Perform Fraction Operations - HSPT Math

1. A piece of ribbon is $5\frac{1}{3}$ feet long. If $\frac{2}{5}$ of it is cut off, and then $\frac{1}{4}$ of what remains is cut off again, what is the length of the final piece?
   Answer: $2\frac{2}{5}$ feet When you encounter multi-step fraction problems involving "cutting off" portions, work through each step sequentially, always being clear about what quantity you're taking the fraction of. Start with the original length: $5\frac{1}{3} = \frac{16}{3}$ feet. First, $\frac{2}{5}$ of the ribbon is cut off. Calculate: $\frac{2}{5} \times \frac{16}{3} = \frac{32}{15}$ feet removed. The remaining length is $\frac{16}{3} - \frac{32}{15} = \frac{80}{15} - \frac{32}{15} = \frac{48}{15} = \frac{16}{5}$ feet. Next, $\frac{1}{4}$ of what remains is cut off. This means $\frac{1}{4}$ of $\frac{16}{5}$: $\frac{1}{4} \times \frac{16}{5} = \frac{16}{20} = \frac{4}{5}$ feet removed. The final length is $\frac{16}{5} - \frac{4}{5} = \frac{12}{5} = 2\frac{2}{5}$ feet. Choice A ($2\frac{2}{5}$ feet) is correct. Choice B ($2\frac{1}{3}$ feet) likely results from calculation errors in the fraction arithmetic. Choice C ($2\frac{2}{3}$ feet) might come from incorrectly adding fractions or working with the wrong remaining amount after the first cut. Choice D ($2\frac{1}{5}$ feet) could result from misunderstanding what "of what remains" means in the second step—perhaps taking $\frac{1}{4}$ of the original length instead of the remaining length. Always convert mixed numbers to improper fractions for easier calculation, and remember that "of what remains" means you're working with the new, reduced quantity, not the original amount.
2. Simplify the expression: $3 \frac{1}{5} \times (4 \frac{1}{2} - 2 \frac{2}{3})$
   Answer: $5 \frac{13}{15}$ First, evaluate the expression inside the parentheses. Convert the mixed numbers to improper fractions.\ $4 \frac{1}{2} - 2 \frac{2}{3} = \frac{9}{2} - \frac{8}{3}$.\ The common denominator is 6. $\frac{27}{6} - \frac{16}{6} = \frac{11}{6}$.\ Next, multiply this result by $3 \frac{1}{5}$.\ Convert $3 \frac{1}{5}$ to an improper fraction: $\frac{16}{5}$.\ Multiply: $\frac{16}{5} \times \frac{11}{6} = \frac{176}{30}$.\ Simplify the resulting fraction: $\frac{176 \div 2}{30 \div 2} = \frac{88}{15}$.\ Convert the improper fraction back to a mixed number: $88 \div 15 = 5$ with a remainder of 13. So, the answer is $5 \frac{13}{15}$.
3. A canister of oil was $\frac{7}{8}$ full. A cook used $\frac{1}{3}$ of the oil in the canister, then added 0.5 liters of oil. If the canister is now $\frac{3}{4}$ full, what is the total capacity of the canister in liters?
   Answer: 3.0 L Let C be the total capacity of the canister. \ Initial amount of oil: $\frac{7}{8}C$.\ Amount of oil used: $\frac{1}{3}$ of $\frac{7}{8}C$, which is $\frac{1}{3} \times \frac{7}{8}C = \frac{7}{24}C$.\ Amount remaining after use: $\frac{7}{8}C - \frac{7}{24}C = \frac{21}{24}C - \frac{7}{24}C = \frac{14}{24}C = \frac{7}{12}C$.\ After adding 0.5 liters, the amount is $\frac{7}{12}C + 0.5$.\ This new amount is equal to $\frac{3}{4}C$.\ Set up the equation: $\frac{7}{12}C + 0.5 = \frac{3}{4}C$.\ To solve for C, first isolate the terms with C: $0.5 = \frac{3}{4}C - \frac{7}{12}C$.\ Find a common denominator: $0.5 = \frac{9}{12}C - \frac{7}{12}C = \frac{2}{12}C = \frac{1}{6}C$.\ So, $\frac{1}{2} = \frac{1}{6}C$. Multiply both sides by 6 to find C: $C = 6 \times \frac{1}{2} = 3$.\ The total capacity is 3.0 liters.
