When x is subtracted from each of the numbers 22, 39, 56 and 107, then the resulting numbers, in this order, are in proportion. What is the mean proportional between (x+3) and (3x−7)?

According to the problem, the terms (22−x), (39−x), (56−x), and (107−x) are in proportion.
39−x
22−x
​
=
107−x
56−x
​

Using the components/differences trick:

(39−x)−(22−x)
22−x
​
=
(107−x)−(56−x)
56−x
​

17
22−x
​
=
51
56−x
​

Since 51=17×3, cross-multiply by 3 on the left side:

3(22−x)=56−x
66−3x=56−x
66−56=3x−x
10=2x⟹x=5
Now, calculate the values for the mean proportional:
First term = x+3=5+3=8
Second term = 3x−7=3(5)−7=15−7=8

The mean proportional between two numbers a and b is given by
ab

​
:

Mean Proportional=
8×8

​
=8