jmc

Question

The formula for converting a Fahrenheit temperature F to the corresponding Celsius temperature C is C = 59(F-32). An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer. For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?

Accepted Answer

Examine F - 32 modulo 9. If F - 32 0 9, then we can define 9x = F - 32. This shows that F = [95[59(F-32)] + 32] F = [95(5x) + 32] F = 9x + 32. This case works. If F - 32 1 9, then we can define 9x + 1 = F - 32. This shows that F = [95[59(F-32)] + 32] F = [95(5x + 1) + 32] = [9x + 95+ 32 ] F = 9x + 34. So this case doesn't work. Generalizing this, we define that 9x + k = F - 32. Thus, F = [95[59(9x + k)] + 32] F = [95(5x + [59k]) + 32] F = [95 [59k ] ] + 9x + 32. We need to find all values 0 k 8 that [ 95 [ 59 k ] ] = k. Testing every value of k shows that k = 0, 2, 4, 5, 7, so 5 of every 9 values of k work. There are 1000 - 329 = 107 cycles of 9, giving 5 107 = 535 numbers that work. Of the remaining 6 numbers from 995 onwards, 995, 997, 999, 1000 work, giving us 535 + 4 = 539 as the solu…