18: $\frac{1}{2} = \frac{9}{18}$ and $\frac{5}{9} = \frac{10}{18}$. Since $\frac{10}{18} > \frac{9}{18}$, flour is larger. The difference is $\frac{10}{18} - \frac{9}{18} = \frac{1}{18}$ cup. Choice A reverses which ingredient is larger. Choice C and D use $\frac{1}{9}$ as the difference, which would be correct if comparing $\frac{10}{18}$ and $\frac{8}{18}$, but that's not our calculation.

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