HSPT Math Quiz: Perform Fraction Operations

What this quiz covers

This quiz focuses on Perform Fraction Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 2.4 L
2. 3.0 L (correct answer)
3. 4.0 L
4. 6.0 L

Explanation: 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.

Question 2

1. $1\frac{2}{3}$ liters
2. $2$ liters (correct answer)
3. $2\frac{1}{3}$ liters
4. $2\frac{2}{3}$ liters

Explanation: 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.

Question 3

1. $\frac{7}{16}$
2. $\frac{14}{33}$ (correct answer)
3. $\frac{7}{18}$
4. $\frac{11}{24}$

Explanation: 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.

Question 4

1. $\frac{17}{24}$ cup (correct answer)
2. $\frac{19}{24}$ cup
3. $\frac{21}{24}$ cup
4. $\frac{23}{24}$ cup

Explanation: This problem tests your ability to add mixed numbers and fractions, then divide the result. When you see a multi-step word problem involving fractions, break it down: first find the total amount, then perform the final operation. Start by adding all three ingredients. Convert the mixed numbers to improper fractions with a common denominator of 8: $1\frac{7}{8} = \frac{15}{8}$, $1\frac{3}{4} = \frac{14}{8}$, and $\frac{5}{8} = \frac{5}{8}$. Adding these gives $\frac{15}{8} + \frac{14}{8} + \frac{5}{8} = \frac{34}{8} = \frac{17}{4}$ cups total. Now divide this total among 6 containers: $\frac{17}{4} \div 6 = \frac{17}{4} \times \frac{1}{6} = \frac{17}{24}$ cups per container. Looking at the wrong answers: Choice B ($\frac{19}{24}$) likely comes from miscalculating the total mixture, perhaps adding incorrectly or making an error when converting mixed numbers. Choice C ($\frac{21}{24}$) suggests an error in the division step, possibly dividing by 4 instead of 6. Choice D ($\frac{23}{24}$) indicates multiple calculation errors, possibly in both the addition and division phases. The correct answer is A: $\frac{17}{24}$ cup. Strategy tip: For multi-step fraction problems, work methodically and check each step. Convert mixed numbers to improper fractions early, find common denominators for addition, and remember that dividing by a whole number means multiplying by its reciprocal.

Question 5

1. $8.40 (correct answer)
2. $9.20
3. $8.60
4. $9.00

Explanation: 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.

Question 6

1. $2\frac{1}{3}$ feet (correct answer)
2. $2\frac{2}{5}$ feet
3. $2\frac{1}{5}$ feet
4. $2\frac{1}{15}$ feet

Explanation: 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.

Question 7

1. 2 trucks
2. 3 trucks (correct answer)
3. 4 trucks
4. 1 truck

Explanation: 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.

Question 8

1. $2\frac{2}{5}$ feet (correct answer)
2. $2\frac{1}{3}$ feet
3. $2\frac{2}{3}$ feet
4. $2\frac{1}{5}$ feet

Explanation: 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.

Question 9

1. 5 batches
2. 6 batches (correct answer)
3. 7 batches
4. 4 batches

Explanation: 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.

Question 10

1. $\frac{1}{3}$ (correct answer)
2. $\frac{81}{250}$
3. $\frac{10}{33}$
4. $\frac{18}{55}$

Explanation: 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: $\frac{5}{18} \times \frac{6}{5} = \frac{30}{90} = \frac{1}{3}$.

Question 11

1. $1$
2. $-1$ (correct answer)
3. $\frac{36}{25}$
4. $-\frac{25}{36}$

Explanation: Let's evaluate the expressions in both sets of parentheses.
Numerator: $\frac{4}{5} - 2 = \frac{4}{5} - \frac{10}{5} = -\frac{6}{5}$.
Denominator: $2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5}$.
The problem becomes $(-\frac{6}{5}) \div (\frac{6}{5})$.
Any non-zero number divided by its opposite is -1. So, $-\frac{6}{5} \div \frac{6}{5} = -1$. Alternatively, $-\frac{6}{5} \times \frac{5}{6} = -1$.

Question 12

1. $\frac{11}{12}$
2. $1$
3. $\frac{29}{30}$ (correct answer)
4. $\frac{61}{30}$

Explanation: 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, $\frac{3}{10} + \frac{2}{3} = \frac{9}{30} + \frac{20}{30} = \frac{29}{30}$.

Question 13

1. $22 \frac{14}{15}$
2. $6 \frac{2}{5}$
3. $5 \frac{13}{15}$ (correct answer)
4. $3 \frac{11}{30}$

Explanation: 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 $5 \frac{13}{15}$.

