📝 Find Equivalent Fractions in Algebra (35 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 35 questions available

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What is Find Equivalent Fractions in Algebra?

Definition:
Finding equivalent fractions in algebra involves determining fractions that represent the same value as a given fraction, obtained by multiplying or dividing both the numerator and denominator by the same non-zero integer, which is essential for simplifying or comparing fractions in algebraic contexts.

Working:
Multiply or divide the numerator and denominator by the same number (e.g., $\frac{a}{b} = \frac{a \times k}{b \times k}$ ); to find equivalent fractions, choose any $k \neq 0$ ; this process is also used to find common denominators for addition/subtraction.

Example:
Find two equivalent fractions for $\frac{2}{3}$ .
Solution: Multiply numerator and denominator by 2: $\frac{2 \times 2}{3 \times 2} = \frac{4}{6}$ ; multiply by 3: $\frac{6}{9}$ .

Reason:
Equivalent fractions are fundamental for simplifying algebraic fractions, solving proportions, and adding/subtracting fractions with different denominators, ensuring that expressions are written in their simplest or most useful form.

Easy

Medium

Hard

📝 All Find Equivalent Fractions in Algebra MCQs

Q1. A student claims that $\frac{6}{10}$ and $\frac{15}{25}$ are equivalent because both fractions can be reduced. Which reasoning best evaluates the claim?

A.They are equivalent because both numerators are even

B.They are equivalent because both reduce to

\frac{3}{5}

✅

C.They are not equivalent because their denominators are different

D.They are equivalent only when their numerators are equal

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Reducing both fractions gives the same simplest form: $\frac{6}{10}=\frac{3}{5}$ and $\frac{15}{25}=\frac{3}{5}$ . Therefore, they represent the same quantity even though their numerators and denominators are different.

Q2. A recipe uses $\frac{3}{4}$ cup of flour. A measuring tool shows amounts in eighths. Which fraction should represent the same amount, and why?

\frac{4}{8}

, because 4 is one more than 3

\frac{5}{8}

, because both denominators are even

\frac{6}{8}

, because multiplying numerator and denominator by 2 preserves the value ✅

\frac{8}{12}

, because both fractions can be reduced

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: To change fourths into eighths, multiply the denominator by 2. The numerator must also be multiplied by 2, giving $\frac{3\times2}{4\times2}=\frac{6}{8}$ . Multiplying both parts by the same nonzero number preserves the fraction's value.

Q3. Two students use different methods to create a fraction equivalent to $\frac{5}{7}$ . Ali writes $\frac{10}{14}$ , while Sara writes $\frac{15}{21}$ . What conclusion is most accurate?

A.Only Ali is correct because 10 is twice 5

B.Only Sara is correct because 21 is divisible by 7

C.Both are correct because each fraction multiplies numerator and denominator by the same number ✅

D.Neither is correct because equivalent fractions must have the same denominator

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Ali multiplies both parts of $\frac{5}{7}$ by 2, producing $\frac{10}{14}$ . Sara multiplies both parts by 3, producing $\frac{15}{21}$ . Since each method changes numerator and denominator by the same factor, both fractions remain equivalent.

Q4. A student changes $\frac{8}{12}$ into $\frac{4}{6}$ by dividing only the numerator by 2. The student then says the fractions are equivalent. What is the error?

A.Dividing the numerator is never allowed

B.The denominator should also be divided by 2 ✅

C.The numerator should be multiplied by 2 instead

D.The fraction should always have denominator 12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An equivalent fraction can be created by multiplying or dividing both numerator and denominator by the same nonzero number. Here, $8\div2=4$ , but $12$ must also be divided by 2, giving $6$ . Thus $\frac{4}{6}$ is equivalent to $\frac{8}{12}$ .

Q5. A number-line model marks the same point using $\frac{3}{5}$ and an unknown fraction $\frac{x}{20}$ . Both labels point to the same location. What value of $x$ makes the labels equivalent?

A.8

B.10

C.12 ✅

D.15

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The denominator changes from 5 to 20, which is a multiplication by 4. To keep the same position on the number line, the numerator must also be multiplied by 4: $3\times4=12$ . Therefore, $\frac{3}{5}=\frac{12}{20}$ .

Q6. A school has two identical trays. One tray is divided into 6 equal sections with 4 shaded, and another is divided into 15 equal sections with 10 shaded. A student says the second tray represents more because 10 is greater than 4. How should the comparison be resolved?

