Which of the following fractions has a value greater than 3/4 but less than 5/6?

Introduction / Context:
This problem checks your understanding of comparing fractions and locating them on the number line. Rather than converting every fraction into a decimal with a calculator, the question encourages you to use basic fraction comparison techniques and reasoning about sizes of fractions relative to familiar benchmarks like three quarters and five sixths.

Given Data / Assumptions:

• We are given two bounds: 3/4 and 5/6.
• We must identify which option lies strictly between these two fractions.
• The options are 3/5, 4/5, 7/5, 9/5, and 5/7.
• All fractions are positive rational numbers.

Concept / Approach:
There are two main ways to compare fractions: either convert to decimals or bring them to a common denominator. Here, 3/4 and 5/6 already share small denominators, so we can convert to decimals quickly or find approximate values. We also note that fractions greater than 1 cannot lie between 3/4 and 5/6, so we can eliminate such options immediately.

Step-by-Step Solution:
First, compute approximate decimal values of the bounds.3/4 = 0.75.5/6 ≈ 0.8333.Now consider each option and compare.3/5 = 0.6, which is less than 0.75, so it is too small.4/5 = 0.8, which satisfies 0.75 < 0.8 < 0.8333.7/5 = 1.4 and 9/5 = 1.8, which are both greater than 1, so they cannot lie between 3/4 and 5/6.5/7 ≈ 0.714, less than 0.75, also too small.Thus, only 4/5 lies strictly between 3/4 and 5/6.

Verification / Alternative check:
Alternatively, we can compare 4/5 without decimals by using cross multiplication. To check if 3/4 < 4/5, compare 3 × 5 and 4 × 4. We get 15 and 16, and since 15 < 16, 3/4 < 4/5. To check if 4/5 < 5/6, compare 4 × 6 and 5 × 5. We get 24 and 25, and since 24 < 25, 4/5 < 5/6. Both inequalities hold, confirming that 4/5 is indeed between the two given fractions.

Why Other Options Are Wrong:
3/5 and 5/7 are both less than 3/4 and therefore cannot satisfy the condition. 7/5 and 9/5 exceed 1, which is much larger than 5/6, so they are automatically excluded. None of these fractions meet the requirement of being simultaneously greater than 3/4 and less than 5/6, leaving 4/5 as the only valid choice.

Common Pitfalls:
Some students misinterpret the phrase greater than 3/4 and less than 5/6 and only check one of the two inequalities. Others compare fractions by looking only at numerators or denominators without considering the full fraction. Forgetting that values greater than 1 cannot sit between two proper fractions less than 1 is another frequent error. Consistent use of cross multiplication or decimal conversion helps prevent these mistakes.

Final Answer:
The fraction that lies between 3/4 and 5/6 is 4/5.