Fractions Between Fractions - ISEE Lower Level Mathematics Achievement

Sarah has a piece of ribbon that is longer than $\frac{1}{4}$ yard but shorter than the total length of a $\frac{1}{3}$ yard piece and a $\frac{1}{6}$ yard piece combined. Which of the following could be the length of Sarah's ribbon?
Answer: $\frac{2}{5}$ yard First, we need to find the upper boundary for the ribbon's length by adding $\frac{1}{3}$ and $\frac{1}{6}$ . To add these, we need a common denominator, which is 6. $\frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ yard. The ribbon's length is between $\frac{1}{4}$ yard and $\frac{1}{2}$ yard. We can convert these to decimals to compare: $\frac{1}{4} = 0.25$ and $\frac{1}{2} = 0.5$ . We need to find which answer choice falls between 0.25 and 0.5. * (A) $\frac{1}{5} = 0.2$ , which is less than 0.25. * (B) $\frac{2}{5} = 0.4$ , which is between 0.25 and 0.5. * (C) $\frac{3}{5} = 0.6$ , which is greater than 0.5. * (D) $\frac{1}{6} \approx 0.167$ , which is less than 0.25. The correct answer is $\frac{2}{5}$ yard.
Jenny's house is $\frac{5}{8}$ of a mile from school. Leo's house is $\frac{3}{4}$ of a mile from school. A park is located at a distance from the school that is between Jenny's and Leo's distances. Which could be the park's distance from the school?
Answer: $\frac{11}{16}$ mile We need to find a fraction between $\frac{5}{8}$ and $\frac{3}{4}$ . Let's convert the fractions to have a common denominator, such as 16. * Jenny's distance: $\frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16}$ mile. * Leo's distance: $\frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16}$ mile. The park's distance must be between $\frac{10}{16}$ and $\frac{12}{16}$ of a mile. Let's check the answer choices. * (A) $\frac{1}{2} = \frac{8}{16}$ , which is less than $\frac{10}{16}$ . * (B) $\frac{9}{16}$ is less than $\frac{10}{16}$ . * (C) $\frac{11}{16}$ is between $\frac{10}{16}$ and $\frac{12}{16}$ . * (D) $\frac{7}{8} = \frac{14}{16}$ , which is greater than $\frac{12}{16}$ . The correct answer is $\frac{11}{16}$ mile.
Which of the following fractions is between $\frac{2}{7}$ and $\frac{2}{5}$ ?
Answer: $\frac{1}{3}$ We are looking for a fraction between $\frac{2}{7}$ and $\frac{2}{5}$ . When comparing fractions with the same numerator (in this case, 2), the fraction with the smaller denominator is larger. So, $\frac{2}{5}$ is larger than $\frac{2}{7}$ . We need a fraction, let's call it $\frac{2}{x}$ , where the denominator x is between 5 and 7. The whole number between 5 and 7 is 6. So, the fraction $\frac{2}{6}$ is between $\frac{2}{7}$ and $\frac{2}{5}$ . The fraction $\frac{2}{6}$ simplifies to $\frac{1}{3}$ . Let's check the choices: (A) $\frac{2}{9}$ : Since 9 is greater than 7, $\frac{2}{9}$ is less than $\frac{2}{7}$ . (B) $\frac{1}{4} = \frac{2}{8}$ : Since 8 is greater than 7, $\frac{2}{8}$ is less than $\frac{2}{7}$ . (C) $\frac{1}{2} = \frac{2}{4}$ : Since 4 is less than 5, $\frac{2}{4}$ is greater than $\frac{2}{5}$ . (D) $\frac{1}{3} = \frac{2}{6}$ : Since 6 is between 5 and 7, $\frac{2}{6}$ is between $\frac{2}{7}$ and $\frac{2}{5}$ .
There are four boxes of crayons. Box A is $\frac{2}{3}$ full. Box B is $\frac{3}{5}$ full. Box C is $\frac{1}{2}$ full. Box D is fuller than Box B but less full than Box A. Which fraction could represent how full Box D is?