4. Which of the following expressions has the greatest value?
   Answer: $6 \div (\frac{1}{2} - \frac{1}{4})$ Evaluate each expression:\ A: $(\frac{3}{6} + \frac{2}{6}) \div \frac{1}{6} = \frac{5}{6} \div \frac{1}{6} = \frac{5}{6} \times 6 = 5$.\ B: $(\frac{1}{2} \times 3) \times 6 = \frac{3}{2} \times 6 = 9$.\ C: $(\frac{3}{6} - \frac{2}{6}) \times 30 = \frac{1}{6} \times 30 = 5$.\ D: $6 \div (\frac{2}{4} - \frac{1}{4}) = 6 \div \frac{1}{4} = 6 \times 4 = 24$.\ Comparing the values: 5, 9, 5, and 24. The greatest value is 24.
5. If a number is first multiplied by $\frac{5}{6}$ and then 1.25 is subtracted from the product, the result is $2 \frac{1}{2}$. What is the original number?
   Answer: 4.5 Let the original number be $x$. We can write the equation: $\frac{5}{6}x - 1.25 = 2 \frac{1}{2}$.\ To solve for $x$, we should work backward and reverse the operations.\ First, convert all numbers to a common format, either fractions or decimals. Let's use fractions. $1.25 = \frac{5}{4}$ and $2 \frac{1}{2} = \frac{5}{2}$.\ The equation becomes $\frac{5}{6}x - \frac{5}{4} = \frac{5}{2}$.\ 1. Reverse the subtraction by adding $\frac{5}{4}$ to both sides: $\frac{5}{6}x = \frac{5}{2} + \frac{5}{4}$.\ Find a common denominator: $\frac{5}{6}x = \frac{10}{4} + \frac{5}{4} = \frac{15}{4}$.\ 2. Reverse the multiplication by dividing both sides by $\frac{5}{6}$: $x = \frac{15}{4} \div \frac{5}{6}$.\ Multiply by the reciprocal: $x = \frac{15}{4} \times \frac{6}{5}$.\ Simplify and calculate: $x = \frac{3 \times 5}{2 \times 2} \times \frac{2 \times 3}{5} = \frac{3 \times 3}{2} = \frac{9}{2}$.\ Convert the result to a decimal: $x = 4.5$.
6. A carpenter cuts a board that is $8\frac{2}{3}$ feet long into three pieces. The first piece is $2\frac{5}{6}$ feet long, and the second piece is $1\frac{1}{2}$ times as long as the third piece. What is the length of the third piece?
   Answer: $2\frac{1}{3}$ feet When you encounter word problems involving parts of a whole, set up an equation that accounts for all the pieces. Here, you have a board cut into three pieces where the relationships between the pieces create the constraint. Let's call the third piece $x$ feet. Since the second piece is $1\frac{1}{2}$ times as long as the third piece, the second piece equals $1\frac{1}{2}x = \frac{3}{2}x$ feet. All three pieces must add up to the original board length: $2\frac{5}{6} + \frac{3}{2}x + x = 8\frac{2}{3}$ Convert to improper fractions: $\frac{17}{6} + \frac{3}{2}x + x = \frac{26}{3}$ Combine the $x$ terms: $\frac{17}{6} + \frac{5}{2}x = \frac{26}{3}$ Subtract $\frac{17}{6}$ from both sides: $\frac{5}{2}x = \frac{26}{3} - \frac{17}{6} = \frac{52}{6} - \frac{17}{6} = \frac{35}{6}$ Solve for $x$: $x = \frac{35}{6} \div \frac{5}{2} = \frac{35}{6} \times \frac{2}{5} = \frac{7}{3} = 2\frac{1}{3}$ Choice A gives $2\frac{1}{3}$ feet, which matches our calculation. Choice B ($2\frac{2}{5}$) might result from calculation errors in fraction arithmetic. Choice C ($2\frac{1}{5}$) could come from incorrectly setting up the relationship between pieces. Choice D ($2\frac{1}{15}$) likely stems from errors in finding common denominators. Always verify your answer: $2\frac{5}{6} + 3\frac{1}{2} + 2\frac{1}{3} = 8\frac{2}{3}$ ✓ When solving multi-step fraction problems, work systematically with improper fractions to avoid conversion errors, and always check that your pieces sum to the original whole.
