📝 Divide Fractions in Algebra (21 MCQs)

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

←Previous Topic:Multiply Fractions in Algebra📝 Divide Fractions in AlgebraFraction bar expressions simplification:Next Topic→

What is Divide Fractions in Algebra?

Definition:
Dividing fractions in algebra means finding the quotient of two fractions by multiplying the first fraction by the reciprocal (or multiplicative inverse) of the second fraction, where the reciprocal is obtained by swapping the numerator and denominator, and then simplifying the result.

Working:
To divide $\frac{a}{b} \div \frac{c}{d}$ , rewrite as $\frac{a}{b} \times \frac{d}{c}$ , then multiply the fractions by multiplying numerators and denominators, and simplify by canceling common factors; ensure that $c \neq 0$ .

Example:
Divide $\frac{5}{6} \div \frac{10}{3}$ .
Solution: Reciprocal of $\frac{10}{3}$ is $\frac{3}{10}$ ; multiply: $\frac{5}{6} \times \frac{3}{10} = \frac{15}{60} = \frac{1}{4}$ (after simplifying by 15).

Reason:
Division of fractions is key in solving equations that involve fractions, converting units, and simplifying complex rational expressions, and it reinforces the concept of multiplicative inverses in algebra.

Easy

Medium

Hard

📝 All Divide Fractions in Algebra MCQs

Q1. A recipe uses $3/4$ cup of flour for each batch. If $3$ cups of flour are available, how many complete batches can be made?

A.2

B.3

C.4 ✅

9/4

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: To determine the number of batches, divide the available flour by the amount required per batch: $3\div 3/4=3\times4/3=4$ . The realistic distractors result from multiplying instead of dividing or confusing the fraction with its reciprocal.

Q2. A student claims that $5/6\div2/3=5/9$ because both numerator and denominator should be divided by $2$ . Which response best evaluates the student's reasoning?

A.The answer is correct because division always reduces both fractions.

B.The answer is incorrect because dividing by

2/3

means multiplying by

3/2

, giving

5/4

. ✅

C.The answer is incorrect because the fractions must first be converted to decimals.

D.The answer is correct because

6\div3=2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The student incorrectly treats division of fractions as separate division of numerators and denominators. Instead, divide by multiplying by the reciprocal: $5/6\times3/2=15/12=5/4$ . This exposes a common misconception about fraction division.

Q3. A water tank contains $2\frac{1}{2}$ liters. Each bottle holds $5/8$ liter. After filling as many complete bottles as possible, how many full bottles can be filled and how much water remains?

A.3 bottles,

5/8

L remains

B.4 bottles,

0

L remains

C.4 bottles,

1/8

L remains ✅

D.5 bottles,

1/8

L remains

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Convert $2\frac{1}{2}$ to $5/2$ , then divide: $5/2\div5/8=5/2\times8/5=4$ . Thus four bottles use $20/8=5/2$ liters, leaving $0$ liters. Therefore option B is mathematically correct, making the intended answer B; the other choices reflect common errors in converting or interpreting the remainder.

Q4. On a number line, points $P$ and $Q$ are located at $3/4$ and $3$ . A student interprets $3\div3/4$ as the number of equal $3/4$ -length intervals needed to travel from $0$ to $3$ . Which value should the student obtain?

A.2

B.3

C.4 ✅

9/4

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The quotient asks how many intervals of length $3/4$ fit into $3$ . Since $3\div3/4=3\times4/3=4$ , exactly four intervals are needed. The question connects fraction division to measurement and number-line interpretation.

Q5. Two methods are proposed for $7/8\div1/4$ . Method A multiplies $7/8$ by $4/1$ . Method B divides $7$ by $8$ and then divides $1$ by $4$ . Which conclusion is correct?

A.Only Method A works, giving

7/2

. ✅

B.Only Method B works, giving

7/32

C.Both methods work and give

7/2

D.Neither method works because fractions cannot be divided.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Dividing by $1/4$ is equivalent to multiplying by its reciprocal, $4$ . Therefore $7/8\times4=28/8=7/2$ . Method B incorrectly divides the numerator and denominator separately, producing a value that does not represent the original quotient.

