On my way to work, I pass by a digital thermometer outside a bank that displays the temperature to the nearest integer, alternating back and forth between Fahrenheit and Celsius at four-second intervals.

The other day, when I was going to work, I noticed that the temperatures displayed in Fahrenheit and Celsius were simply reverse digits of each other.

Well, it warmed up nicely during the day, and when I was on my way home, I looked at the same thermometer.
Imagine my shock when, again, the Fahrenheit and Celsius temperature displays were simply reversed digits of each other!

What was the temperature when I came in to work, and what was it when I was on my way home?

Note: To convert from degrees Fahrenheit to Celsius, subtract 32 then multiply by 5/9. To convert the other way, simply do the opposite (multiply by 9/5 and add 32). Ignore leading zeroes.

This solution requires a bit trial and error.

Fahrenheit: 10a + b
Celsius: (10a + b - 32)*5/9 = 10b + a

We can solve this further:
50/9a + 5/9b - 160/9 = 10b + a
<=> 41a - 160 = 85b

Now you have to try values for a:
a=9 -> b ~ 2.46
a=8 -> b ~ 1.98
a=7 -> b ~ 1.49
a=6 -> b ~ 1.01
a=5 -> b ~ 0.53
a=4 -> b ~ 0.05
a=3 -> b ~ -0.44 (we can stop here)

Now you take the values for which b is closest to an integer.
Those are:
a=8 and b=2 -> 82F and 28C°
a=6 and b=1 -> 61F and 16C°
a=4 and b=0 -> 40F and 04C° (which is not valid if we assume that there is no 0 for first digit)

The two first are correct solutions.
Proof:
(82-32)*5/9 = 27 7/9 (28 nearest integer)
(61-32)*5/9 = 16 1/9 (16 nearest integer)

Posted by abc on 2003-09-29 10:36:22