7. A baker uses $\frac{3}{4}$ cup of flour for each batch of cookies. If she has $6\frac{1}{8}$ cups of flour and wants to save $1\frac{1}{2}$ cups for bread, how many complete batches of cookies can she make?
   Answer: 6 batches This is a multi-step word problem involving mixed numbers and fractions. When you encounter problems like this, identify what you have, what you need to save or use, and what remains for your main calculation. First, convert the mixed numbers to improper fractions for easier calculation. The baker has $6\frac{1}{8} = \frac{49}{8}$ cups of flour total. She wants to save $1\frac{1}{2} = \frac{3}{2}$ cups for bread. Next, find how much flour is available for cookies by subtracting what she's saving: $\frac{49}{8} - \frac{3}{2}$. Convert $\frac{3}{2}$ to eighths: $\frac{3}{2} = \frac{12}{8}$. So $\frac{49}{8} - \frac{12}{8} = \frac{37}{8}$ cups available for cookies. Since each batch requires $\frac{3}{4}$ cup, divide the available flour by the flour per batch: $\frac{37}{8} ÷ \frac{3}{4} = \frac{37}{8} × \frac{4}{3} = \frac{148}{24} = \frac{37}{6} = 6\frac{1}{6}$ Since you can only make complete batches, she can make 6 batches. Choice A (5 batches) underestimates the result—you might get this if you made an arithmetic error in the subtraction. Choice C (7 batches) overestimates—this could happen if you forgot to subtract the flour saved for bread. Choice D (4 batches) is too low and might result from incorrectly handling the mixed number conversions. Remember: in "complete batches" problems, always round down to the nearest whole number, even if the decimal portion is large.
8. A tank contains $18\frac{3}{4}$ gallons of water. If $\frac{3}{5}$ of the water is used for irrigation and $\frac{1}{8}$ of the remaining water evaporates, how many gallons are left in the tank?
   Answer: $6\frac{9}{16}$ gallons When you encounter multi-step fraction problems like this, work through each operation systematically, keeping track of what remains after each step. Start with $18\frac{3}{4}$ gallons. First, convert to an improper fraction: $18\frac{3}{4} = \frac{75}{4}$ gallons. Step 1: Find how much water is used for irrigation. $\frac{3}{5}$ of the total is used, so multiply: $\frac{3}{5} \times \frac{75}{4} = \frac{225}{20} = \frac{45}{4}$ gallons used. Step 2: Calculate what remains after irrigation. Subtract the amount used from the original: $\frac{75}{4} - \frac{45}{4} = \frac{30}{4} = \frac{15}{2}$ gallons remaining. Step 3: Find how much evaporates. $\frac{1}{8}$ of the remaining water evaporates: $\frac{1}{8} \times \frac{15}{2} = \frac{15}{16}$ gallons evaporate. Step 4: Calculate the final amount. Subtract evaporation from what remained: $\frac{15}{2} - \frac{15}{16}$. Convert to common denominators: $\frac{120}{16} - \frac{15}{16} = \frac{105}{16} = 6\frac{9}{16}$ gallons. Choice A ($5\frac{5}{16}$) likely results from calculation errors in the fraction arithmetic. Choice B ($6\frac{1}{4}$) probably comes from incorrectly handling the evaporation step. Choice C ($5\frac{15}{16}$) suggests an error in the irrigation calculation or final subtraction. The answer is D: $6\frac{9}{16}$ gallons. Study tip: In multi-step fraction problems, convert mixed numbers to improper fractions early, then convert your final answer back to mixed number form for easier comparison with answer choices.
9. What is $\frac{5}{6}+\frac{7}{9}$ expressed in simplest form?
   Answer: $\frac{29}{18}$ To add fractions, rewrite them with a common denominator. The least common denominator of 6 and 9 is 18. $\frac{5}{6} = \frac{15}{18}$ and $\frac{7}{9} = \frac{14}{18}$. Add the numerators: $\frac{15}{18} + \frac{14}{18} = \frac{29}{18}$. Since 29 is prime and does not divide 18, this is already in simplest form. Adding the tops and bottoms separately gives $\frac{5 + 7}{6 + 9} = \frac{12}{15}$, which is not how fractions add; the sum of two fractions near 1 cannot be less than 1. A quick estimate also helps: $\frac{5}{6} \approx 0.83$ and $\frac{7}{9} \approx 0.78$, so the sum is about 1.61. Values like $\frac{19}{18} \approx 1.06$ and $\frac{17}{15} \approx 1.13$ are far too small. The sum is $\frac{29}{18}$.
