When x is added to each of 2, 3, 30 and 35, then the numbers obtained in this order, are in proportion. What is the mean proportional between (x + 7) and (x - 2)?
Order after adding x,
2 +Â x, 3 +Â x, 30 + x, 35 + x
Condition of proportion,
$$\frac{2 + x}{3 + x} = \frac{30 +Â x}{35 +Â x}$$
$$(2 + x)(35 +Â x)Â = (3 +Â x)(30 +Â x)$$
$$70 +Â 2x +Â 35x +Â x^2 = 90 +Â 3x +Â 30x +Â x^2$$
37x = 20 +Â 33x
x = 5
Mean proportional between (x + 7) and (x - 2) = $$\sqrt{(x + 7)(x - 2)}$$
=Â $$\sqrt{(5 + 7)(5 - 2)}$$
= $$\sqrt{36}$$
= 6
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