📝 Add or Subtract Fractions with Different Denominators (21 MCQs)

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

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What is Add or Subtract Fractions with Different Denominators?

Definition:
Adding or subtracting fractions with different denominators requires finding a common denominator (preferably the Least Common Denominator, LCD) so that fractions can be rewritten as equivalent fractions with the same denominator, then combining the numerators and simplifying the result.

Working:
Find the LCD by finding the LCM of the denominators; rewrite each fraction with the LCD as denominator by multiplying numerator and denominator by appropriate factors; then add or subtract numerators, keep LCD as denominator, and simplify the final fraction.

Example:
Subtract $\frac{5}{6} - \frac{1}{4}$ .
Solution: LCD of 6 and 4 is 12; $\frac{5}{6} = \frac{10}{12}$ , $\frac{1}{4} = \frac{3}{12}$ ; subtract: $\frac{10-3}{12} = \frac{7}{12}$ .

Reason:
This skill is essential when denominators are not the same, which is common in algebra, and it allows us to combine fractions in equations, simplify rational expressions, and solve real-world problems involving parts with different unit sizes.

Easy

Medium

Hard

📝 All Add or Subtract Fractions with Different Denominators MCQs

Q1. A recipe uses $3/4$ cup of flour and later adds $2/3$ cup. How much flour is used altogether?

5/7

cup

17/12

cups

13/12

cups ✅

11/12

cups

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The denominators 4 and 3 must be replaced by a common denominator, 12. Thus $3/4=9/12$ and $2/3=8/12$ . Adding gives $17/12$ , so the correct answer is $17/12$ , not $13/12$ .

Q2. Two students solve $5/6-3/8$ . Student A changes both denominators to 48 and gets $40/48-18/48$ . Student B changes them to 24 and gets $20/24-9/24$ . Which conclusion is correct?

A.Only Student A is correct

B.Only Student B is correct

C.Both are correct because both use equivalent fractions ✅

D.Neither is correct because subtraction requires equal numerators

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both students correctly create equivalent fractions with a common denominator. Student A obtains $22/48$ , while Student B obtains $11/24$ . Since $22/48=11/24$ , their results are equivalent.

Q3. A tank is $7/10$ full. After using $1/4$ of the tank's total capacity, how much of the tank remains full?

9/20

11/20

✅

3/5

6/14

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The situation requires subtracting $1/4$ from $7/10$ . A common denominator of 20 gives $14/20-5/20=9/20$ . Therefore the remaining amount is $9/20$ , making option A correct.

Q4. A student claims that $2/3+3/5=5/8$ because the numerators and denominators can simply be added. What is the best analysis of the error?

A.The student should multiply both numerators only

B.The student should subtract the fractions instead

C.The student added unlike fractional parts without creating a common denominator ✅

D.The student should multiply the denominators and leave the numerators unchanged

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Fractions with different denominators represent different-sized parts, so their numerators cannot be combined directly. Using denominator 15 gives $10/15+9/15=19/15$ , exposing why $5/8$ is incorrect.

Q5. On a number line, point $P$ is at $1/2$ , and moving right by $2/3$ places the point at $Q$ . Which coordinate best represents $Q$ ?

5/6

7/6

✅

3/5

4/6

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Moving right means addition, so $Q=1/2+2/3$ . Using denominator 6 gives $3/6+4/6=7/6$ . The graph therefore places $Q$ one-sixth unit beyond 1, which corresponds to $7/6$ .

Q6. A student wants to calculate $3/4+5/6-1/3$ . They first combine $5/6-1/3$ , then add the result to $3/4$ . Which value should they obtain?

5/4

✅

7/6

11/12

13/12

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: First, $5/6-1/3=5/6-2/6=3/6=1/2$ . Then $3/4+1/2=3/4+2/4=5/4$ . This multi-step approach is valid because subtraction and addition are being performed carefully with compatible denominators.

Q7. Without fully calculating each expression, which expression has the greatest value?

5/8+1/6

3/4-1/5

2/3+1/4

✅

7/10-1/8

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Estimate each expression: A is about $0.79$ , B about $0.55$ , C about $0.92$ , and D about $0.58$ . Therefore C is greatest. Exact common-denominator calculations confirm that $2/3+1/4=11/12$ , exceeding the other choices.

Q8. What is the least common denominator for the fractions $5/12$ and $7/18$ ?

A.24

B.30

C.36 ✅

D.72

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The LCD must be the least common multiple of 12 and 18. Their multiples include 36, which is divisible by both denominators, while no smaller positive common multiple works. Therefore the LCD is $36$ .

Q9. A student says the LCD of $3/8$ and $5/12$ is $96$ because $8\times12=96$ . Which evaluation is most accurate?

A.The student is correct because multiplying denominators always gives the LCD

B.The student is incorrect because

24

is the least common denominator ✅

C.The student is incorrect because

20

is the least common denominator

D.The student is correct only when the numerators are relatively prime

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplying denominators always produces a common denominator, but not necessarily the least one. Since $8$ and $12$ both divide $24$ , the LCD is $24$ . Using $96$ works but creates unnecessary computation.

Q10. A school activity requires combining $2/15$ of a class for one task with $1/10$ for another. Before adding the fractions, which denominator would make the calculation most efficient?

A.20

B.25

C.30 ✅

D.150

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The denominators are 15 and 10. Their least common multiple is 30 because $30\div15=2$ and $30\div10=3$ . Thus converting both fractions to denominator 30 minimizes the size of the equivalent fractions.

