When x is subtracted from each of 21, 22, 60 and 64, the numbers so obtained in this order are in proportion. What is the mean proportional between $$(x+4)$$ and $$(\frac{x}{2}-1)$$ ?

The four numbers obtained after subtracting $$x$$ are $$21-x$$, $$22-x$$, $$60-x$$ and $$64-x$$, and they are said to be in proportion.

For four quantities $$a,b,c,d$$ to be in proportion, we must have $$\frac{a}{b}=\frac{c}{d}$$.
Therefore,

$$\frac{21-x}{22-x}=\frac{60-x}{64-x}\;.-(1)$$

Cross-multiplying in $$(1)$$:

$$(21-x)(64-x)=(22-x)(60-x).$$

Expand both sides:
$$21\cdot64-21x-64x+x^{2}=22\cdot60-22x-60x+x^{2}.$$

Simplify each side:
Left side: $$1344-85x+x^{2}$$
Right side: $$1320-82x+x^{2}.$$

The $$x^{2}$$ terms cancel. Hence

$$1344-85x=1320-82x.$$

Bring variables to one side and constants to the other:

$$1344-1320=85x-82x$$
$$24=3x.$$

Thus $$x=8.$

Now compute the two given numbers:
$$x+4=8+4=12,$$
$$$$\frac{x}{2}$$-1=$$\frac{8}{2}$$-1=4-1=3.$$

The mean proportional (geometric mean) between $$a$$ and $$b$$ is $$$$\sqrt{ab}$$$$. Therefore,

Mean proportional $$=$$\sqrt{12\cdot3}=\sqrt{36}$$=6.$$

Hence the required mean proportional is 6.

Option C which is: 6