📝 Multiply Fractions in Algebra (21 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 21 questions available

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What is Multiply Fractions in Algebra?

Definition:
Multiplying fractions in algebra involves finding the product of two or more fractions by multiplying their numerators together to get the new numerator and their denominators together to get the new denominator, simplifying the result by canceling common factors before or after multiplication.

Working:
Multiply the numerators: $a \times c$ for $\frac{a}{b} \times \frac{c}{d}$ , and denominators: $b \times d$ , then simplify by dividing common factors; it is often easier to cancel factors diagonally (cross-cancel) before multiplying to reduce large numbers.

Example:
Multiply $\frac{2}{3} \times \frac{9}{4}$ .
Solution: Cancel 3 and 9 (common factor 3): $\frac{2}{1} \times \frac{3}{4}$ ; then multiply: $\frac{2 \times 3}{1 \times 4} = \frac{6}{4} = \frac{3}{2}$ .

Reason:
This operation is used in scaling, proportions, and algebraic rational expressions, and understanding multiplication of fractions is essential for more advanced topics like operations with rational functions and solving equations with fractions.

Easy

Medium

Hard

📝 All Multiply Fractions in Algebra MCQs

Q1. A student calculates $\frac{3}{5}\times\frac{10}{9}$ . Which result is correct after simplifying efficiently before multiplying?

\frac{2}{3}

✅

\frac{30}{45}

\frac{13}{14}

\frac{3}{2}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The factors can be simplified before multiplication: $10\div5=2$ and $9\div3=3$ . This gives $\frac{1\times2}{1\times3}=\frac{2}{3}$ . This method reduces the numbers while preserving the value of the product.

Q2. Two fractions represent portions of the same quantity. A student claims that multiplying $\frac{2}{3}$ by $\frac{3}{4}$ should produce a number larger than both fractions because multiplication usually makes numbers larger. Which reasoning best evaluates the claim?

A.The claim is correct because multiplication always increases a number.

B.The claim is incorrect because multiplying two positive fractions less than

1

produces a result smaller than either factor. ✅

C.The claim is correct only when the denominators are different.

D.The claim is incorrect because fractions cannot be multiplied when both are less than

1

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The claim confuses multiplication by whole numbers with multiplication by fractions. Since $\frac{2}{3}$ and $\frac{3}{4}$ are both less than $1$ , their product $\frac{1}{2}$ is smaller than either factor. This reflects scaling by a quantity below one.

Q3. A recipe uses $\frac{3}{4}$ cup of flour for one batch. A baker prepares $\frac{2}{3}$ of a batch. How much flour is needed, and why is multiplication appropriate?

\frac{1}{2}

cup, because

\frac{2}{3}

scales the original amount ✅

\frac{5}{7}

cup, because the fractions should be added

\frac{9}{8}

cups, because both quantities are multiplied by their denominators

\frac{3}{8}

cup, because the denominators must be multiplied only

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The amount needed is $\frac{3}{4}\times\frac{2}{3}=\frac{6}{12}=\frac{1}{2}$ cup. Multiplication models taking a fractional portion of an existing quantity, so $\frac{2}{3}$ acts as a scale factor for the full-batch amount.

Q4. A student computes $\frac{4}{7}\times\frac{14}{15}$ as $\frac{56}{105}$ and then says the answer is already simplest because 56 and 105 are different numbers. What is the best correction?

A.The answer should be

\frac{4}{15}

because 56 and 105 share a common factor of 7. ✅

B.The answer should be

\frac{18}{35}

because the numerators must be added.

C.The answer is correct because fractions with different numerator and denominator are always simplified.

D.The answer should be

\frac{8}{15}

because 14 cancels with 7 after multiplication.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The multiplication $\frac{4}{7}\times\frac{14}{15}=\frac{56}{105}$ is correct initially, but both 56 and 105 are divisible by 7. Dividing by 7 gives $\frac{8}{15}$ , so the student's conclusion about simplification is incorrect.

Q5. A number line represents the product $\frac{3}{4}\times\frac{2}{5}$ by first locating $\frac{3}{4}$ and then taking $\frac{2}{5}$ of that distance from zero. Which point should represent the product?

\frac{1}{5}

\frac{3}{10}

✅

\frac{5}{9}

\frac{7}{10}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Taking $\frac{2}{5}$ of $\frac{3}{4}$ means multiplying them: $\frac{2}{5}\times\frac{3}{4}=\frac{6}{20}=\frac{3}{10}$ . On the number line, the product must lie between zero and $\frac{3}{4}$ , consistent with multiplying by a factor less than one.

Q6. A rectangular garden has length $\frac{5}{6}$ km and width $\frac{3}{10}$ km. A planner calculates its area as $\frac{8}{16}$ square km by adding the numerators and denominators. Which analysis identifies the correct approach and result?