Answer: $\frac{5}{8}$ We are looking for a fraction between $\frac{3}{5}$ (Box B) and $\frac{2}{3}$ (Box A). Let's convert these to decimals to make comparison easier. * $\frac{3}{5} = 0.6$ * $\frac{2}{3} \approx 0.667$ We need to find an answer choice with a value between 0.6 and 0.667. * (A) $\frac{4}{7} \approx 0.571$ . This is less than 0.6. * (B) $\frac{5}{8} = 0.625$ . This is between 0.6 and 0.667. * (C) $\frac{3}{4} = 0.75$ . This is greater than 0.667. * (D) $\frac{7}{10} = 0.7$ . This is greater than 0.667. The fraction $\frac{5}{8}$ is the only one that falls between $\frac{3}{5}$ and $\frac{2}{3}$ .
A recipe for a fruit smoothie calls for an amount of yogurt that is more than $\frac{2}{3}$ cup but less than $\frac{3}{4}$ cup. Which of the following amounts of yogurt could be used in the recipe?
Answer: $\frac{7}{10}$ cup To solve this problem, we need to find a fraction that is between $\frac{2}{3}$ and $\frac{3}{4}$ . A good way to compare these fractions is to find a common denominator. The least common multiple of 3, 4, and the denominators in the answer choices (2, 8, 10, 6) is 120. Let's convert the given fractions and the choices to have a denominator of 120. * The lower bound is $\frac{2}{3} = \frac{2 \times 40}{3 \times 40} = \frac{80}{120}$ . * The upper bound is $\frac{3}{4} = \frac{3 \times 30}{4 \times 30} = \frac{90}{120}$ . We need to find a fraction between $\frac{80}{120}$ and $\frac{90}{120}$ . * (A) $\frac{1}{2} = \frac{60}{120}$ , which is less than $\frac{80}{120}$ . * (B) $\frac{5}{8} = \frac{5 \times 15}{8 \times 15} = \frac{75}{120}$ , which is less than $\frac{80}{120}$ . * (C) $\frac{7}{10} = \frac{7 \times 12}{10 \times 12} = \frac{84}{120}$ , which is between $\frac{80}{120}$ and $\frac{90}{120}$ . * (D) $\frac{5}{6} = \frac{5 \times 20}{6 \times 20} = \frac{100}{120}$ , which is greater than $\frac{90}{120}$ . Therefore, $\frac{7}{10}$ cup is a valid amount.
Which of the following fractions is greater than $\frac{1}{8}$ but less than $\frac{1}{6}$ ?
Answer: $\frac{7}{48}$ When comparing fractions, you need to determine which values fall between two given boundaries. Here, you're looking for a fraction between $\frac{1}{8}$ and $\frac{1}{6}$ . The most efficient approach is to convert all fractions to a common denominator. Since you're comparing with $\frac{1}{8}$ and $\frac{1}{6}$ , and several answer choices have denominator 48, let's use 48 as our common denominator. First, convert the boundaries: $\frac{1}{8} = \frac{6}{48}$ and $\frac{1}{6} = \frac{8}{48}$ . So you need a fraction between $\frac{6}{48}$ and $\frac{8}{48}$ . Choice A, $\frac{7}{48}$ , falls perfectly between these values since $6 < 7 < 8$ . This means $\frac{6}{48} < \frac{7}{48} < \frac{8}{48}$ , confirming that $\frac{7}{48}$ is between $\frac{1}{8}$ and $\frac{1}{6}$ . Choice B, $\frac{1}{10} = \frac{4.8}{48}$ , is less than $\frac{6}{48}$ , making it smaller than $\frac{1}{8}$ . Choice C, $\frac{1}{4} = \frac{12}{48}$ , is much larger than $\frac{8}{48}$ , exceeding $\frac{1}{6}$ . Choice D, $\frac{5}{48}$ , is less than $\frac{6}{48}$ , so it's also smaller than $\frac{1}{8}$ . Strategy tip: When comparing fractions with different denominators, convert to a common denominator first. Look for the largest denominator among your choices—it often works well as your common denominator and simplifies the comparison process.
Which of the following fractions has a value between $0.65$ and $0.8$ ?
Answer: $\frac{2}{3}$ We need a fraction with a decimal value between 0.65 and 0.8. Let's convert each answer choice to a decimal. * (A) $\frac{3}{5} = 0.6$ , which is less than 0.65. * (B) $\frac{4}{5} = 0.8$ , which is equal to the upper boundary, not between the boundaries. * (C) $\frac{2}{3} = 2 \div 3 \approx 0.667$ , which is between 0.65 and 0.8. * (D) $\frac{5}{8} = 5 \div 8 = 0.625$ , which is less than 0.65. Therefore, $\frac{2}{3}$ is the only fraction that falls between 0.65 and 0.8.
