When will $1$ dollar equal $1$ yen?

Question

Suppose the value of \$1 in Japanese yen decreases at 2% per year. Starting from \$1 = Y240, when will $1$ dollar equal $1$ yen?

The equation should be $240e^{-0.02t}=1$, which gives $t=50\ln240$. But the key says something different: $t=\frac{\ln(1/240)}{\ln(.98)}$. My calculator tells me that my answer is $274$ years but the key is $271$ years. Why the discrepancy?

Answer 1

Basically the value keeps decreasing by $2$%, that is, the value of the yen after one year is $98$% of the value it was in the previous year. Translating this mathematically, we can get $$\text{Value of Yen after one year} = Y_1 = 240\times \frac{98}{100}$$ Similarly, we can show that $$\text{Value of Yen after two years}= Y_2 = 240\times \left(\frac{98}{100}\right)^2$$ $$\Rightarrow \text{Value of Yen after n years} = Y_n = 240\times \left(\frac{98}{100}\right)^n$$ Now, we have $$Y_n = 1\implies 240\times \left(\frac{98}{100}\right)^n = 1$$ giving us, $$\left(\frac{98}{100}\right)^n =\frac{1}{240}$$ Taking $\log$ on both sides, $$n\log(0.98) = \log\left(\frac{1}{240}\right) \Rightarrow n=\frac{\log\left(\frac{1}{240}\right)}{\log 0.98} \approx 271 \text{years}$$ Hope it helps.

Answer 2

The equation you set up is incorrect: As it decreases by $2$% each year it should read $$240\times(1-0.02)^t=1$$ because at the end of each year its value will be $98$% of the value it started with.