If a = − $\frac{3}{4}$ and b = $\frac{5}{6}$, then which of the following does not lie between a and b ?

The correct answer is − 7/9.

To determine which of the given numbers does not lie between a and b, let us first convert all given fractions into decimal form or compare them with a common denominator.

The given values are:

$a=-\frac{3}{4}=-0.75$

$b=\frac{5}{6}\approx 0.833$

Thus, any number that lies between a and b must satisfy the inequality:

$-0.75<x<0.833$

Now, let us evaluate each of the given options:

1. 0:
0 lies clearly between −0.75 and 0.833 since −0.75 < 0 < 0.833.

2. − 1/2:
$-\frac{1}{2}=-0.5$
Since −0.75 < −0.5 < 0.833, this number lies between a and b.

3. − 2/5:
$-\frac{2}{5}=-0.4$
Since −0.75 < −0.4 < 0.833, this number also lies between a and b.

4. − 7/9:
$-\frac{7}{9}\approx -\mathrm{0.777...}$
Comparing −0.777... with −0.75, we find that −0.777... is less than −0.75. Therefore, −7/9 is smaller than a and does not lie between a and b.

Hence, − 7/9 does not lie between a and b.