What fraction is exactly halfway between $\frac{1}{3}$ and $\frac{1}{2}$ ?
Answer: $\frac{5}{12}$ To find the fraction exactly halfway between two numbers, we find their average. This means we add the two fractions and then divide the sum by 2. Step 1: Add the fractions. Find a common denominator for $\frac{1}{3}$ and $\frac{1}{2}$ , which is 6. $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$ Step 2: Divide the sum by 2. $\frac{5}{6} \div 2 = \frac{5}{6} \times \frac{1}{2} = \frac{5}{12}$ The fraction exactly halfway between $\frac{1}{3}$ and $\frac{1}{2}$ is $\frac{5}{12}$ . Distractor (A) is a common mistake made by adding the numerators (1+1=2) and the denominators (3+2=5).
A chocolate bar is divided into 12 equal squares. Maya eats more than $\frac{1}{4}$ of the bar but less than $\frac{1}{2}$ of the bar. How many squares could Maya have eaten?
Answer: 5 First, we need to determine the number of squares that represent $\frac{1}{4}$ and $\frac{1}{2}$ of the 12-square bar. * The lower limit is $\frac{1}{4}$ of 12, which is $\frac{1}{4} \times 12 = 3$ squares. * The upper limit is $\frac{1}{2}$ of 12, which is $\frac{1}{2} \times 12 = 6$ squares. The problem states that Maya ate more than 3 squares and less than 6 squares. This means the number of squares eaten must be a whole number greater than 3 and less than 6. The possible whole numbers are 4 and 5. Looking at the answer choices: * (A) 3 is not more than 3. * (B) 5 is a possible number of squares. * (C) 6 is not less than 6. * (D) 7 is more than 6. Therefore, Maya could have eaten 5 squares.
A pitcher contains some water. The amount of water is more than $\frac{3}{4}$ of a liter but less than $\frac{4}{5}$ of a liter. Which could be the amount of water in the pitcher?
Answer: $\frac{31}{40}$ liter When you see a problem asking which value falls within a given range, you need to compare fractions by finding a common way to evaluate them all. The water amount must be between $\frac{3}{4}$ and $\frac{4}{5}$ of a liter. To compare these fractions with the answer choices, convert everything to decimals or find a common denominator. Using decimals: $\frac{3}{4} = 0.75$ and $\frac{4}{5} = 0.80$ , so you need a value between 0.75 and 0.80. Let's check each option: Choice A: $\frac{31}{40} = 0.775$ . This falls perfectly between 0.75 and 0.80, so this works. Choice B: $\frac{2}{3} = 0.667$ . This is less than 0.75, so it's too small to be in our range. Choice C: $\frac{15}{20} = \frac{3}{4} = 0.75$ . The problem states the amount must be more than $\frac{3}{4}$ , so this exact value doesn't qualify. Choice D: $\frac{7}{10} = 0.70$ . This is also less than 0.75, making it too small. Only choice A gives us a value that's actually between the two boundaries, not equal to them or outside the range. Strategy tip: When comparing fractions, converting to decimals often makes the relationships clearer than finding common denominators. Also, pay close attention to whether the problem uses "more than/less than" versus "at least/at most" — this determines whether boundary values are included.
Three of the following fractions have a value between $\frac{1}{5}$ and $\frac{3}{5}$ . Which fraction does NOT have a value between $\frac{1}{5}$ and $\frac{3}{5}$ ?
Answer: $\frac{2}{3}$ We need to compare each answer choice to the given range of $\frac{1}{5}$ to $\frac{3}{5}$ . Using decimals can be helpful here: $\frac{1}{5} = 0.2$ and $\frac{3}{5} = 0.6$ . The question asks for the fraction that is NOT between 0.2 and 0.6. * (A) $\frac{1}{4} = 0.25$ . This is between 0.2 and 0.6. * (B) $\frac{1}{3} \approx 0.333$ . This is between 0.2 and 0.6. * (C) $\frac{1}{2} = 0.5$ . This is between 0.2 and 0.6. * (D) $\frac{2}{3} \approx 0.666$ . This is greater than 0.6. Therefore, $\frac{2}{3}$ is the fraction that is not between $\frac{1}{5}$ and $\frac{3}{5}$ .