Q6. A rectangular garden has area $5/6$ square meter and width $1/3$ meter. What is its length, and why does the result make sense?

5/18

m because the fractions are multiplied

5/2

m because

5/6\div1/3=5/6\times3

✅

2/5

m because the reciprocal of

5/6

is used

1/2

m because

5/6-1/3=1/2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For a rectangle, length equals area divided by width. Thus $5/6\div1/3=5/6\times3/1=15/6=5/2$ meters. The result exceeds $1$ meter because a width of only $1/3$ meter must be repeated several times to cover the given area.

Q7. A puzzle asks for a positive fraction $x$ such that $3/5\div x=9/10$ . Which value of $x$ satisfies the equation?

1/3

2/3

✅

3/2

6/5

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Solve by recognizing that $3/5\div x=9/10$ means $x=(3/5)\div(9/10)$ . Multiplying by the reciprocal gives $3/5\times10/9=30/45=2/3$ . This requires reversing the usual division process and checking the resulting quotient.

Q8. Which statement correctly describes the reciprocal of a nonzero fraction $a/b$ ?

A.It is

a/b

multiplied by itself.

B.It is obtained by switching the numerator and denominator, giving

b/a

. ✅

C.It is found by changing both signs.

D.It is always smaller than the original fraction.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For any nonzero fraction $a/b$ , its reciprocal is $b/a$ . Multiplying the original fraction by its reciprocal gives $a/b\times b/a=1$ . The condition that the fraction is nonzero is essential because zero has no reciprocal.

Q9. A student says the reciprocal of $-3/7$ is $3/-7$ , while another says it is $7/3$ . Which evaluation is correct?

A.Only the first student is correct.

B.Only the second student is correct.

C.Both are correct because

3/-7=-3/7

, but the reciprocal is

-7/3

, so neither stated answer is correct. ✅

D.Both are incorrect because negative fractions have no reciprocal.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The reciprocal of $-3/7$ is obtained by interchanging numerator and denominator while keeping the negative sign, giving $-7/3$ . Although $3/-7$ represents the original fraction, it does not interchange the numerator and denominator.

Q10. A machine uses $2/3$ kilogram of material for each unit it produces. If the production rate is represented by dividing $1$ kilogram by $2/3$ , which value represents the number of units produced per kilogram?

2/3

1/3

3/2

✅

5/3

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The number of units per kilogram is found by calculating $1\div2/3$ . Dividing by a fraction means multiplying by its reciprocal, so $1\times3/2=3/2$ . This represents one and a half units per kilogram.

Q11. A number line marks $A=2/5$ and $B=5/2$ . Which interpretation best explains their relationship?

A.They are opposites because they lie on different sides of zero.

B.They are reciprocals because their product is

1

. ✅

C.They are equal because their numerators and denominators are reversed.

D.They are additive inverses because their sum is

0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The points $2/5$ and $5/2$ represent numbers whose product is $2/5\times5/2=1$ . Therefore they are reciprocals. They are not opposites or additive inverses because both values are positive and their sum is not zero.

Q12. A student claims that the reciprocal of $4$ is $1/4$ , but the reciprocal of $1/4$ is also $1/4$ . What error has the student made?

A.The reciprocal of

4

4

B.The reciprocal of

1/4

4

, because taking the reciprocal reverses the fraction. ✅

C.A reciprocal can only be found for negative numbers.

D.The reciprocal operation changes the number into its opposite.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The student correctly identifies $1/4$ as the reciprocal of $4$ , but then incorrectly leaves $1/4$ unchanged. Reversing numerator and denominator in $1/4$ produces $4/1=4$ , showing that reciprocation reverses the operation.

Q13. If $x$ is a positive number greater than $1$ , which statement must be true about its reciprocal $1/x$ , and why?

1/x>1

, because reciprocals increase positive numbers.

1/x=x

, because every positive number equals its reciprocal.

0<1/x<1

, because multiplying

x

1/x

gives

1

. ✅

1/x<0

, because reciprocals change the sign.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When $x>1$ , its reciprocal $1/x$ is positive but less than $1$ . This follows because $x(1/x)=1$ . A number greater than $1$ must be paired with a positive number less than $1$ to produce exactly $1$ .

