At an inflation rate of 7 percent, the purchasing power of $3 would be cut in half in 10.25 years. how long to the nearest year would it take the purchasing power of $9 to be cut to one-fourth if the inflation rate were only 6.5 percent

The applicable formula is;
A = P(1-r)^n

Where;
A = Final purchasing power
P = Current purchasing power
r = inflation
n = Number of years when P changes to A

Confirming the first claim:
A = 1/2P (to be confirmed)
P = $3
r = 7% = 0.07
n = 10.25 years

Using the formula;
A = 3(1-0.07)^10.25 = 3(0.475) ≈ 3(0.5) = $1.5
And therefore, A = 1/2P after 10.25 years.

Now, give;
P = $9
A = 1/4P = $9/4 = $2.25
r = 6.5% = 0.065
n = ? (nearest year).

Substituting;
2.25 = 9(1-0.065)^n
2.25/9 = (1-0.065)^n
0.25 = (1-0.065)^n
ln (0.25)= n ln(1-0.065)
-1.3863 = -0.0672n
n = (-1.3863)/(-0.0672) = 20.63 years

To nearest year;
n = 21 years

Therefore, it would take approximately 21 years fro purchasing power to reduce by 4. That is, from $9 to $2.25.