A.The second tray is larger because 10 is greater than 4

B.The first tray is larger because 6 is smaller than 15

C.They represent the same amount because

\frac{4}{6}=\frac{10}{15}=\frac{2}{3}

✅

D.There is not enough information because the denominators differ

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The raw numerators cannot be compared without considering the denominators. Simplifying gives $\frac{4}{6}=\frac{2}{3}$ and $\frac{10}{15}=\frac{2}{3}$ . Therefore, the shaded portions represent exactly the same quantity despite having different numbers of sections.

Q7. A student wants to find three different fractions equivalent to $\frac{2}{3}$ , but each new denominator must be greater than 20 and less than 40. Which set is possible?

\frac{22}{33},\frac{24}{36},\frac{26}{39}

✅

\frac{12}{18},\frac{18}{27},\frac{20}{30}

\frac{21}{30},\frac{28}{35},\frac{32}{39}

\frac{14}{21},\frac{22}{33},\frac{30}{45}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Equivalent fractions of $\frac{2}{3}$ are formed by multiplying both parts by the same integer. Using multipliers 11, 12, and 13 gives $\frac{22}{33},\frac{24}{36},\frac{26}{39}$ . Their denominators all fall between 20 and 40, satisfying every condition.

Q8. In the fraction $\frac{7}{12}$ , a student says that 12 tells how many parts are selected. Which correction best explains the roles of both numbers?

A.7 shows the total number of equal parts and 12 shows selected parts

B.7 shows selected parts and 12 shows the total number of equal parts ✅

C.Both numbers show selected parts

D.12 shows selected parts while 7 shows the total number of parts

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: In $\frac{7}{12}$ , the numerator 7 tells how many equal parts are being considered, while the denominator 12 tells how many equal parts make up the whole. Confusing these roles reverses the meaning of the fraction.

Q9. A chocolate bar is divided into 10 equal pieces, and 6 pieces are eaten. Which fraction correctly models the portion eaten, and what do its numbers represent?

\frac{10}{6}

; 10 eaten pieces and 6 total pieces

\frac{6}{10}

; 6 eaten pieces and 10 total pieces ✅

\frac{4}{10}

; 4 eaten pieces and 10 total pieces

\frac{6}{4}

; 6 eaten pieces and 4 remaining pieces

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because the whole chocolate bar contains 10 equal pieces and 6 are eaten, the fraction eaten is $\frac{6}{10}$ . The numerator represents the selected or eaten pieces, while the denominator represents all equal pieces in the whole.

Q10. A student sees $\frac{3}{8}$ and claims that 3 must be the denominator because it appears first when reading the fraction. How should the reasoning be evaluated?

A.Correct, because the first number is always the denominator

B.Correct, because smaller numbers are denominators

C.Incorrect, because the bottom number represents the total equal parts ✅

D.Incorrect, because fractions cannot have a numerator smaller than the denominator

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The position of a number determines its role in a fraction: the numerator is written above the fraction bar and the denominator below it. In $\frac{3}{8}$ , 8 represents the total equal parts, while 3 represents selected parts.

Q11. Two identical pizzas are divided differently. Pizza A has 8 equal slices with 5 selected, while Pizza B has 12 equal slices with 7 selected. Which statement correctly compares the information represented by their fractions?

\frac{5}{8}

and

\frac{7}{12}

have the same denominator

B.The numerator always describes the whole pizza

C.The denominators show the total number of equal slices in each pizza ✅

D.The larger numerator automatically means Pizza B has the larger selected portion

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For Pizza A, $\frac{5}{8}$ means 5 selected slices out of 8 equal slices. For Pizza B, $\frac{7}{12}$ means 7 selected slices out of 12 equal slices. Thus, each denominator describes the total equal divisions.

Q12. A diagram shows a rectangle divided into 9 equal sections, with 4 shaded. A student writes $\frac{9}{4}$ . What mistake has been made, and what fraction should be written?

A.The student reversed numerator and denominator; the correct fraction is

\frac{4}{9}

✅

B.The student counted the shaded sections twice; the correct fraction is

\frac{9}{4}

C.The denominator must always be smaller; the correct fraction is

\frac{4}{5}

D.The numerator should describe the entire rectangle; the correct fraction is

\frac{5}{9}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The denominator represents the total number of equal sections, which is 9. The numerator represents the shaded sections, which is 4. Therefore, the correct fraction is $\frac{4}{9}$ , and the student's error comes from reversing their roles.

