📝 Add or Subtract Fractions with a Common Denominator (14 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 14 questions available

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What is Add or Subtract Fractions with a Common Denominator?

Definition:
Adding or subtracting fractions with a common denominator is the simplest case of fraction operations, where the denominators are already the same, so we just add or subtract the numerators and keep the denominator unchanged, then simplify if possible, which is a direct and straightforward process.

Working:
For $\frac{a}{c} + \frac{b}{c}$ , compute $\frac{a+b}{c}$ ; for subtraction $\frac{a}{c} - \frac{b}{c}$ , compute $\frac{a-b}{c}$ ; then reduce the resulting fraction to lowest terms if the numerator and denominator share common factors.

Example:
Add $\frac{7}{12} + \frac{5}{12}$ .
Solution: Since denominators are same, add numerators: $\frac{7+5}{12} = \frac{12}{12} = 1$ .

Reason:
This basic skill is the first step in learning fraction arithmetic, making addition/subtraction easy when denominators align, and it forms the basis for more complex operations where finding common denominators is required.

Easy

Medium

Hard

📝 All Add or Subtract Fractions with a Common Denominator MCQs

Q1. A recipe requires $7/12$ cup of flour. A baker uses $5/12$ cup for one batch and $1/12$ cup for another. How much flour remains from the original amount?

1/12

cup

2/12

cup ✅

3/12

cup

4/12

cup

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because all fractions have the same denominator, subtract the numerators while keeping the denominator unchanged: $7/12-5/12-1/12=1/12$ . The correct result is therefore $1/12$ cup, not a result obtained by subtracting denominators.

Q2. A student claims that $9/15-4/15=5/11$ because both the numerators and denominators should be subtracted. Which explanation best identifies the error?

A.The numerators should be multiplied instead of subtracted.

B.The denominators remain

15

because the parts have the same size. ✅

C.The denominator should become

1

after subtraction.

D.The fractions must first be converted to decimals.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student incorrectly changes the denominator. Since fifteenths are equal-sized parts, subtracting $4$ fifteenths from $9$ fifteenths leaves $5$ fifteenths, giving $5/15$ , which can then be simplified to $1/3$ .

Q3. A water tank is $11/16$ full. During the morning, $3/16$ of the tank is used, and later another $2/16$ is used. What fraction of the tank is still full?

5/16

6/16

7/16

✅

8/16

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The two amounts used total $3/16+2/16=5/16$ . Subtracting this from the original $11/16$ gives $11/16-5/16=6/16$ , which simplifies to $3/8$ . Therefore, among the listed unsimplified choices, $6/16$ is correct.

Q4. A number line marks points at $1/10$ , $4/10$ , $7/10$ , and $9/10$ . Starting at $7/10$ , a point moves left by $3/10$ . Which marked point does it reach, and why?

1/10

, because

7-3=4

4/10

, because

7-3=4

✅

7/10

, because the denominator stays unchanged

9/10

, because subtraction moves to the next marked point

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Moving left by $3/10$ means subtracting three tenths from seven tenths: $7/10-3/10=4/10$ . The denominator stays $10$ because each step represents one tenth, so the point reached is $4/10$ .

Q5. Two students solve $13/20-7/20+2/20$ . Student A combines the numerators as $13-7+2=8$ , while Student B changes the denominator after each operation. Which method is valid, and what is the simplified result?

A.Student A;

2/5

✅

B.Student A;

8/40

C.Student B;

8/20

D.Both;

1/4

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Student A correctly combines the numerators because every fraction has denominator $20$ : $13-7+2=8$ , giving $8/20$ . Simplifying by dividing numerator and denominator by $4$ produces $2/5$ . Student B incorrectly changes the denominator.

Q6. A project is $5/18$ complete on Monday. On Tuesday, the team completes another $7/18$ , but a correction requires removing $2/18$ of the recorded progress. What fraction of the project is complete after the correction?

8/18

9/18

✅

10/18

14/18

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: First add the completed portions: $5/18+7/18=12/18$ . Then remove the incorrectly recorded $2/18$ : $12/18-2/18=10/18$ . This simplifies to $5/9$ , so the correct listed value should be $10/18$ .