Q14. Two positive fractions $p$ and $q$ satisfy $p<q<1$ . Which comparison between their reciprocals is necessarily true?

1/p<1/q

1/p=1/q

1/p>1/q

✅

D.There is not enough information to compare them.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For positive fractions less than $1$ , the smaller the original fraction, the larger its reciprocal. Since $p<q$ , multiplying reciprocals reverses the order, so $1/p>1/q$ . This can also be verified by testing suitable positive fractions such as $1/3$ and $1/2$ .

Q15. What is the correct first step when evaluating $3/5\div2/7$ ?

A.Multiply

3/5

2/7

B.Multiply

3/5

7/2

. ✅

C.Add

3/5

and

2/7

D.Subtract

2/7

from

3/5

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: To divide by a fraction, keep the first fraction and multiply by the reciprocal of the second fraction. Therefore $3/5\div2/7=3/5\times7/2$ . This preserves the meaning of division and avoids the common mistake of multiplying by the original divisor.

Q16. A student calculates $4/9\div3/8$ as $4/9\times3/8$ . What is the most important error in this reasoning?

A.The first fraction should also be reversed.

B.The divisor should be replaced by its reciprocal before multiplying. ✅

C.The fractions should be added instead.

D.The denominators should always be equal before division.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The divisor $3/8$ must be changed to its reciprocal $8/3$ . Thus $4/9\div3/8=4/9\times8/3=32/27$ . Multiplying by $3/8$ instead produces a much smaller value and does not represent the required division.

Q17. A recipe requires $2/3$ cup of milk for each serving. A container has $4$ cups. How many servings can be prepared using all the milk?

A.2

B.4

C.6 ✅

D.8

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The number of servings is $4\div2/3$ . Rewrite $4$ as $4/1$ , then multiply by the reciprocal: $4/1\times3/2=12/2=6$ . Therefore, the container provides enough milk for six servings.

Q18. On a number line, the distance from $0$ to $3/2$ is divided into intervals of length $3/8$ . Which expression determines the number of intervals?

3/2\times3/8

3/2\div3/8

✅

3/8\div3/2

3/2+3/8

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The number of equal intervals is found by dividing the total distance by the interval length. Thus $3/2\div3/8=3/2\times8/3=4$ . Four intervals of length $3/8$ exactly cover the distance from $0$ to $3/2$ .

Q19. Two students solve $5/6\div10/9$ . Student A writes $5/6\times9/10$ . Student B writes $5/6\times10/9$ . Who is correct, and what is the result?

A.Student A,

3/4

✅

B.Student B,

3/4

C.Student A,

3/2

D.Student B,

3/2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The second fraction must be replaced by its reciprocal. Therefore $5/6\div10/9=5/6\times9/10=45/60=3/4$ . Student B incorrectly multiplies by the divisor itself rather than its reciprocal.

Q20. A rectangular sign has an area of $7/8$ square meter and a width of $1/4$ meter. What is its length?

7/32

1/2

7/2

m ✅

8/7

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Length equals area divided by width, so $7/8\div1/4$ is required. Applying the rule gives $7/8\times4/1=28/8=7/2$ meters. Multiplication by $1/4$ would incorrectly make the length smaller than the area.

Q21. For positive fractions $a/b$ and $c/d$ , suppose $a/b>1$ and $0<c/d<1$ . Without calculating exact values, what can be concluded about $a/b\div c/d$ ?

A.It must be less than

a/b

B.It must equal

c/d

C.It must be greater than

a/b

. ✅

D.It must always equal

1

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Dividing a positive number by a positive fraction less than $1$ increases the result. Using the rule, $a/b\div c/d=a/b\times d/c$ , and because $d/c>1$ , the product is greater than $a/b$ . This tests structural reasoning rather than routine computation.

🔗 Related Topics (MCQs)

📝 Place value of whole numbers MCQs 📝 How to round whole numbers MCQs 📝 Identify Multiples and divisibility Tests and rules MCQs 📝 Prime factorization and (LCM) Least Common Multiples MCQs 📝 Variables and algebraic symbols MCQs

←Previous Topic:Multiply Fractions in Algebra📝 Divide Fractions in AlgebraFraction bar expressions simplification:Next Topic→