📝 Simplify Fractions in Algebra (21 MCQs)

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

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What is Simplify Fractions in Algebra?

Definition:
Simplifying fractions in algebra means reducing a fraction to its lowest terms by dividing the numerator and denominator by their greatest common factor (GCF), so that they have no common factors other than $1$ , resulting in a fraction that is easier to work with in expressions and equations.

Working:
Find the GCF of the numerator and denominator, then divide both by that number; for algebraic fractions, factor both numerator and denominator completely, then cancel any common factors (variables or numbers) from both.

Example:
Simplify $\frac{18}{24}$ .
Solution: GCF of 18 and 24 is 6; divide both by 6: $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$ .

Reason:
Simplification makes fractions less cumbersome, helps in comparing sizes, and is crucial in algebra for solving equations, combining terms, and performing operations like multiplication and division of rational expressions.

Easy

Medium

Hard

📝 All Simplify Fractions in Algebra MCQs

Q1. A student simplifies $\frac{84}{126}$ by dividing both numbers by $6$ , obtaining $\frac{14}{21}$ . What should the student do next to reach simplest form?

A.Divide numerator and denominator by

2

✅

B.Multiply numerator and denominator by

3

C.Subtract

7

from both numbers

D.The fraction is already simplest

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The fraction $\frac{14}{21}$ still has a common factor of $7$ . Dividing both numerator and denominator by $7$ gives $\frac{2}{3}$ , so dividing by $2$ is not the correct next step. The key is to continue reducing using a common factor.

Q2. Two students simplify $\frac{96}{144}$ . Student A divides both terms by $12$ , while Student B divides both terms by $24$ . Which conclusion is most accurate?

A.Only Student A can obtain simplest form

B.Only Student B can obtain simplest form

C.Both methods can produce the same simplest fraction ✅

D.Neither method preserves the fraction's value

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both $12$ and $24$ are common factors of $96$ and $144$ . Dividing by $12$ gives $\frac{8}{12}$ , which reduces to $\frac{2}{3}$ , while dividing directly by $24$ gives $\frac{4}{6}$ , which also reduces to $\frac{2}{3}$ .

Q3. A recipe uses $\frac{18}{24}$ cup of an ingredient for one batch. The cook wants an equivalent amount written with the smallest possible whole-number numerator and denominator. Which form should be used?

\frac{9}{12}

\frac{6}{8}

\frac{3}{4}

✅

\frac{18}{24}

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: To simplify $\frac{18}{24}$ , divide both numerator and denominator by their greatest common factor, $6$ . This produces $\frac{3}{4}$ . The other choices are equivalent fractions but are not fully simplified because their numerator and denominator still share a common factor.

Q4. A student claims that $\frac{45}{60}=\frac{4}{5}$ because $45$ and $60$ can both be divided by $15$ , but the student writes the denominator incorrectly. What is the correct simplified fraction?

\frac{3}{4}

✅

\frac{4}{5}

\frac{15}{20}

\frac{5}{4}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Dividing both $45$ and $60$ by $15$ gives $3$ and $4$ , respectively. Therefore the correct result is $\frac{3}{4}$ . The error comes from incorrectly calculating or recording the denominator after applying the same divisor to both parts.

Q5. A number line marks two points representing $\frac{12}{18}$ and $\frac{2}{3}$ . The points appear at exactly the same location. Which explanation best accounts for this observation?

A.The fractions have different values but happen to overlap

\frac{12}{18}

simplifies to

\frac{2}{3}

✅

C.The denominator can be ignored when comparing fractions

D.Only fractions with denominator

3

can appear on a number line

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Dividing $12$ and $18$ by their greatest common factor, $6$ , gives $\frac{2}{3}$ . Therefore both fractions represent the same numerical value and must occupy the same position on a correctly scaled number line.

Q6. A school records that $\frac{72}{108}$ of collected forms were complete. One student reduces it to $\frac{6}{9}$ , while another reduces it to $\frac{2}{3}$ . Which statement best compares their work?

A.Only

\frac{6}{9}

is equivalent to the original

B.Only

\frac{2}{3}

is equivalent to the original

C.Both are equivalent, but only

\frac{2}{3}

is fully simplified ✅

D.Neither result preserves the original value

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Dividing $72$ and $108$ by $12$ gives $\frac{6}{9}$ , so that result is equivalent to the original. However, $6$ and $9$ share a factor of $3$ , allowing another reduction to $\frac{2}{3}$ , which is simplest.

Q7. A puzzle asks for a fraction equivalent to $\frac{54}{90}$ whose numerator and denominator have no common factor greater than $1$ . A student divides only the numerator by $18$ . What is the best correction?