10. A solution contains $40\%$ acid. How many liters of pure water must be added to $3\frac{1}{3}$ liters of this solution to create a mixture that is $25\%$ acid?
    Answer: $2$ liters This is a mixture problem involving concentration percentages. When you see questions about diluting solutions, focus on tracking the amount of pure substance (acid) before and after mixing. Start by finding how much pure acid is in the original solution. You have $3\frac{1}{3} = \frac{10}{3}$ liters at 40% concentration, so the pure acid amount is $\frac{10}{3} \times 0.40 = \frac{4}{3}$ liters of acid. When you add pure water, the amount of acid stays the same, but the total volume increases. If you add $x$ liters of water, your new total volume becomes $\frac{10}{3} + x$ liters, still containing $\frac{4}{3}$ liters of acid. Set up the equation for 25% concentration: $\frac{\frac{4}{3}}{\frac{10}{3} + x} = 0.25$ Solving: $\frac{4}{3} = 0.25(\frac{10}{3} + x)$ $\frac{4}{3} = \frac{2.5}{3} + 0.25x$ $\frac{4}{3} - \frac{2.5}{3} = 0.25x$ $\frac{1.5}{3} = 0.25x$ $0.5 = 0.25x$ $x = 2$ The answer is B) 2 liters. Choice A ($1\frac{2}{3}$) would result in too high a concentration. Choice C ($2\frac{1}{3}$) and choice D ($2\frac{2}{3}$) would both dilute the solution too much, creating concentrations below 25%. For mixture problems, always identify what stays constant (the pure substance amount) versus what changes (total volume), then set up your equation based on the desired final concentration.
11. A number $N$ is equal to $0.2\overline{7}$. What is $N \div \frac{5}{6}$?
    Answer: $\frac{1}{3}$ First, convert the repeating decimal $0.2\overline{7}$ to a fraction.\ Let $x = 0.2\overline{7}$.\ Then $10x = 2.\overline{7}$ and $100x = 27.\overline{7}$.\ Subtract the two equations: $100x - 10x = 27.\overline{7} - 2.\overline{7}$\ This gives $90x = 25$, so $x = \frac{25}{90} = \frac{5}{18}$.\ Now perform the division: $N \div \frac{5}{6} = \frac{5}{18} \div \frac{5}{6}$.\ To divide by a fraction, multiply by its reciprocal: $\frac{5}{18} \times \frac{6}{5} = \frac{30}{90} = \frac{1}{3}$.
12. Evaluate the expression: $\frac{1.5 - \frac{2}{3}}{1.5 \times \frac{2}{3}}$
    Answer: $\frac{5}{6}$ First, convert the decimal 1.5 to a fraction, which is $\frac{3}{2}$. Then evaluate the numerator and denominator. Numerator: $1.5 - \frac{2}{3} = \frac{3}{2} - \frac{2}{3}$. The common denominator is 6. $\frac{9}{6} - \frac{4}{6} = \frac{5}{6}$. Denominator: $1.5 \times \frac{2}{3} = \frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1$. The expression simplifies to $\frac{5/6}{1}$, which is $\frac{5}{6}$.
13. Evaluate: $0.3 + \frac{4}{5} \times (1 \frac{1}{2} - \frac{2}{3})$
    Answer: $\frac{29}{30}$ Following the order of operations (PEMDAS), first calculate the expression in the parentheses.\ 1. Parentheses: $1 \frac{1}{2} - \frac{2}{3} = \frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}$.\ 2. Multiplication: $\frac{4}{5} \times \frac{5}{6} = \frac{20}{30} = \frac{2}{3}$.\ 3. Addition: Convert 0.3 to a fraction, $\frac{3}{10}$. Then, $\frac{3}{10} + \frac{2}{3} = \frac{9}{30} + \frac{20}{30} = \frac{29}{30}$.
14. Marcus bought $3\frac{1}{4}$ pounds of apples at $2.40 per pound and $2\frac{3}{8}$ pounds of oranges at $1.60 per pound. If he pays with a $20 bill, how much change will he receive?
