The Present Worth of $169 due in 2 years at 4% Per Annum Compound Interest

Question

The present worth of $\$169$ due in 2 years at $4\%$ per annum compound interest is

The choices are as follow:

$\$150.50$

$\$154.75$

$\$156.25$

$\$158$

I tried to solve this by multiplying $169$ to ($0.96$) and ($0.96$) and got $155.7504$ which is not the same as in the answer key I have which says 156.25. The solution in my answer key translates $26/25$ (which is $1.04$ in fraction) into $25/26$ and the answer 156.25 follows exactly from the equation.

$$=169(25/26)(25/26)$$

I want to know why $26/25$ was converted to $25/26$.

PS I am a college student having troubles with word problems.

Answer 1

Put $\$P$ in the bank today and leave it there for two years. It will get multiplied by $1 + 0.04$ each year, so you'll have $\$P \cdot 1.04^2$. If that is $\$169$ then you have $$ \$P \cdot 1.04^2 = \$169, $$ so $$ P = \frac{169}{1.04^2}. $$ That is the present value.

Since $1.04=\dfrac{26}{25}$, we can write this as $$ P\cdot\left(\frac{26}{25}\right)^2 = 169, $$ so $$ P = \frac{169}{\left(\frac{26}{25}\right)^2} = 169\cdot\left(\frac{25}{26}\right)^2. $$

Answer 2

If going forward, you multiply by $\frac{26}{25}$, then going in reverse, you divide by that. Which is the same as multiplying by $\frac{25}{26}$

For a more extreme example, look at an investment that gains 50% in a year ($\times\frac{3}{2}$). Going in reverse would be ($\times\frac{2}{3}$)