Q13. On a number line from 0 to 1, a point lies at the fourth mark when the interval is divided into 7 equal parts. Which fraction labels the point, assuming the first mark after 0 is $\frac{1}{7}$ ?

\frac{7}{4}

\frac{3}{7}

\frac{4}{7}

✅

\frac{4}{3}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Dividing the interval from 0 to 1 into 7 equal parts makes each step equal to $\frac{1}{7}$ . The fourth mark therefore represents four of those equal parts, so its value is $\frac{4}{7}$ .

Q14. A teacher asks students to create a fraction representing 5 selected equal parts from a whole divided into $n$ equal parts. One student writes $\frac{n}{5}$ , while another writes $\frac{5}{n}$ . Which reasoning is correct for any valid $n$ ?

A.The first student is correct because the denominator always represents selected parts

B.The second student is correct because the numerator represents selected parts and

n

represents the whole division ✅

C.Both are correct because fractions can be reversed without changing meaning

D.Neither is correct because a numerator must always be larger than its denominator

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: When a whole is divided into $n$ equal parts and 5 of those parts are selected, the fraction is $\frac{5}{n}$ . The numerator counts selected parts, while the denominator describes the total equal parts forming the whole.

Q15. Which expression has a value of exactly 1, assuming all variables are nonzero?

\frac{x+3}{x+3}

✅

\frac{x}{x+1}

\frac{x-2}{x}

\frac{2x}{x+1}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The numerator and denominator of $\frac{x+3}{x+3}$ are identical, so their quotient is 1 whenever $x+3\neq0$ . The other expressions have different numerator and denominator, so they are not automatically equal to 1.

Q16. A student simplifies $\frac{5y}{5y}$ to 5 because the 5s appear in both the numerator and denominator. What is the best evaluation of this reasoning?

A.Correct, because common factors are added

B.Incorrect, because the entire numerator and denominator are equal, making the quotient 1 when

y\neq0

✅

C.Correct, because

y

cancels before division

D.Incorrect, because fractions with variables cannot equal 1

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The complete numerator $5y$ is identical to the complete denominator $5y$ , so $\frac{5y}{5y}=1$ provided $y\neq0$ . The student incorrectly treats the common factor 5 as the final value instead of recognizing the entire quotient.

Q17. A rectangular garden has an area expression $12x$ square meters and is divided into $12x$ equal-area units. What fraction represents the entire garden, assuming $x>0$ ?

\frac{1}{12x}

\frac{12}{x}

\frac{12x}{12x}=1

✅

12x-12x=0

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The number of units represented by the whole garden is $12x$ , and the total number of units is also $12x$ . Therefore the fraction describing the entire garden is $\frac{12x}{12x}=1$ , since $x>0$ ensures the denominator is nonzero.

Q18. If $a\neq0$ , which transformation preserves the value of an expression $\frac{a}{a}$ without changing its meaning?

A.Replace it with

a

B.Replace it with 0

C.Replace it with 1 ✅

D.Replace it with

a^2

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: When the numerator and denominator are the same nonzero quantity, their quotient is exactly 1. Thus $\frac{a}{a}=1$ . Replacing the expression with $a$ , 0, or $a^2$ generally changes its value.

Q19. A graph shows the function $y=\frac{x}{x}$ at every allowed input. A student says the graph should be the line $y=x$ because the numerator contains $x$ . Which conclusion is mathematically correct?

A.The graph is

y=x

for all real

x

B.The graph is the horizontal line

y=1

, except at

x=0

, where the expression is undefined ✅

C.The graph is the horizontal line

y=0

for all

x

D.The graph has no points because variables cannot be divided

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For every nonzero $x$ , $\frac{x}{x}=1$ , so the graph consists of points on the horizontal line $y=1$ . At $x=0$ , the original expression becomes $0/0$ , which is undefined, so that input cannot be included.

Q20. A company calculates an efficiency ratio as $\frac{8m}{8m}$ , where $m$ represents a positive number of operating hours. Another calculation gives $\frac{8m+8}{8m}$ . Which statement best compares them?