A.Divide only the denominator by

18

B.Divide both numerator and denominator by

18

✅

C.Subtract

18

from both numbers

D.Multiply both numbers by

18

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Equivalent fractions require the same nonzero factor to be applied to both numerator and denominator. Dividing both $54$ and $90$ by $18$ gives $\frac{3}{5}$ , whose numerator and denominator have no common factor greater than $1$ . Dividing only one part changes the fraction's value.

Q8. A student says $\frac{42}{56}$ is simplified because both numbers are even but no further work is needed. Which response correctly evaluates the claim?

A.The claim is correct because even numbers cannot be reduced twice

B.The claim is incorrect because both numbers can be divided by

14

, giving

\frac{3}{4}

✅

C.The claim is incorrect because the numerator must become

1

D.The claim is correct because

42

and

56

are consecutive even multiples

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The claim is incorrect because $42$ and $56$ share a greatest common factor of $14$ . Dividing both by $14$ gives $\frac{3}{4}$ , and $3$ and $4$ have no common factor greater than $1$ , so the fraction is fully simplified.

Q9. A school survey shows that $48$ out of $72$ students prefer a particular activity. Which simplified fraction represents the same proportion, and why?

\frac{3}{5}

, because

48\div16=3

and

72\div16=5

\frac{2}{3}

, because

48

and

72

can both be divided by

24

✅

\frac{4}{6}

, because

48

and

72

can both be divided by

12

\frac{24}{36}

, because

24

is the difference between

72

and

48

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The greatest common factor of $48$ and $72$ is $24$ . Dividing both parts by $24$ produces $\frac{2}{3}$ . Although $\frac{4}{6}$ and $\frac{24}{36}$ are equivalent, neither is fully simplified because both still have a common factor.

Q10. A recipe uses $\frac{36}{60}$ of a container of an ingredient. One cook divides both numbers by $6$ , while another divides both by $12$ . Which comparison is correct?

A.Only the first cook preserves the value

B.Only the second cook preserves the value

C.Both preserve the value, but only the second immediately reaches simplest form ✅

D.Neither preserves the value because different divisors were used

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Dividing numerator and denominator by the same nonzero number preserves the fraction's value. The first cook obtains $\frac{6}{10}$ , which still reduces to $\frac{3}{5}$ . The second obtains $\frac{3}{5}$ immediately because $12$ is the greatest common factor.

Q11. A student simplifies $\frac{63}{81}$ by dividing the numerator by $9$ and the denominator by $3$ , obtaining $\frac{7}{27}$ . What is the student's main error?

A.The student should subtract the same number from both terms

B.The student used different divisors, so the original ratio was changed ✅

C.The numerator should always be divided by

3

D.The denominator should always remain unchanged

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: To preserve the value of a fraction, the numerator and denominator must be divided by the same nonzero factor. Dividing $63$ by $9$ but $81$ by $3$ changes the ratio and produces an incorrect result. Dividing both by $9$ gives $\frac{7}{9}$ .

Q12. On a number line, point $P$ is located at the same position as $\frac{18}{24}$ . Which fraction could represent point $P$ in simplest form?

\frac{2}{3}

\frac{3}{5}

\frac{4}{5}

✅

\frac{5}{6}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Simplifying $\frac{18}{24}$ requires dividing both numbers by their greatest common factor, $6$ . This gives $\frac{3}{4}$ , not $\frac{4}{5}$ . Therefore none of the listed choices correctly represents the point; the intended simplest value is $\frac{3}{4}$ .

Q13. Three teams complete $84$ of $126$ assigned tasks. Team A writes $\frac{14}{21}$ , Team B writes $\frac{2}{3}$ , and Team C writes $\frac{4}{6}$ . Which evaluation is most accurate?

A.Only Team A is correct because it uses the largest numbers

B.Teams A, B, and C are all correct and all are simplified

C.All are equivalent, but only Team B gives the fraction in simplest form ✅

D.Teams B and C are incorrect because their denominators are smaller

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Dividing $84$ and $126$ by $6$ gives $\frac{14}{21}$ , so Team A has an equivalent fraction. Further reduction gives $\frac{2}{3}$ . Team C's $\frac{4}{6}$ is also equivalent but not simplified because $4$ and $6$ share a factor of $2$ .

Q14. A fraction $\frac{a}{b}$ is already simplified, where $a$ and $b$ are positive integers. Another student multiplies both by $5$ , obtaining $\frac{5a}{5b}$ , and claims the new fraction is also simplified. Which conclusion is necessarily true?