Question 14

1. $\frac{41}{54}$ (correct answer)
2. $\frac{12}{15}$
3. $\frac{17}{15}$
4. $\frac{19}{18}$

Explanation: When adding fractions with different denominators, you need to find a common denominator before combining the numerators. This requires finding the least common multiple (LCM) of the denominators. To add $\frac{5}{6}+\frac{7}{9}$, first find the LCM of 6 and 9. The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54... The multiples of 9 are: 9, 18, 27, 36, 45, 54... The LCM is 18. Convert each fraction: $\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$ and $\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}$ Now add: $\frac{15}{18} + \frac{14}{18} = \frac{29}{18}$ Wait—this doesn't match any answer choice! Let me recalculate using 54 as the common denominator (which is also a common multiple): $\frac{5}{6} = \frac{5 \times 9}{6 \times 9} = \frac{45}{54}$ and $\frac{7}{9} = \frac{7 \times 6}{9 \times 6} = \frac{42}{54}$ Adding: $\frac{45}{54} + \frac{42}{54} = \frac{87}{54}$ Since $\frac{87}{54} = \frac{29}{18}$, and this still doesn't match, let me check if $\frac{41}{54}$ could be correct by working backwards: $\frac{41}{54} ≈ 0.759$, while $\frac{29}{18} ≈ 1.611$. Actually, choice A) $\frac{41}{54}$ appears to be incorrect based on proper fraction addition. Choice B) $\frac{12}{15}$ incorrectly adds denominators. Choice C) $\frac{17}{15}$ incorrectly adds both numerators and denominators. Choice D) $\frac{19}{18}$ comes from adding numerators incorrectly. Always double-check your common denominator calculation and verify your final answer makes sense compared to the original fractions.

Question 15

1. $\frac{5}{6}$ (correct answer)
2. $1$
3. $\frac{13}{6}$
4. $\frac{5}{4}$

Explanation: First, convert the decimal 1.5 to a fraction, which is $\frac{3}{2}$. Then evaluate the numerator and denominator.\nNumerator: $1.5 - \frac{2}{3} = \frac{3}{2} - \frac{2}{3}$. The common denominator is 6. $\frac{9}{6} - \frac{4}{6} = \frac{5}{6}$.\nDenominator: $1.5 \times \frac{2}{3} = \frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1$.\nThe expression simplifies to $\frac{5/6}{1}$, which is $\frac{5}{6}$.

Question 16

1. $\frac{19}{48}$
2. $\frac{19}{32}$ (correct answer)
3. $\frac{59}{32}$
4. $\frac{38}{9}$

Explanation: 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: $\frac{19}{4 \times 3} \times \frac{3}{8} = \frac{19}{4 \times 8} = \frac{19}{32}$.

Question 17

1. $1$
2. $-\frac{2}{5}$
3. $-\frac{38}{125}$
4. $\frac{1}{5}$ (correct answer)

Explanation: 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: $\frac{3}{5} - \frac{2}{5} = \frac{1}{5}$.

Question 18

1. 4.5 (correct answer)
2. 2.25
3. 3.125
4. 1.5

Explanation: 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: $x = 4.5$.

Question 19

1. $(\frac{1}{2} + \frac{1}{3}) \div \frac{1}{6}$
2. $(\frac{1}{2} \div \frac{1}{3}) \times 6$
3. $(\frac{1}{2} - \frac{1}{3}) \times 30$
4. $6 \div (\frac{1}{2} - \frac{1}{4})$ (correct answer)

Explanation: 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.

Question 20

1. $\frac{3}{4}$ cup (correct answer)
2. $\frac{2}{3}$ cup
3. $1 \frac{1}{16}$ cups
4. $4 \frac{11}{16}$ cups

Explanation: To find the amount of flour for one dozen muffins, you need to divide the total amount of flour by the number of dozens.
Amount of flour = $1 \frac{7}{8}$ cups.
Number of dozens = $2 \frac{1}{2}$.
The calculation is $1 \frac{7}{8} \div 2 \frac{1}{2}$.
First, convert the mixed numbers to improper fractions: $\frac{15}{8} \div \frac{5}{2}$.
To divide, multiply by the reciprocal of the second fraction: $\frac{15}{8} \times \frac{2}{5}$.
Simplify before multiplying: $\frac{15}{5} = 3$ and $\frac{2}{8} = \frac{1}{4}$.
So, the expression becomes $\frac{3}{4} \times 1 = \frac{3}{4}$.
Exactly $\frac{3}{4}$ cup of flour is needed for one dozen muffins.