When x is subtracted from each of the numbers 64,81,85 and 109, the numbers so obtained in this order are in proportion. What is the mean proportional between (x+3) and (2x−1)?

According to the problem, the modified numbers are in proportion:
81−x
64−x
​
=
109−x
85−x
​

Using the cross-multiplication method:

(64−x)(109−x)=(81−x)(85−x)
6976−64x−109x+x
2
=6885−81x−85x+x
2

6976−173x=6885−166x
6976−6885=173x−166x
91=7x
x=13
Now, we need to find the mean proportional between (x+3) and (2x−1).
Substitute x=13:

1
st
 term=x+3=13+3=16
2
nd
 term=2x−1=2(13)−1=25
The mean proportional between two numbers a and b is
ab

​
:

Mean Proportional=
16×25

​
=4×5=20