Q11. On a number line, one point is labeled $1/6$ and another is labeled $1/4$ . A student wants to divide the interval from 0 to 1 into equal smaller sections that can represent both fractions exactly. What is the smallest number of sections needed?

A.8

B.10

C.12 ✅

D.24

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The section count must be divisible by both 6 and 4. The least common multiple of 6 and 4 is 12, so twelve equal sections allow $1/6$ to be represented by 2 sections and $1/4$ by 3 sections.

Q12. A student finds the LCD of $4/9$ , $5/12$ , and $7/18$ by multiplying $9\times12\times18=1944$ . What is the better approach and result?

A.Use the greatest denominator, giving LCD

18

B.Find the least common multiple of the denominators, giving LCD

36

✅

C.Add the denominators, giving LCD

39

D.Multiply only the two smallest denominators, giving LCD

108

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The product of all denominators is a common denominator but is unnecessarily large. Factoring gives $9=3^2$ , $12=2^2\cdot3$ , and $18=2\cdot3^2$ . Taking the highest powers gives $2^2\cdot3^2=36$ .

Q13. Which pair of fractions can be rewritten using the smallest common denominator?

5/14

and

3/21

✅

7/16

and

5/24

2/15

and

7/20

4/18

and

5/27

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For option A, the denominators 14 and 21 have LCD 42. The other pairs have LCDs 48, 60, and 54 respectively. Therefore option A requires the smallest common denominator among the choices.

Q14. Three students choose different common denominators for $5/6$ and $7/15$ : Student A uses 30, Student B uses 60, and Student C uses 90. Who selected the LCD, and why?

A.Student A, because 30 is divisible by both 6 and 15 ✅

B.Student B, because larger common denominators are always better

C.Student C, because 90 contains more factors

D.All three, because every common denominator is automatically an LCD

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The LCD is the smallest positive integer divisible by every denominator. Since $30\div6=5$ and $30\div15=2$ , 30 is a common denominator, and no smaller positive integer is divisible by both 6 and 15. Thus Student A is correct.

Q15. Which sequence correctly explains how to add $3/8$ and $5/12$ ?

A.Add numerators and denominators directly, then simplify

B.Find the LCD, rewrite both fractions with that denominator, add numerators, then simplify ✅

C.Multiply the numerators and denominators, then add the results

D.Subtract the smaller numerator from the larger numerator before finding a common denominator

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The denominators represent different-sized parts, so they must first be converted to equal-sized parts. The LCD of 8 and 12 is 24, giving $3/8=9/24$ and $5/12=10/24$ , so the sum is $19/24$ .

Q16. A student calculates $7/10-1/6$ as $6/4$ by subtracting the denominators and numerators separately. What is the best diagnosis?

A.The student should subtract denominators only after simplifying

B.The student correctly subtracted because both fractions are positive

C.The student ignored the need for a common denominator before subtracting ✅

D.The student should multiply the numerators before subtracting

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Subtracting numerators and denominators separately changes the meaning of the fractions. The LCD of 10 and 6 is 30, so $7/10=21/30$ and $1/6=5/30$ . Therefore the correct difference is $16/30=8/15$ .

Q17. A water container is $5/6$ full. Workers remove $1/4$ of the container's total capacity and then add $1/3$ of the total capacity. What fraction of the container is full now?

7/12

11/12

✅

13/12

5/8

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The changes must be applied to the original whole container. Calculate $5/6-1/4+1/3$ . Using LCD 12 gives $10/12-3/12+4/12=11/12$ . The result remains below one full container, so $11/12$ is reasonable.

Q18. Two methods are used for $2/9+5/12$ . Method A uses LCD 36, while Method B uses 108. Which statement is most accurate?

A.Only Method A can produce a correct answer

B.Only Method B can produce a correct answer

C.Both can produce the same correct value, but Method A is more efficient ✅

D.Neither method works because denominators must be prime

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Both 36 and 108 are common denominators because each is divisible by 9 and 12. However, 36 is the least common denominator, so it requires smaller equivalent numerators and less computation. Both methods can ultimately give $13/36$ .

Q19. A number-line model shows a point at $1/3$ . Moving right by $1/2$ places the point at which location?

2/5

3/5

5/6

✅

1/6

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Moving right represents addition, so the new position is $1/3+1/2$ . Using denominator 6, $1/3=2/6$ and $1/2=3/6$ . Their sum is $5/6$ , which lies to the right of both original fractions.

Q20. A recipe needs $3/4$ cup of oats and $2/5$ cup of nuts. The measuring cup shows markings that allow twentieths. How many cups are needed altogether?

17/20

19/20

✅

23/20

11/20

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The common denominator 20 allows both ingredients to be measured using the same-sized fractional parts. Since $3/4=15/20$ and $2/5=8/20$ , adding gives $23/20$ . Thus the recipe requires $1\frac{3}{20}$ cups altogether.

Q21. Without calculating every option completely, which expression has the greatest value?

5/8+1/6

7/10-1/5

3/4+1/9

2/3+1/4

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Estimate the expressions: A is about $0.79$ , B is $0.50$ , C is about $0.86$ , and D is about $0.92$ . Therefore D is greatest. Exactly, $2/3+1/4=8/12+3/12=11/12$ , confirming the estimate.

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