Converting Improper Fractions & Mixed Numbers - Accuplacer Arithmetic

1. A plank of wood is $5\frac{1}{4}$ feet long. A section measuring $\frac{11}{4}$ feet is cut off. What is the length of the remaining piece of wood?
   Answer: $2\frac{1}{2}$ feet First, convert the mixed number $5\frac{1}{4}$ to an improper fraction: $5 \times 4 + 1 = 21$, so it's $\frac{21}{4}$. Now subtract the length of the cut section: $\frac{21}{4} - \frac{11}{4} = \frac{10}{4}$. To express this as a mixed number, divide 10 by 4, which is 2 with a remainder of 2. The result is $2\frac{2}{4}$, which simplifies to $2\frac{1}{2}$.
2. A jug contains $3\frac{1}{2}$ liters of water. This amount is poured into glasses that each hold $\frac{1}{2}$ of a liter. How many glasses can be completely filled?
   Answer: 7 The problem asks how many $\frac{1}{2}$-liter portions are in $3\frac{1}{2}$ liters. First, convert the mixed number $3\frac{1}{2}$ to an improper fraction. Multiply the whole number by the denominator ($3 \times 2 = 6$) and add the numerator ($6 + 1 = 7$). The improper fraction is $\frac{7}{2}$. This fraction means there are 7 portions of $\frac{1}{2}$ liter. Therefore, 7 glasses can be filled.
3. Which of the following improper fractions is equivalent to $8\frac{7}{12}$?
   Answer: $\frac{103}{12}$ To convert the mixed number $8\frac{7}{12}$ to an improper fraction, multiply the whole number (8) by the denominator (12), and then add the numerator (7). The calculation is $(8 \times 12) + 7 = 96 + 7 = 103$. The denominator remains 12. So the improper fraction is $\frac{103}{12}$.
4. A movie is 135 minutes long. How long is the movie in hours, expressed as a mixed number in simplest form? (There are 60 minutes in 1 hour.)
   Answer: $2\frac{1}{4}$ hours To convert minutes to hours, divide the total minutes by 60. This gives the fraction $\frac{135}{60}$. To convert this improper fraction to a mixed number, divide 135 by 60. $135 \div 60 = 2$ with a remainder of 15. This gives the mixed number $2\frac{15}{60}$. To simplify the fractional part, divide both the numerator and denominator by their greatest common divisor, which is 15. $\frac{15 \div 15}{60 \div 15} = \frac{1}{4}$. The final answer is $2\frac{1}{4}$ hours.
5. What is the sum of $2\frac{3}{5}$ and $\frac{9}{5}$, expressed as a mixed number?
   Answer: $4\frac{2}{5}$ First, convert the mixed number $2\frac{3}{5}$ to an improper fraction: $2 \times 5 + 3 = 13$, so it is $\frac{13}{5}$. Now add the two fractions: $\frac{13}{5} + \frac{9}{5} = \frac{22}{5}$. Finally, convert $\frac{22}{5}$ back to a mixed number. $22 \div 5 = 4$ with a remainder of 2. So the answer is $4\frac{2}{5}$.
6. An improper fraction was correctly converted to the mixed number $6\frac{3}{7}$. What was the numerator of the original improper fraction?
   Answer: 45 To find the numerator of the original improper fraction, you must convert the mixed number $6\frac{3}{7}$ back to an improper fraction. Multiply the whole number (6) by the denominator (7) and add the numerator (3). The calculation is $(6 \times 7) + 3 = 42 + 3 = 45$. The numerator was 45.
7. Over three days, a student studied for $\frac{5}{2}$ hours, $\frac{7}{2}$ hours, and $\frac{3}{2}$ hours. What is the total study time in hours, expressed as a mixed number?