Three friends ran part of a mile. Maria ran $\frac{5}{12}$ of a mile. Sam ran $\frac{3}{8}$ of a mile. Chloe ran a distance that was between Maria's distance and Sam's distance. What could have been the distance Chloe ran?
Answer: $\frac{19}{48}$ mile First, we need to compare the distances run by Maria and Sam to determine the range for Chloe's distance. We'll use a common denominator for $\frac{5}{12}$ and $\frac{3}{8}$ . The least common multiple of 12 and 8 is 24. * Maria's distance: $\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}$ mile. * Sam's distance: $\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$ mile. Chloe's distance is between $\frac{9}{24}$ and $\frac{10}{24}$ . To find a fraction between these, we can use a larger common denominator, like 48. * Sam: $\frac{9}{24} = \frac{18}{48}$ . Maria: $\frac{10}{24} = \frac{20}{48}$ . Chloe's distance is between $\frac{18}{48}$ and $\frac{20}{48}$ . Let's check the choices. * (A) $\frac{1}{3} = \frac{16}{48}$ , which is less than $\frac{18}{48}$ . * (B) $\frac{19}{48}$ is between $\frac{18}{48}$ and $\frac{20}{48}$ . * (C) $\frac{7}{16} = \frac{21}{48}$ , which is greater than $\frac{20}{48}$ . * (D) $\frac{1}{2} = \frac{24}{48}$ , which is greater than $\frac{20}{48}$ .
Liam is thinking of a fraction that is greater than $\frac{2}{5}$ and less than $\frac{2}{3}$ . Which fraction could Liam be thinking of?
Answer: $\frac{1}{2}$ We need to find a fraction between $\frac{2}{5}$ and $\frac{2}{3}$ . Let's find a common denominator for the given fractions and the answer choices. A common denominator for 5, 3, 4, 2 is 60. * The lower bound is $\frac{2}{5} = \frac{2 \times 12}{5 \times 12} = \frac{24}{60}$ . * The upper bound is $\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}$ . We need to find a fraction between $\frac{24}{60}$ and $\frac{40}{60}$ . * (A) $\frac{1}{3} = \frac{20}{60}$ . This is less than $\frac{24}{60}$ . * (B) $\frac{3}{4} = \frac{45}{60}$ . This is greater than $\frac{40}{60}$ . * (C) $\frac{1}{2} = \frac{30}{60}$ . This is between $\frac{24}{60}$ and $\frac{40}{60}$ . * (D) $\frac{4}{5} = \frac{48}{60}$ . This is greater than $\frac{40}{60}$ . Therefore, the fraction Liam could be thinking of is $\frac{1}{2}$ .
Which of the following fractions is located on a number line between $\frac{3}{8}$ and $\frac{5}{6}$ ?
Answer: $\frac{2}{3}$ To find a fraction between $\frac{3}{8}$ and $\frac{5}{6}$ , we should first convert them to fractions with a common denominator. The least common multiple of 8 and 6 is 24. * $\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$ * $\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$ Now we need to find which answer choice is between $\frac{9}{24}$ and $\frac{20}{24}$ . Let's convert the answer choices to have a denominator of 24. * (A) $\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$ . This is less than $\frac{9}{24}$ . * (B) $\frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24}$ . This is less than $\frac{9}{24}$ . * (C) $\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}$ . This is between $\frac{9}{24}$ and $\frac{20}{24}$ . * (D) $\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}$ . This is greater than $\frac{20}{24}$ . The correct answer is $\frac{2}{3}$ .
The fraction $\frac{4}{9}$ is between which of the following pairs of fractions?
Answer: $\frac{2}{5}$ and $\frac{1}{2}$ We need to check each pair of fractions to see if $\frac{4}{9}$ lies between them. Using decimals is an efficient method. $\frac{4}{9} = 4 \div 9 \approx 0.444$ . (A) $\frac{1}{3} \approx 0.333$ and $\frac{2}{5} = 0.4$ . The number 0.444 is not between 0.333 and 0.4. (B) $\frac{2}{5} = 0.4$ and $\frac{1}{2} = 0.5$ . The number 0.444 is between 0.4 and 0.5. This is the correct answer. (C) $\frac{1}{2} = 0.5$ and $\frac{3}{5} = 0.6$ . The number 0.444 is not between 0.5 and 0.6. (D) $\frac{3}{8} = 0.375$ and $\frac{7}{16} = 0.4375$ . The number 0.444 is not between 0.375 and 0.4375. Therefore, $\frac{4}{9}$ is between $\frac{2}{5}$ and $\frac{1}{2}$ .
