Given below are two statements: Statement I: A fraction can be reduced to its…
2022
Given below are two statements:
Statement I: A fraction can be reduced to its lowest form by dividing the numerator and denominator by a common factor.
Statement II: Two fractions can be compared by comparing their numerators if their positive denominators are equal.
In the light of the above statements, choose the correct answer from the options given below:
Answer: A. Both Statement I and Statement II are true — ConceptEquivalent fractions result when the numerator and denominator are divided by the same nonzero common factor. A fraction is in lowest terms when its…
- A.
Both Statement I and Statement II are true
- B.
Both Statement I and Statement II are false
- C.
Statement I is true but Statement II is false
- D.
Statement I is false but Statement II is true
Attempted by 4 students.
Show answer & explanation
Correct answer: A
Concept
Equivalent fractions result when the numerator and denominator are divided by the same nonzero common factor. A fraction is in lowest terms when its numerator and denominator are coprime; dividing by their greatest common factor reaches that form directly.
For positive equal denominators, ordering fractions is equivalent to ordering their numerators because division by the same positive number preserves order.
Application
For Statement I, write the numerator and denominator as ka and kb, where k > 1 is a common factor. Dividing both by k gives a/b because (ka)/(kb) = a/b. Repeating this process, or dividing once by the greatest common factor, produces coprime terms; hence Statement I is true.
For Statement II, write the fractions as a/d and b/d with d > 0. If a > b, then a/d > b/d, and if a < b, then a/d < b/d. Thus comparing the numerators is sufficient when the denominators are equal; hence Statement II is true.
Contrast
Both Statement I and Statement II are true: this agrees with the common-factor reduction rule and the equal-denominator comparison rule.
Both Statement I and Statement II are false: this rejects both established fraction rules.
Statement I is true but Statement II is false: this rejects order preservation under a shared positive denominator.
Statement I is false but Statement II is true: this rejects value preservation when both terms are divided by the same common factor.
Cross-check
As an independent check, 12/18 becomes 2/3 after dividing both terms by 6, and 3/7 < 5/7 because 3 < 5. The two examples confirm the two rules separately.
Result
Therefore, both Statement I and Statement II are true.