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Question

# The formula$I_t=\frac{I_f}{1-T}$gives the taxable interest rate $I_t$ equivalent to the tax-free interest rate $I_f$ for a person in the $(100 \cdot T) \%$ tax bracket. Solve for $T$.

Solution

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Step 1
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Our goal is to have only $T$ on one side.

\begin{align*} (1-T)I_{t}&=\frac{I_{f}}{1-T}\cdot(1-T)\\ I_{t}-TI_{t}&=I_{f}\\ I_{t}-TI_{t}-I_{t}&=I_{f}-I_{t}\\ -TI_{t}&=I_{f}-I_{t}\\ \frac{-TI_{t}}{-I_{t}}&=\frac{I_{f}-I_{t}}{-I_{t}}\\ T&=\frac{I_{t}-I_{f}}{I_{t}} \end{align*}

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