QuestionJune 13, 2025

Solve PV=(PMT(1-(1+r/m)^-mt))/(r/m) for the technology formula of mt, the format to input into the calculator. mt=square

Solve PV=(PMT(1-(1+r/m)^-mt))/(r/m) for the technology formula of mt, the format to input into the calculator. mt=square
Solve PV=(PMT(1-(1+r/m)^-mt))/(r/m) for the technology formula of mt, the format to input into the calculator. mt=square

Solution
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Answer

mt = -\frac{\ln(1 - \frac{PV \cdot r}{PMT \cdot m})}{\ln(1 + r/m)} Explanation 1. Isolate the term with mt Multiply both sides by \frac{r}{m} to isolate the fraction: PV \cdot \frac{r}{m} = PMT(1-(1+r/m)^{-mt}). 2. Simplify the equation Divide both sides by PMT: \frac{PV \cdot r}{PMT \cdot m} = 1 - (1 + r/m)^{-mt}. 3. Rearrange for (1 + r/m)^{-mt} Subtract 1 from both sides: (1 + r/m)^{-mt} = 1 - \frac{PV \cdot r}{PMT \cdot m}. 4. Solve for -mt Take the natural logarithm of both sides: -mt = \ln(1 - \frac{PV \cdot r}{PMT \cdot m}) / \ln(1 + r/m). 5. Solve for mt Multiply both sides by -1: mt = -\ln(1 - \frac{PV \cdot r}{PMT \cdot m}) / \ln(1 + r/m).

Explanation

1. Isolate the term with $mt$<br /> Multiply both sides by $\frac{r}{m}$ to isolate the fraction: $PV \cdot \frac{r}{m} = PMT(1-(1+r/m)^{-mt})$.<br />2. Simplify the equation<br /> Divide both sides by $PMT$: $\frac{PV \cdot r}{PMT \cdot m} = 1 - (1 + r/m)^{-mt}$.<br />3. Rearrange for $(1 + r/m)^{-mt}$<br /> Subtract 1 from both sides: $(1 + r/m)^{-mt} = 1 - \frac{PV \cdot r}{PMT \cdot m}$.<br />4. Solve for $-mt$<br /> Take the natural logarithm of both sides: $-mt = \ln(1 - \frac{PV \cdot r}{PMT \cdot m}) / \ln(1 + r/m)$.<br />5. Solve for $mt$<br /> Multiply both sides by -1: $mt = -\ln(1 - \frac{PV \cdot r}{PMT \cdot m}) / \ln(1 + r/m)$.
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