A.Add the fractions to obtain

\frac{8}{16}

, which equals

\frac{1}{2}

B.Multiply the fractions:

\frac{5}{6}\times\frac{3}{10}=\frac{1}{4}

square km. ✅

C.Subtract the width from the length to obtain

\frac{8}{15}

square km.

D.Multiply only the denominators to obtain

\frac{1}{60}

square km.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Area of a rectangle is found by multiplying its length and width. Thus $\frac{5}{6}\times\frac{3}{10}=\frac{15}{60}=\frac{1}{4}$ square km. Adding numerators and denominators is not a valid multiplication strategy and produces an unrelated value.

Q7. Without calculating every product separately, compare $\frac{7}{12}\times\frac{8}{21}$ and $\frac{5}{9}\times\frac{6}{25}$ . Which conclusion is correct?

A.The first product is larger. ✅

B.The second product is larger.

C.The products are equal.

D.Their relationship cannot be determined without decimal conversion.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first product simplifies to $\frac{7}{12}\times\frac{8}{21}=\frac{2}{9}$ , while the second becomes $\frac{5}{9}\times\frac{6}{25}=\frac{2}{15}$ . Since $\frac{2}{9}>\frac{2}{15}$ , the first product is larger. Strategic cancellation avoids unnecessary computation.

Q8. A student wants to multiply $\frac{4}{9}$ by $\frac{3}{8}$ . Which expression correctly applies the multiplication rule before simplifying?

\frac{4\times3}{9\times8}

✅

\frac{4+3}{9+8}

\frac{4\times8}{9\times3}

\frac{4+3}{9\times8}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For multiplying fractions, multiply the numerators together and multiply the denominators together. Therefore, $\frac{4}{9}\times\frac{3}{8}=\frac{4\times3}{9\times8}$ . The resulting fraction can then be simplified to obtain the final value.

Q9. A student claims that $\frac{2}{5}\times\frac{15}{8}=\frac{30}{13}$ because multiplying fractions means multiplying the numerators and adding the denominators. Which response best identifies the error?

A.The denominators should also be multiplied, giving

\frac{30}{40}

. ✅

B.The numerators should be added, giving

\frac{17}{40}

C.The denominators should be subtracted, giving

\frac{30}{3}

D.The fractions should first be converted to decimals before multiplying.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The student has confused fraction multiplication with an invalid operation involving denominator addition. The correct rule multiplies corresponding parts: $\frac{2}{5}\times\frac{15}{8}=\frac{30}{40}=\frac{3}{4}$ .

Q10. A tank is $\frac{3}{5}$ full. A machine removes $\frac{2}{3}$ of the water currently in the tank. What fraction of the tank's total capacity is removed?

\frac{2}{5}

\frac{6}{15}

only, which cannot be simplified

\frac{1}{5}

✅

\frac{5}{6}

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The machine removes $\frac{2}{3}$ of the amount already present, so multiplication models the situation: $\frac{2}{3}\times\frac{3}{5}=\frac{6}{15}=\frac{2}{5}$ . Thus the removed amount is $\frac{2}{5}$ , not $\frac{1}{5}$ .

Q11. On a number line, a point is located at $\frac{5}{6}$ . Another operation takes $\frac{3}{10}$ of the distance from zero to that point. Which location represents the resulting product?

\frac{1}{4}

✅

\frac{1}{2}

\frac{4}{5}

\frac{13}{16}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Taking $\frac{3}{10}$ of $\frac{5}{6}$ requires multiplication: $\frac{3}{10}\times\frac{5}{6}=\frac{15}{60}=\frac{1}{4}$ . Because the scale factor $\frac{3}{10}$ is less than one, the resulting point must lie between zero and $\frac{5}{6}$ .

Q12. Two students use different methods for $\frac{7}{12}\times\frac{18}{35}$ . Student A multiplies directly and simplifies afterward. Student B cancels common factors before multiplying. Which conclusion is most accurate?

A.Only Student A can obtain the correct answer.

B.Only Student B can obtain the correct answer.

C.Both methods can produce the same answer if cancellation is valid and factors are handled correctly. ✅

D.Neither method works because fractions must first have a common denominator.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both approaches are mathematically valid. Direct multiplication gives $\frac{126}{420}$ , while valid cross-cancellation reduces the numbers before multiplication. Both must lead to the same simplified result because cancellation preserves the value of the original factors.

Q13. A rectangular sign has length $\frac{7}{8}$ meter and width $\frac{4}{9}$ meter. A worker incorrectly calculates its area as $\frac{11}{17}$ square meters. What should the worker do instead?

A.Add the fractions because area combines two measurements.

B.Multiply the fractions to obtain

\frac{7}{8}\times\frac{4}{9}=\frac{7}{18}

square meters. ✅

C.Subtract the width from the length to obtain

\frac{5}{72}

square meters.