   Answer: $7\frac{1}{2}$ hours To find the total study time, add the three improper fractions: $\frac{5}{2} + \frac{7}{2} + \frac{3}{2}$. Since they have a common denominator, add the numerators: $5 + 7 + 3 = 15$. The total is $\frac{15}{2}$ hours. To express this as a mixed number, divide 15 by 2. $15 \div 2 = 7$ with a remainder of 1. The mixed number is $7\frac{1}{2}$ hours.
8. A point on a number line is located at $\frac{17}{3}$. Between which two consecutive whole numbers does this point lie?
   Answer: 5 and 6 To find the location on the number line, convert the improper fraction $\frac{17}{3}$ to a mixed number. Divide the numerator (17) by the denominator (3). $17 \div 3 = 5$ with a remainder of 2. This means $\frac{17}{3} = 5\frac{2}{3}$. The value $5\frac{2}{3}$ is greater than 5 but less than 6, so it lies between the whole numbers 5 and 6.
9. Which of the following fractions has the greatest value?
   Answer: $\frac{19}{5}$ To compare the values, convert all options to the same format, such as mixed numbers. A) $\frac{19}{5} = 3\frac{4}{5}$. B) $3\frac{3}{4}$. C) $\frac{27}{8} = 3\frac{3}{8}$. D) $3\frac{1}{2}$. Now compare the fractional parts: $\frac{4}{5}$, $\frac{3}{4}$, $\frac{3}{8}$, and $\frac{1}{2}$. In decimal form, these are 0.8, 0.75, 0.375, and 0.5. The largest fractional part is $\frac{4}{5}$, so $\frac{19}{5}$ is the greatest value.
10. The number $5\frac{7}{8}$ is written as an improper fraction. What is that fraction?
    Answer: $\frac{47}{8}$ To convert the mixed number $5\frac{7}{8}$ to an improper fraction, multiply the whole number (5) by the denominator (8) and add the numerator (7). Calculation: $(5 \times 8) + 7 = 40 + 7 = 47$. The denominator remains 8. The improper fraction is $\frac{47}{8}$.
11. Two lengths of fabric measure $2\frac{1}{6}$ yards and $3\frac{5}{6}$ yards. What is the total length of the fabric, expressed as an improper fraction?
    Answer: $\frac{36}{6}$ First, add the two mixed numbers: $2\frac{1}{6} + 3\frac{5}{6}$. Add the whole numbers: $2+3=5$. Add the fractions: $\frac{1}{6} + \frac{5}{6} = \frac{6}{6} = 1$. The total is $5 + 1 = 6$. To express 6 as an improper fraction with a denominator of 6, multiply the whole number by the denominator: $6 \times 6 = 36$. So the answer is $\frac{36}{6}$. Alternatively, convert $2\frac{1}{6} = \frac{13}{6}$ and $3\frac{5}{6} = \frac{23}{6}$, then add $\frac{13}{6} + \frac{23}{6} = \frac{36}{6}$.
12. Which mixed number is equivalent to $\frac{113}{9}$?
    Answer: $12\frac{5}{9}$ To convert $\frac{113}{9}$ to a mixed number, divide 113 by 9. $113 \div 9 = 12$ with a remainder of 5. The quotient (12) is the whole number, the remainder (5) is the numerator, and the denominator (9) stays the same. The equivalent mixed number is $12\frac{5}{9}$.
13. The improper fraction $\frac{53}{6}$ is equivalent to the mixed number $X\frac{5}{6}$. What is the value of X?
    Answer: 8 To find the whole number part (X) of the mixed number, divide the numerator of the improper fraction (53) by its denominator (6). $53 \div 6 = 8$ with a remainder of 5. The quotient, 8, is the whole number X. The mixed number is $8\frac{5}{6}$.