A.Both ratios are always 1

B.The first ratio is 1, while the second is greater than 1 for

m>0

✅

C.The first ratio is greater than 1, while the second is 1

D.Both ratios are always greater than 1

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Because $m>0$ , $8m$ is nonzero and $\frac{8m}{8m}=1$ . The second ratio has a larger numerator than denominator: $8m+8>8m$ , so $\frac{8m+8}{8m}>1$ . This demonstrates why identical numerator and denominator matter.

Q21. A student claims that $\frac{a}{a}=1$ is true even when $a=0$ , because any quantity divided by itself should equal 1. Which response gives the strongest mathematical analysis?

A.The claim is always true because zero is a number

B.The claim is false only when

a

is negative

C.The claim is true because

0/0

equals 1

D.The rule requires

a\neq0

; when

a=0

\frac{a}{a}

becomes

0/0

, which is undefined ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The property $\frac{a}{a}=1$ applies only when $a\neq0$ . If $a=0$ , the expression becomes $0/0$ , which is undefined. Therefore, treating every self-division as 1 ignores the essential restriction that the denominator cannot be zero.

Q22. Which fraction is equivalent to $\frac{4}{7}$ and has a denominator of 21?

\frac{8}{21}

\frac{12}{21}

✅

\frac{14}{21}

\frac{16}{21}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The denominator 7 must be multiplied by 3 to become 21. To preserve the fraction's value, the numerator must also be multiplied by 3. Thus $4\times3=12$ , giving $\frac{12}{21}$ , which represents the same quantity as $\frac{4}{7}$ .

Q23. A student says $\frac{6}{8}$ and $\frac{9}{12}$ are not equivalent because their numerators and denominators are different. Which response best evaluates the student's reasoning?

A.The student is correct because equivalent fractions must have identical numbers

B.The student is incorrect because both fractions simplify to

\frac{3}{4}

✅

C.The student is correct because 8 is smaller than 12

D.The student is incorrect because only denominators determine equivalence

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Equivalent fractions do not need identical numerators or denominators. Dividing both parts of $\frac{6}{8}$ by 2 gives $\frac{3}{4}$ , while dividing both parts of $\frac{9}{12}$ by 3 also gives $\frac{3}{4}$ . Therefore, they represent the same value.

Q24. A water tank is $\frac{5}{6}$ full. A sensor records the fraction using 18 equal levels. Which reading correctly represents the same amount of water?

\frac{10}{18}

\frac{12}{18}

\frac{15}{18}

✅

\frac{16}{18}

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: To change sixths into eighteenthths, multiply the denominator by 3. The numerator must also be multiplied by 3: $5\times3=15$ . Therefore $\frac{5}{6}=\frac{15}{18}$ , so the sensor should show 15 of its 18 equal levels.

Q25. A student transforms $\frac{7}{10}$ into $\frac{21}{20}$ by multiplying the numerator by 3 and the denominator by 2. What is the student's error?

A.The numerator should not be changed

B.The denominator should not be changed

C.Different multiplication factors change the value of the fraction ✅

D.The original fraction cannot have an equivalent fraction

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: To create an equivalent fraction, the numerator and denominator must be multiplied or divided by the same nonzero factor. Here, 7 is multiplied by 3 while 10 is multiplied by 2, so the resulting fraction $\frac{21}{20}$ does not have the same value as $\frac{7}{10}$ .

Q26. On a number line from 0 to 1, one point is labeled $\frac{3}{4}$ . The same point is divided into twelfths. Which label should appear at that location?

\frac{6}{12}

\frac{8}{12}

\frac{9}{12}

✅

\frac{10}{12}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The denominator changes from 4 to 12, which requires multiplying by 3. The numerator must also be multiplied by 3, so $3\times3=9$ . Therefore $\frac{3}{4}=\frac{9}{12}$ , and both labels identify the same point on the number line.

Q27. Two students compare $\frac{8}{12}$ and $\frac{10}{15}$ . One reduces the first fraction to $\frac{2}{3}$ , while the other converts the second to $\frac{8}{12}$ . What can be concluded?

A.Only the first method proves equivalence

B.Only the second method proves equivalence

C.Both methods show that the fractions are equivalent ✅

D.Neither method is valid because denominators must match originally

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The first fraction simplifies as $\frac{8}{12}=\frac{2}{3}$ . The second fraction can also be transformed by multiplying $\frac{2}{3}$ by 4, producing $\frac{8}{12}$ . Since both methods lead to the same value, the fractions are equivalent.