D.Multiply only the denominators to obtain

\frac{1}{72}

square meters.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The area of a rectangle is modeled by multiplying its length and width. Applying the fraction rule gives $\frac{7\times4}{8\times9}=\frac{28}{72}=\frac{7}{18}$ square meters. The incorrect addition ignores the meaning of the measurements.

Q14. Without fully expanding the numbers, determine which product is greater: $\frac{11}{18}\times\frac{9}{22}$ or $\frac{5}{12}\times\frac{8}{15}$ .

A.The first product is greater. ✅

B.The second product is greater.

C.The two products are equal.

D.There is insufficient information to compare them.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first product simplifies through cancellation: $\frac{11}{18}\times\frac{9}{22}=\frac{1}{4}$ . The second simplifies to $\frac{5}{12}\times\frac{8}{15}=\frac{2}{9}$ . Since $\frac{1}{4}>\frac{2}{9}$ , the first product is greater.

Q15. What is the value of $-\frac{3}{4}\times 8$ ?

-6

✅

B.6

-\frac{3}{32}

\frac{32}{3}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Rewrite the integer as $\frac{8}{1}$ . Then multiply $-\frac{3}{4}\times\frac{8}{1}=-\frac{24}{4}=-6$ . The product is negative because one factor is negative and the other is positive, so the sign must be negative.

Q16. A student calculates $-\frac{5}{6}\times-\frac{9}{10}$ as $-\frac{45}{60}$ . What is the most important error in the student's reasoning?

A.The numerators should be added instead of multiplied.

B.Two negative factors produce a positive product, so the result should be

\frac{3}{4}

. ✅

C.The denominator must be negative because both fractions are negative.

D.The fractions cannot be multiplied because both are negative.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student correctly multiplied the magnitudes but incorrectly assigned the sign. A negative factor multiplied by another negative factor gives a positive product. Thus $\frac{45}{60}=\frac{3}{4}$ , not $-\frac{3}{4}$ .

Q17. A temperature changes by $-\frac{3}{2}$ degrees each hour for $4$ hours. What is the total temperature change?

-6

degrees ✅

B.6 degrees

-\frac{3}{8}

degrees

\frac{8}{3}

degrees

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The repeated change is modeled by $4\times\left(-\frac{3}{2}\right)$ . Rewriting $4$ as $\frac{4}{1}$ gives $-\frac{12}{2}=-6$ . Therefore, the temperature decreases by 6 degrees overall.

Q18. A number line shows a point at $-\frac{2}{3}$ . A transformation multiplies every position by $-\frac{3}{2}$ . Where should this point move?

-1

B.1 ✅

\frac{1}{9}

-\frac{4}{3}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The transformed position is $-\frac{2}{3}\times-\frac{3}{2}$ . The two negative signs produce a positive result, and the factors cancel to give $\frac{6}{6}=1$ . Thus the point moves to $1$ .

Q19. A company records a loss of $-\frac{2}{5}$ million dollars each quarter. If this same loss occurs for $3\frac{1}{2}$ quarters, what is the total change?

-\frac{7}{5}

million dollars ✅

-\frac{6}{5}

million dollars

\frac{7}{5}

million dollars

-\frac{5}{7}

million dollars

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Convert $3\frac{1}{2}$ to $\frac{7}{2}$ . Then multiply $-\frac{2}{5}\times\frac{7}{2}=-\frac{14}{10}=-\frac{7}{5}$ . The negative sign represents a loss, while the fractional duration scales the quarterly loss.

Q20. A student compares $-\frac{4}{7}\times 3$ with $-\frac{4}{7}\times(-3)$ and says the first product is greater because both products contain the same fraction. Which conclusion is correct?

A.The first is greater because

-\frac{12}{7}> \frac{12}{7}

B.The second is greater because

\frac{12}{7}>-\frac{12}{7}

. ✅

C.They are equal because the integer has the same magnitude.

D.The comparison cannot be determined without decimal conversion.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The first product is $-\frac{12}{7}$ , while the second is $\frac{12}{7}$ . Changing the sign of the integer changes the sign of the product. Since every positive number is greater than a corresponding negative number, the second product is greater.

Q21. A sequence is defined by multiplying each term by $-\frac{2}{3}$ . Starting with $9$ , which pair correctly gives the second and third terms?

-6,\ 4

✅

6,\ -4

-6,\ -4

4,\ -\frac{8}{3}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Multiply successively by $-\frac{2}{3}$ : $9\times-\frac{2}{3}=-6$ , then $-6\times-\frac{2}{3}=4$ . The alternating signs occur because the multiplier is negative, while the magnitude is reduced by a factor of $\frac{2}{3}$ each step.

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