Joanna claims 2/3 is equal to 4/6. Kyle thinks 2/3 is equal to 8/12. Who is correct?
The Correct Answer and Explanation is :
Both Joanna and Kyle are correct in their claims, but their reasoning is based on different forms of equivalent fractions. Let’s explain this concept.
Joanna’s Claim: ( \frac{2}{3} = \frac{4}{6} )
Joanna’s claim is correct. To verify, we can simplify the fraction ( \frac{4}{6} ). The greatest common divisor (GCD) of 4 and 6 is 2. Dividing both the numerator and the denominator of ( \frac{4}{6} ) by 2:
[
\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}
]
Therefore, ( \frac{2}{3} = \frac{4}{6} ), confirming that Joanna is correct. This is an example of multiplying both the numerator and denominator of a fraction by the same number (in this case, 2), which results in an equivalent fraction.
Kyle’s Claim: ( \frac{2}{3} = \frac{8}{12} )
Kyle’s claim is also correct. To verify, we can simplify the fraction ( \frac{8}{12} ). The greatest common divisor (GCD) of 8 and 12 is 4. Dividing both the numerator and the denominator of ( \frac{8}{12} ) by 4:
[
\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}
]
So, ( \frac{2}{3} = \frac{8}{12} ), confirming that Kyle is also correct. This is another example of equivalent fractions, where the numerator and denominator are both multiplied by 4.
Conclusion
Both Joanna and Kyle are correct because ( \frac{2}{3} ) is equal to both ( \frac{4}{6} ) and ( \frac{8}{12} ). In each case, the fractions represent the same value, and we can obtain one fraction from the other by multiplying the numerator and denominator by a common factor. Thus, fractions can be equivalent if they can be simplified or scaled up while maintaining the same ratio between the numerator and denominator.