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 + 1) and (7x + 8) ?

As per the question,

$$\Rightarrow \dfrac{21-x}{22-x}=\dfrac{60-x}{64-x}$$

$$\Rightarrow (21-x)(64-x)=(22-x)(60-x)$$

$$\Rightarrow 21\times 64-64x-21x+x^2=22\times 60 -22x-60x+x^2$$

$$\Rightarrow 1344-85x+x^2=1320-82x+x^2$$

$$\Rightarrow 85x-82x=1344-1320=24$$

$$\Rightarrow 3x=24$$

So, $$x=8$$

Now, substituting the values,

$$\Rightarrow (x+1) =8+1=9$$ and$$ (7x+8)=7\times 8+8=64$$

Hence, the required mean $$=\sqrt{9\times 64}=3\times 8=24$$