    Answer: $8.40 This problem tests your ability to work with mixed numbers in a multi-step money calculation. When you see word problems involving purchases and change, break them down into clear steps: calculate each cost, find the total, then subtract from the amount paid. First, convert the mixed numbers to decimals or improper fractions for easier calculation. Marcus bought $3\frac{1}{4} = 3.25$ pounds of apples at $2.40 per pound, costing $3.25 \times 2.40 = 7.80$. He also bought $2\frac{3}{8} = 2.375$ pounds of oranges at $1.60 per pound, costing $2.375 \times 1.60 = 3.80$. The total purchase is $7.80 + 3.80 = 11.60$. His change from a $20 bill is $20.00 - 11.60 = 8.40$. Looking at the wrong answers: Choice B ($9.20) likely comes from miscalculating one of the weights—perhaps using 2.5 instead of 2.375 for the oranges. Choice C ($8.60) might result from rounding errors or using 3.5 instead of 3.25 for the apples. Choice D ($9.00) could come from multiple calculation mistakes or using rounded values throughout. The correct answer is A) $8.40. When working with mixed numbers in word problems, always convert them carefully to decimals—remember that $\frac{3}{8} = 0.375$, not 0.3 or 0.5. Double-check your multiplication and addition, especially when money is involved, since small errors compound quickly in multi-step problems.
15. What is the result of $[ (2.5 - \frac{3}{4}) \div \frac{1}{2} ] - 1 \frac{1}{8}$?
    Answer: $2 \frac{3}{8}$ Follow the order of operations, starting with the innermost parentheses.\ 1. Convert to fractions: $2.5 = 2 \frac{1}{2} = \frac{5}{2}$. The expression is $[ (\frac{5}{2} - \frac{3}{4}) \div \frac{1}{2} ] - 1 \frac{1}{8}$.\ 2. Innermost parentheses: $\frac{5}{2} - \frac{3}{4} = \frac{10}{4} - \frac{3}{4} = \frac{7}{4}$.\ 3. The expression is now $[ \frac{7}{4} \div \frac{1}{2} ] - 1 \frac{1}{8}$.\ 4. Division within brackets: $\frac{7}{4} \div \frac{1}{2} = \frac{7}{4} \times 2 = \frac{14}{4} = \frac{7}{2}$.\ 5. The expression is now $\frac{7}{2} - 1 \frac{1}{8}$.\ 6. Final subtraction: $\frac{7}{2} - \frac{9}{8} = \frac{28}{8} - \frac{9}{8} = \frac{19}{8}$.\ 7. Convert to a mixed number: $\frac{19}{8} = 2 \frac{3}{8}$.
16. A recipe calls for $1\frac{3}{8}$ cups of sugar, but Jessica only has $\frac{7}{12}$ cup available. If she makes the recipe with the amount she has, what fraction of the original sugar amount is she using?
    Answer: $\frac{14}{33}$ When you encounter a fraction problem asking "what fraction of the original amount," you're looking for a ratio. You need to divide the amount Jessica has by the amount the recipe calls for. First, convert the mixed number to an improper fraction: $1\frac{3}{8} = \frac{8 + 3}{8} = \frac{11}{8}$ cups (the original amount). Now divide Jessica's amount by the original amount: $\frac{7}{12} \div \frac{11}{8}$ To divide fractions, multiply by the reciprocal: $\frac{7}{12} \times \frac{8}{11} = \frac{7 \times 8}{12 \times 11} = \frac{56}{132}$ Simplify by finding the GCD of 56 and 132. Since $56 = 8 \times 7$ and $132 = 4 \times 33 = 4 \times 3 \times 11$, the GCD is 4. $\frac{56}{132} = \frac{56 \div 4}{132 \div 4} = \frac{14}{33}$ This confirms answer B is correct. Looking at the wrong answers: A) $\frac{7}{16}$ likely comes from incorrectly using denominators 12 and 16 (doubling 8). C) $\frac{7}{18}$ might result from adding denominators incorrectly (12 + 6 instead of proper fraction operations). D) $\frac{11}{24}$ could come from mixing up the numerators and denominators during division. Study tip: In "what fraction of" problems, always divide the actual amount by the target amount. Remember to convert mixed numbers to improper fractions first, and when dividing fractions, multiply by the reciprocal of the second fraction.
