Find the single quantity that must be removed from each of the values 8, 10, 11, and 14 so that the four resulting terms form a proportion.

The correct answer is Option 4: 2.

To find the single quantity that must be subtracted from each of the numbers 8, 10, 11, and 14 so that the remaining numbers form a proportion, we can set up an equation.

Let the required quantity to be removed be $x$.

Subtracting $x$ from each of the given numbers gives the new numbers as:

$(8-x)$, $(10-x)$, $(11-x)$, and $(14-x)$

For four numbers to be in proportion, the ratio of the first two numbers must equal the ratio of the last two numbers. Therefore, we can write the proportion as:

$\frac{8-x}{10-x}=\frac{11-x}{14-x}$

Now, we solve for $x$ by cross-multiplying:

$(8-x)(14-x)=(10-x)(11-x)$

Expanding both sides of the equation:

$112-8x-14x+x^2=110-10x-11x+x^2$

Simplifying the like terms on both sides:

$112-22x+x^2=110-21x+x^2$

Subtracting $x^2$ from both sides gives:

$112-22x=110-21x$

Rearranging the terms to isolate $x$:

$112-110=22x-21x$

$x=2$

Verification:
If we subtract 2 from each of the numbers:
8 - 2 = 6
10 - 2 = 8
11 - 2 = 9
14 - 2 = 12
Checking the ratios: 6 / 8 = 3 / 4 and 9 / 12 = 3 / 4. Since 3 / 4 = 3 / 4, the resulting terms form a valid proportion.

Hence, the quantity that must be removed is 2.