Mr. Chen's class is collecting canned goods for a food drive. They collected more than $2\frac{1}{3}$ boxes but fewer than $2\frac{1}{2}$ boxes. Which of the following amounts could they have collected?
Answer: $2\frac{2}{5}$ boxes The whole number part of the mixed numbers is 2 for both boundaries and all answer choices, so we only need to compare the fractional parts. We need to find a fraction that is between $\frac{1}{3}$ and $\frac{1}{2}$ . Let's convert $\frac{1}{3}$ and $\frac{1}{2}$ to decimals to compare: $\frac{1}{3} \approx 0.333$ and $\frac{1}{2} = 0.5$ . We are looking for a fraction between 0.333 and 0.5. * (A) The fractional part is $\frac{1}{4} = 0.25$ , which is less than 0.333. * (B) The fractional part is $\frac{2}{5} = 0.4$ , which is between 0.333 and 0.5. * (C) The fractional part is $\frac{3}{5} = 0.6$ , which is greater than 0.5. * (D) The fractional part is $\frac{1}{6} \approx 0.167$ , which is less than 0.333. Therefore, the class could have collected $2\frac{2}{5}$ boxes.
A plant grew more than $\frac{3}{10}$ of a meter but less than $\frac{4}{5}$ of a meter in one month. Which of the following could be the growth of the plant?
Answer: $\frac{2}{3}$ meter We need to find a fraction that lies between $\frac{3}{10}$ and $\frac{4}{5}$ . First, let's make the denominators the same for the boundary fractions. The least common multiple of 10 and 5 is 10. $\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$ So, the plant's growth is between $\frac{3}{10}$ and $\frac{8}{10}$ . Now we check the answer choices. * (A) $\frac{1}{5} = \frac{2}{10}$ . This is less than $\frac{3}{10}$ . * (B) $\frac{9}{10}$ . This is greater than $\frac{8}{10}$ . * (C) $\frac{1}{4}$ . To compare with $\frac{3}{10}$ , we can use cross-multiplication: $1 \times 10 = 10$ and $4 \times 3 = 12$ . Since $10 < 12$ , $\frac{1}{4} < \frac{3}{10}$ . * (D) $\frac{2}{3}$ . To compare with $\frac{3}{10}$ , cross-multiply: $2 \times 10 = 20$ and $3 \times 3 = 9$ . Since $20 > 9$ , $\frac{2}{3} > \frac{3}{10}$ . To compare with $\frac{8}{10}$ (or $\frac{4}{5}$ ), cross-multiply: $2 \times 5 = 10$ and $3 \times 4 = 12$ . Since $10 < 12$ , $\frac{2}{3} < \frac{4}{5}$ . So, $\frac{2}{3}$ is between the two fractions.
Which fraction has a value between $\frac{4}{7}$ and $\frac{5}{8}$ ?
Answer: $\frac{17}{28}$ To find a fraction between $\frac{4}{7}$ and $\frac{5}{8}$ , we must first compare them using a common denominator. The least common multiple of 7 and 8 is 56. * $\frac{4}{7} = \frac{4 \times 8}{7 \times 8} = \frac{32}{56}$ * $\frac{5}{8} = \frac{5 \times 7}{8 \times 7} = \frac{35}{56}$ We are looking for a fraction with a value between $\frac{32}{56}$ and $\frac{35}{56}$ . Now, we convert the answer choices to have a denominator of 56. * (A) $\frac{1}{2} = \frac{28}{56}$ . This is less than $\frac{32}{56}$ . * (B) $\frac{9}{14} = \frac{9 \times 4}{14 \times 4} = \frac{36}{56}$ . This is greater than $\frac{35}{56}$ . * (C) $\frac{2}{3} = \frac{2}{3} \times \frac{56}{56} \approx \frac{37.3}{56}$ . This is greater than $\frac{35}{56}$ . * (D) $\frac{17}{28} = \frac{17 \times 2}{28 \times 2} = \frac{34}{56}$ . This is between $\frac{32}{56}$ and $\frac{35}{56}$ . Thus, $\frac{17}{28}$ is the correct answer.

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