If a = -3/4 and b = 5/6, then which of the following does not lie between a…

Concept: To decide whether a value lies between two given numbers a and b (with a < b), every candidate must be compared to BOTH bounds in a single, common form -- a common denominator or decimal -- since fractions with different denominators cannot be ordered by looking at numerators or denominators alone, especially once signs differ.

Application:

1. Take a = -3/4 and b = 5/6. Using the LCM of all the denominators involved (4, 6, 2, 5, 9), which is 180, rewrite the two bounds as a = -135/180 and b = 150/180.

2. Convert 0 to the same base: 0 = 0/180. Since -135/180 < 0/180 < 150/180, this value lies inside the range.

3. Convert -1/2 to the same base: -1/2 = -90/180. Since -135/180 < -90/180 < 150/180, this value lies inside the range.

4. Convert -2/5 to the same base: -2/5 = -72/180. Since -135/180 < -72/180 < 150/180, this value lies inside the range.

5. Convert -7/9 to the same base: -7/9 = -140/180. Comparing to a = -135/180, -140/180 is more negative, so -140/180 < -135/180 -- this value falls below the lower bound and lies outside the range.

Cross-check: Converting the bounds to decimals gives a = -0.75 and b is approximately 0.833. Checking each candidate the same way: 0, -0.5, and -0.4 all sit between -0.75 and 0.833, while -7/9 (approximately -0.778) is less than -0.75 and therefore outside the range -- confirming the common-denominator result independently.

So -7/9 is the value that does not lie between a and b.