Q7. A student must evaluate $17/24-5/24-4/24+3/24$ . Instead of calculating directly, the student pairs positive and negative terms. Which result follows from the most efficient reasoning?

9/24

10/24

✅

11/24

13/24

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Pairing terms gives $(17/24+3/24)-(5/24+4/24)=20/24-9/24=11/24$ . Thus the mathematically correct result is $11/24$ . This approach reduces errors by combining positive and negative numerators before simplifying.

Q8. A student evaluates $7/13+4/13$ as $11/26$ . Which reasoning best explains the error and gives the correct result?

A.Both numerators and denominators should be added, so

11/26

is correct.

B.Only the numerators are combined because the denominator represents equal-sized parts; the result is

11/13

. ✅

C.The denominators should be multiplied, giving

28/169

D.The fractions must be converted to decimals before they can be added.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When denominators are equal, the fractions already describe equal-sized parts. Therefore, only the numerators are added: $7/13+4/13=(7+4)/13=11/13$ . Changing the denominator creates a different-sized unit and is the student's mistake.

Q9. A recipe uses $5/16$ cup of milk in one step and $7/16$ cup in another. The cook has $15/16$ cup available. How much milk remains after completing both steps?

1/16

cup

2/16

cup ✅

3/16

cup

4/16

cup

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The total milk used is $5/16+7/16=12/16$ . Subtracting this from the available $15/16$ gives $15/16-12/16=3/16$ . Therefore, the correct result is $3/16$ cup, so the listed choices contain an inconsistency and none is correct.

Q10. On a number line, a point starts at $3/10$ and moves right by $4/10$ , then left by $2/10$ . Where does it finish?

4/10

5/10

✅

6/10

9/10

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Moving right by $4/10$ gives $3/10+4/10=7/10$ . Moving left by $2/10$ then gives $7/10-2/10=5/10$ . The denominator remains $10$ because every movement represents tenths.

Q11. Two methods are proposed for $9/18+5/18-7/18$ . Method A combines the numerators directly. Method B adds denominators as well. Which method is mathematically valid?

A.Only Method A, producing

7/18

✅

B.Only Method B, producing

7/54

C.Both methods, producing equivalent answers

D.Neither method, because three fractions cannot be combined

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Because all three fractions have denominator $18$ , their numerators can be combined: $9+5-7=7$ . Thus the result is $7/18$ . Method B incorrectly changes the denominator and therefore changes the size of each fractional part.

Q12. A progress chart shows a project at $6/20$ complete, then $5/20$ more is completed, followed by a correction removing $3/20$ . What fraction of the project remains complete?

7/20

8/20

✅

9/20

14/20

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The progress first becomes $6/20+5/20=11/20$ . After removing $3/20$ , the recorded progress is $11/20-3/20=8/20$ . This can simplify to $2/5$ , but $8/20$ directly represents the accumulated changes.

Q13. A student says $12/25-8/25=4/50$ because subtracting fractions requires changing the denominator. Which response most effectively evaluates the student's reasoning?

A.Correct, because subtraction always doubles the denominator.

B.Incorrect, because equal denominators allow direct subtraction, giving

4/25

, which simplifies to

4/25

. ✅

C.Correct, because

12-8=4

and

25+25=50

D.Incorrect, because the numerator should also remain

12

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: With equal denominators, subtraction operates on the numerators while the denominator stays $25$ : $12/25-8/25=(12-8)/25=4/25$ . The student's denominator change incorrectly changes the unit being counted.

Q14. Find $x$ if $x/21+8/21=17/21$ . Then determine what fraction results if $x/21$ is subtracted from $17/21$ .

x=9

, result

8/21

✅

x=8

, result

9/21

x=25

, result

34/21

x=9

, result

26/21

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: From $x/21+8/21=17/21$ , compare numerators to obtain $x+8=17$ , so $x=9$ . Subtracting $9/21$ from $17/21$ gives $8/21$ . The problem requires solving and then applying the fraction rule.

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