Smaller correction—I think you’ve had her buy an extra pair of boots. At $260 she’s already bought one pair, so we apply x↦1.07x−20 thirteen times, then multiply by 1.07 again for the final year’s interest, and she ends with no boots, so that’s $239.41. (Or start with $280 and apply x↦1.07(x−20) fourteen times.)
Not sure why my own result is wrong. Part of it is that I forgot to subtract the money actually spent on boots—I did “the $20 she spends after the first year gets one year’s interest, so that’s $21.40; the $20 she spends after the second year gets two years’ interest, so that’s $22.90...” but actually it’s only $1.40, $2.90 and so on. But even accounting for that, I get $222.58. So let’s see...
Suppose she only needs to buy two pairs of boots. According to your method she goes $40 → $21.40 → $1.50. (Or, $40 and no boots → $20 and boots → $21.40 and no boots a year later → $1.40 and boots → $1.50 and no boots a year later.) According to mine, of her original $40, $20 of it earns no interest and $20 of it earns a years’ interest. But that assumes the interest she earns in that year is withdrawn, she gets to keep it but it doesn’t keep earning interest. So that’s why I got the wrong answer.
I think you’ve had her buy an extra pair of boots.
Ah, true. So, $239.41, at the end.
(Of course, this all assumes that the cheap boots don’t get more expensive over the course of 14 years. Siderea does say that she spends $20 each year on boots, but that’s hard to take seriously over a decade-plus period…)
Huh, thanks for the correction.
Smaller correction—I think you’ve had her buy an extra pair of boots. At $260 she’s already bought one pair, so we apply x↦1.07x−20 thirteen times, then multiply by 1.07 again for the final year’s interest, and she ends with no boots, so that’s $239.41. (Or start with $280 and apply x↦1.07(x−20) fourteen times.)
Not sure why my own result is wrong. Part of it is that I forgot to subtract the money actually spent on boots—I did “the $20 she spends after the first year gets one year’s interest, so that’s $21.40; the $20 she spends after the second year gets two years’ interest, so that’s $22.90...” but actually it’s only $1.40, $2.90 and so on. But even accounting for that, I get $222.58. So let’s see...
Suppose she only needs to buy two pairs of boots. According to your method she goes $40 → $21.40 → $1.50. (Or, $40 and no boots → $20 and boots → $21.40 and no boots a year later → $1.40 and boots → $1.50 and no boots a year later.) According to mine, of her original $40, $20 of it earns no interest and $20 of it earns a years’ interest. But that assumes the interest she earns in that year is withdrawn, she gets to keep it but it doesn’t keep earning interest. So that’s why I got the wrong answer.
Ah, true. So, $239.41, at the end.
(Of course, this all assumes that the cheap boots don’t get more expensive over the course of 14 years. Siderea does say that she spends $20 each year on boots, but that’s hard to take seriously over a decade-plus period…)