A.The new fraction is simplified because multiplication always reduces fractions

B.The new fraction is not simplified because

5

is a common factor of both terms ✅

C.The new fraction must equal

1

D.The new fraction has a smaller denominator than the original

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiplying both numerator and denominator by $5$ creates an equivalent fraction, but it also introduces $5$ as a common factor. Thus $\frac{5a}{5b}$ cannot be in simplest form because both terms are divisible by $5$ , regardless of the original values of $a$ and $b$ .

Q15. A student wants to simplify $\frac{72}{108}$ efficiently. Which first step is most useful for producing the simplest equivalent fraction with one division?

A.Divide both terms by

2

B.Divide both terms by

6

C.Divide both terms by

36

✅

D.Subtract

36

from both terms

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The greatest common factor of $72$ and $108$ is $36$ . Dividing both numerator and denominator by $36$ gives $\frac{2}{3}$ , which cannot be reduced further. Dividing by a smaller factor would preserve the value but require additional simplification.

Q16. A student simplifies $\frac{45}{75}$ by dividing the numerator by $15$ and the denominator by $15$ . Another student divides both by $5$ first and then simplifies again. Which comparison is correct?

A.Only the first method is valid

B.Only the second method is valid

C.Both methods preserve the value, but the first reaches simplest form faster ✅

D.Both methods change the value because division is used

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Dividing both numerator and denominator by the same nonzero number creates an equivalent fraction. The first student divides directly by the greatest common factor $15$ , obtaining $\frac{3}{5}$ . The second method also works, but requires another reduction.

Q17. A sports team won $84$ of $126$ games. A coach wants the win record expressed as a simplified fraction. Which sequence correctly models the simplification?

A.Divide both terms by

7

, then stop at

\frac{12}{18}

B.Divide both terms by

42

, obtaining

\frac{2}{3}

✅

C.Divide the numerator by

42

and denominator by

21

D.Subtract

42

from both terms, obtaining

\frac{42}{84}

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The greatest common factor of $84$ and $126$ is $42$ . Dividing both terms by $42$ produces $\frac{2}{3}$ . The same operation on numerator and denominator preserves the original proportion, while subtracting or using different divisors does not.

Q18. A student claims that $\frac{56}{98}$ becomes $\frac{4}{7}$ because $56\div14=4$ and $98\div14=7$ . Another student says the result is wrong because the denominator changed. Who is correct?

A.The first student, because both terms were divided by the same factor ✅

B.The second student, because denominators cannot change during simplification

C.Neither, because the numerator must always become

1

D.Neither, because multiplication rather than division is required

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The first student is correct. Dividing both numerator and denominator by the same nonzero factor creates an equivalent fraction. Since $56\div14=4$ and $98\div14=7$ , the simplified fraction is $\frac{4}{7}$ . Changing the denominator is expected.

Q19. On a number line, points $P$ and $Q$ appear at exactly the same location. Point $P$ is labeled $\frac{24}{36}$ . Which label could correctly replace $Q$ if it must be in simplest form?

\frac{3}{4}

\frac{2}{3}

✅

\frac{4}{5}

\frac{5}{6}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Simplifying $\frac{24}{36}$ requires dividing both numbers by their greatest common factor, $12$ . This gives $\frac{2}{3}$ . Because equivalent fractions represent the same value, both labels correspond to the same point on the number line.

Q20. A student changes $\frac{90}{120}$ to $\frac{45}{60}$ , then to $\frac{15}{20}$ , and finally to $\frac{3}{4}$ . What does this sequence demonstrate?

A.Only the final division is mathematically valid

B.Repeatedly dividing numerator and denominator by common factors can preserve the value until simplest form is reached ✅

C.A fraction changes value whenever its numerator and denominator become smaller

D.The denominator should be divided before the numerator

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Each step divides the numerator and denominator by the same common factor, so the fraction remains equivalent throughout the process. The sequence eventually reaches $\frac{3}{4}$ , where numerator and denominator have no common factor greater than $1$ .

Q21. Suppose $\frac{a}{b}$ is positive and already simplified. A student multiplies numerator and denominator by $k$ , where $k>1$ , and then says the new fraction is simpler because it is equivalent. Which reasoning best identifies the flaw?

A.Equivalent fractions must always have different values

B.Multiplication by

k

preserves value but creates

k

as a common factor, so the new fraction is not simplest ✅

C.The numerator should be multiplied but the denominator should remain unchanged

D.Any fraction with larger numbers is automatically simpler

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiplying both parts by $k$ preserves the fraction's value, but it makes $k$ a common factor of the new numerator and denominator. Therefore the resulting fraction can be reduced back to $\frac{a}{b}$ , so it is equivalent but not simpler.

🔗 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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