17. What is the value of $\frac{3}{5} - 2.5 \times (-\frac{2}{5})^2$?
    Answer: $\frac{1}{5}$ Following the order of operations (PEMDAS):\ 1. Exponents: $(-\frac{2}{5})^2 = (-\frac{2}{5}) \times (-\frac{2}{5}) = \frac{4}{25}$. Note that the negative sign is eliminated when squared.\ 2. Multiplication: Convert 2.5 to a fraction $\frac{5}{2}$. Then, $\frac{5}{2} \times \frac{4}{25} = \frac{20}{50} = \frac{2}{5}$.\ 3. Subtraction: $\frac{3}{5} - \frac{2}{5} = \frac{1}{5}$.
18. The first term in a sequence is 256. Each subsequent term is found by taking $\frac{3}{4}$ of the previous term. What is the fourth term of this sequence?
    Answer: 108 This is a geometric sequence. We need to find the 4th term.\ Term 1 = 256.\ Term 2 = $\frac{3}{4} \times 256 = 3 \times 64 = 192$.\ Term 3 = $\frac{3}{4} \times 192 = 3 \times 48 = 144$.\ Term 4 = $\frac{3}{4} \times 144 = 3 \times 36 = 108$.\ The fourth term is 108.
19. A construction crew needs to fill a foundation that requires $18\frac{5}{6}$ cubic yards of concrete. They have already poured $7\frac{2}{3}$ cubic yards and plan to pour the remaining amount in 3 equal loads. If each truck can carry a maximum of $4\frac{1}{4}$ cubic yards, how many trucks will be needed for the remaining concrete?
    Answer: 3 trucks Multi-step word problems with mixed numbers require you to break down the problem systematically and perform operations with fractions. Start by identifying what you need to find: the number of trucks required for the remaining concrete. First, find how much concrete is still needed. Convert the mixed numbers to improper fractions or find a common denominator. The total needed is $18\frac{5}{6}$ cubic yards, and $7\frac{2}{3}$ cubic yards have been poured. Converting $7\frac{2}{3}$ to sixths: $7\frac{4}{6}$. The remaining amount is $18\frac{5}{6} - 7\frac{4}{6} = 11\frac{1}{6}$ cubic yards. Next, divide this remaining concrete into 3 equal loads: $11\frac{1}{6} ÷ 3 = \frac{67}{6} ÷ 3 = \frac{67}{18} = 3\frac{13}{18}$ cubic yards per load. Finally, determine trucks needed per load. Each truck carries $4\frac{1}{4} = 4.25$ cubic yards maximum. Since each load is $3\frac{13}{18} ≈ 3.72$ cubic yards, one truck can handle each load. With 3 loads total, you need 3 trucks. Choice A (2 trucks) assumes incorrectly that loads can be combined or miscalculates the division. Choice D (1 truck) ignores that the concrete must be delivered in 3 separate loads, even though the total remaining concrete could fit in one truck. Choice C (4 trucks) likely results from calculation errors in the fraction operations. When solving multi-step fraction problems, work methodically through each step and always check that your final answer makes logical sense in the context.
20. What is the value of the expression $\frac{3 \frac{1}{4} - 1 \frac{2}{3}}{2 \frac{1}{2} + \frac{1}{6}}$?
    Answer: $\frac{19}{32}$ First, evaluate the numerator and the denominator separately. Convert all mixed numbers to improper fractions.\ Numerator: $3 \frac{1}{4} - 1 \frac{2}{3} = \frac{13}{4} - \frac{5}{3}$. The common denominator is 12. $\frac{13 \times 3}{4 \times 3} - \frac{5 \times 4}{3 \times 4} = \frac{39}{12} - \frac{20}{12} = \frac{19}{12}$.\ Denominator: $2 \frac{1}{2} + \frac{1}{6} = \frac{5}{2} + \frac{1}{6}$. The common denominator is 6. $\frac{5 \times 3}{2 \times 3} + \frac{1}{6} = \frac{15}{6} + \frac{1}{6} = \frac{16}{6} = \frac{8}{3}$.\ Finally, divide the numerator by the denominator: $\frac{19/12}{8/3} = \frac{19}{12} \div \frac{8}{3} = \frac{19}{12} \times \frac{3}{8}$. Simplify before multiplying: $\frac{19}{4 \times 3} \times \frac{3}{8} = \frac{19}{4 \times 8} = \frac{19}{32}$.

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