14. What is the difference between $\frac{31}{5}$ and $4\frac{2}{5}$?
    Answer: $1\frac{4}{5}$ First, convert both numbers to the same format. Let's use improper fractions. Convert $4\frac{2}{5}$ to an improper fraction: $(4 \times 5) + 2 = 22$, so it is $\frac{22}{5}$. Now subtract the smaller fraction from the larger one: $\frac{31}{5} - \frac{22}{5} = \frac{9}{5}$. Finally, convert the result $\frac{9}{5}$ to a mixed number. $9 \div 5 = 1$ with a remainder of 4. The difference is $1\frac{4}{5}$.
15. A sack of flour weighs $\frac{37}{4}$ kilograms. How can this weight be expressed as a mixed number?
    Answer: $9\frac{1}{4}$ To convert the improper fraction $\frac{37}{4}$ to a mixed number, divide the numerator (37) by the denominator (4). $37 \div 4 = 9$ with a remainder of 1. The quotient (9) becomes the whole number, the remainder (1) becomes the new numerator, and the denominator (4) stays the same. Thus, the mixed number is $9\frac{1}{4}$.
16. A mixed number is written as $7\frac{5}{8}$. When this number is converted to an improper fraction, what is its numerator?
    Answer: 61 To convert the mixed number $7\frac{5}{8}$ to an improper fraction, multiply the whole number (7) by the denominator (8) and then add the numerator (5). Calculation: $(7 \times 8) + 5 = 56 + 5 = 61$. This result, 61, is the numerator of the improper fraction $\frac{61}{8}$.
17. A carpenter cuts a board into three pieces with lengths $2\frac{3}{16}$ inches, $\frac{41}{16}$ inches, and $1\frac{15}{16}$ inches. What was the total length of the original board expressed as an improper fraction?
    Answer: $\frac{107}{16}$ First, convert all measurements to improper fractions: $2\frac{3}{16} = \frac{32 + 3}{16} = \frac{35}{16}$; $\frac{41}{16}$ is already improper; $1\frac{15}{16} = \frac{16 + 15}{16} = \frac{31}{16}$. Now add: $\frac{35}{16} + \frac{41}{16} + \frac{31}{16} = \frac{35 + 41 + 31}{16} = \frac{107}{16}$.
18. A recipe requires $2\frac{3}{4}$ cups of flour. If the recipe is measured out using only a $\frac{1}{4}$ cup scoop, how many scoops of flour are needed?
    Answer: 11 To solve this, first convert the mixed number $2\frac{3}{4}$ into an improper fraction. Multiply the whole number by the denominator ($2 \times 4 = 8$) and add the numerator ($8 + 3 = 11$). The improper fraction is $\frac{11}{4}$. This fraction represents eleven parts, where each part is $\frac{1}{4}$. Therefore, 11 scoops of $\frac{1}{4}$ cup are needed.
19. A pie is cut into 8 equal slices. A group eats 11 slices, which is one whole pie and 3 slices from a second pie. Which improper fraction represents the total amount of pie eaten?
    Answer: $\frac{11}{8}$ The amount eaten is $1\frac{3}{8}$ pies. To convert this mixed number to an improper fraction, multiply the whole number (1) by the denominator (8) and add the numerator (3): $1 \times 8 + 3 = 11$. The denominator stays 8. Therefore, the improper fraction is $\frac{11}{8}$. Choice A is the mixed number, not the improper fraction. Choice B incorrectly inverts the fraction. Choice D incorrectly adds the denominators of two pies.
20. If $\frac{a}{b} = 4\frac{3}{7}$ where $a$ and $b$ are positive integers and $\gcd(a,b) = 1$, what is the value of $a + b$?
    Answer: $38$ Convert $4\frac{3}{7}$ to an improper fraction: $4\frac{3}{7} = \frac{4 \times 7 + 3}{7} = \frac{28 + 3}{7} = \frac{31}{7}$. Since $\gcd(a,b) = 1$, the fraction $\frac{a}{b}$ is in lowest terms. Since 31 and 7 are both prime and different, $\gcd(31,7) = 1$. Therefore, $a = 31$ and $b = 7$, giving us $a + b = 31 + 7 = 38$.

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