Question

Ms. Giansante asked her students to compare 3/4 and 5/7 and show their work. A student named Timothy compared the fractions as represented in the work shown. Student's work: Comparing 3/4 and 5/7 . The student adds 7 to the numerator and denominator in 3/4 and adds 4 to the numerator and denominator in 5/7 . The student concludes that 10/11 > 9/11 . Ms. Giansante wants to provide a counterexample for Timothy to help him realize his method is not valid and will not always give the correct comparison. Which of the following pairs of fractions provides a counterexample to Timothy's method?

A. 2/3 and 4/5
B. 7/8 and 9/11
C. 5/6 and 3/4
D. 11/12 and 13/15​

Answer 1

Timothy's method of comparing fractions by adding the same number to the numerator and denominator is not always correct. The pair of fractions 5/6 and 3/4 provides a counterexample to Timothy's method.

Step-by-step explanation:

To show that Timothy's method is not always correct, we need to find a pair of fractions for which his method gives an incorrect comparison. Timothy's method involves adding the same number to the numerator and denominator of each fraction and then comparing the results. Let's analyze the given options:

A. 2/3 and 4/5: If we use Timothy's method, we get (3+2)/(5+3) = 5/8, which is incorrect because 5/8 is smaller than 4/5.

B. 7/8 and 9/11: If we use Timothy's method, we get (8+7)/(11+9) = 15/20, which is correct because 15/20 is greater than 9/11.

C. 5/6 and 3/4: If we use Timothy's method, we get (6+5)/(4+3) = 11/7, which is incorrect because 11/7 is greater than 3/4.

D. 11/12 and 13/15: If we use Timothy's method, we get (12+11)/(15+13) = 23/28, which is correct because 23/28 is greater than 13/15.

Therefore, the pair of fractions that provides a counterexample to Timothy's method is 5/6 and 3/4, which is option C.