Q28. Find a fraction equivalent to $\frac{2}{5}$ whose numerator and denominator have a sum of 35. Which fraction satisfies this condition?

\frac{6}{15}

\frac{8}{20}

\frac{10}{25}

✅

\frac{12}{30}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Equivalent fractions of $\frac{2}{5}$ have the form $\frac{2k}{5k}$ . Their numerator and denominator sum to $7k$ . Setting $7k=35$ gives $k=5$ , so the required fraction is $\frac{10}{25}$ . Therefore option C is mathematically correct.

Q29. Which fraction is produced when $\frac{3}{5}$ is multiplied by $\frac{4}{4}$ without changing its value?

\frac{7}{9}

\frac{12}{20}

✅

\frac{12}{9}

\frac{3}{20}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying a fraction by $\frac{4}{4}$ is equivalent to multiplying both its numerator and denominator by 4. Thus $\frac{3}{5}=\frac{3\times4}{5\times4}=\frac{12}{20}$ , because $\frac{4}{4}=1$ does not change the original value.

Q30. A student changes $\frac{7}{9}$ into $\frac{21}{27}$ and claims the value changed because both numbers became larger. Which explanation is most accurate?

A.The value changed because larger numerators always make fractions larger

B.The value stayed the same because both numerator and denominator were multiplied by 3 ✅

C.The value changed because the denominator should remain 9

D.The fractions cannot be compared because their denominators differ

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The numerator and denominator were both multiplied by the same nonzero number, 3. Therefore $\frac{7}{9}=\frac{7\times3}{9\times3}=\frac{21}{27}$ . Although the numbers increased, their ratio remained unchanged.

Q31. A school models $\frac{2}{3}$ of a field as 2 shaded sections out of 3. The field is redrawn using 12 equal sections. How many sections should be shaded to preserve the same proportion?

A.6

B.8 ✅

C.9

D.10

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The denominator changes from 3 to 12, which is multiplication by 4. The numerator must also be multiplied by 4: $2\times4=8$ . Therefore $\frac{2}{3}=\frac{8}{12}$ , so 8 of the 12 sections should be shaded.

Q32. A student transforms $\frac{5}{8}$ into $\frac{15}{32}$ by multiplying the numerator by 3 and denominator by 4. What is the student's main error?

A.The numerator should never be multiplied

B.The denominator should never be multiplied

C.Different factors were used, so the original ratio was not preserved ✅

D.The fraction

\frac{5}{8}

cannot have an equivalent form

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The equivalent-fraction property requires multiplying both numerator and denominator by the same nonzero factor. The student used 3 for the numerator and 4 for the denominator, changing the ratio. A valid transformation would be $\frac{15}{24}$ or $\frac{20}{32}$ .

Q33. On a number line, $\frac{4}{7}$ marks a point. The interval is then divided so that each original seventh becomes 3 smaller equal parts. Which fraction represents the same point?

\frac{7}{12}

\frac{8}{21}

\frac{12}{21}

✅

\frac{14}{21}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Each of the 7 original parts is split into 3 equal pieces, so the denominator becomes $7\times3=21$ . The 4 selected parts become $4\times3=12$ , giving $\frac{12}{21}$ . This identifies exactly the same location on the number line.

Q34. A designer represents $\frac{3}{4}$ of a panel using 16 equal tiles. Later, the designer doubles the number of tiles in each original section. Which calculation correctly preserves the shaded proportion?

\frac{3\times2}{4}=\frac{6}{4}

\frac{3}{4\times2}=\frac{3}{8}

\frac{3\times2}{4\times2}=\frac{6}{8}

✅

\frac{3+2}{4+2}=\frac{5}{6}

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When every original section is divided into 2 smaller equal sections, both the shaded count and total count double. Therefore $\frac{3}{4}$ becomes $\frac{6}{8}$ . Multiplying both parts by the same factor preserves the proportion.

Q35. For nonzero $b$ , suppose $\frac{a}{b}=\frac{ac}{bc}$ . A student wants the new denominator to be $5b$ , then uses the resulting fraction in a comparison. Which numerator is required to preserve the original value?

a+5

a-5

5a

✅

a/5

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: To make the denominator $5b$ , the original denominator $b$ is multiplied by 5. The same factor must be applied to the numerator, producing $5a$ . Thus $\frac{a}{b}=\frac{5a}{5b}$ , provided $b\neq0$ .}

🔗 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

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