AN INSTRUCTOR AND STUDENT RAN INTO UNEXPECTED WX AND RAN OUT OF FUEL.

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MR. X (OWNER OF NY) ASKED ME TO GO ALONG WITH HIM TO RAPID CITY; SD. I AGREED AND TOLD HIM TO FILE AN IFR FLT PLAN. THAT EVENING HE FILED AND RELAYED TO ME THAT WX WAS: CEILINGS WERE 2800 FT AND VISIBILITY WAS OVER 5 MI; SO HE DIDN'T FILE AN ALTERNATE. NEXT DAY; HE CONTACTED GREEN BAY FSS FOR WX UPDATE. HE RELAYED TO ME THAT THE WX FORECAST WAS THE SAME WITH SOME LIGHT ICING RPTED IN THE CLOUDS FROM 8000-11000 FT; WINDS ALOFT WERE AT 360 DEGS AT 5 KTS. HE TOPPED OFF HIS FUEL TANKS AND GOT OUR IFR CLRNC FROM UBE TO RAP. WE DEPARTED UBE AND CLBED TO 10000 FT ON COURSE TO RAP. AT THAT ALT WE WERE ON TOP OF A CLOUD DECK WHICH WAS AT APPROX 6000 FT MSL. WE CLBED TO 12000 FT TO REMAIN ON TOP. WE REMAINED ON TOP UNTIL ABOUT 30 MINS FROM RAP. APPROX 15 MINS FROM RAP WE GOT WX FROM CTR. AT RAP IT WAS APPROX 400 FT AND 1 MI. DUE TO THE FACT THAT THE GS WAS INOP WE DECIDED TO GO TO CASPER; WY; BECAUSE THEY HAD AN ILS AND WX WAS RPTED AT 400 FT AND 1 MI OF VISIBILITY. WE CALCULATED THAT IT WAS 1 HR ENRTE TO CPR AND THAT WE HAD A LITTLE OVER 1.6 HRS OF FUEL REMAINING AND IT AGREED WITH THE FUEL GAUGES. WE FILED IN THE AIR THROUGH ZMP AND THE FSS AND GOT A CLRNC FROM RAP TO CPR. ENRTE TO CPR; WE ENCOUNTERED SOME LIGHT ICING SO WE CLBED TO 14000 FT UNTIL MUDDY MOUNTAIN VOR; DDY. WE WERE THEN CLRED DOWN TO 7000 FT FOR THE ILS TO RWY 8. UPON REACHING 7500 FT; WE TRIED TO INCREASE THE PWR BUT THE ENG WAS RUNNING ROUGH AND LOSING PWR. WE HAD CARB HEAT ON BUT IT DIDN'T HELP MUCH. I THEN TOOK THE PLANE FROM MY STUDENT AND DECLARED AN EMER WITH CASPER TWR. I WAS UNABLE TO MAINTAIN ALT SO I DSNDED AT AS SLOW OF AN AIRSPD AS SAFELY POSSIBLE. I INSTRUCTED MY STUDENT TO WATCH FOR THE GND SO I COULD FLY THE ACFT. AT ABOUT 100 FT AGL WE HAD VISUAL GND CONTACT; WE LANDED GEAR UP IN A VALLEY APPROX 5 MI FROM CPR. THERE WERE NO INJURIES OF ANY KIND. THE ACFT SUSTAINED SUBSTANTIAL DAMAGE TO THE R WING AND TO THE PROP. WE REMAINED AT THE CRASH SITE FOR OVER 2 HRS; THEN DECIDED TO WALK OUT TO A ROAD WHICH WAS ABOUT 2 MI AWAY. WE ENSURED THE ELT WAS ON; TOOK ALL THE MAPS AND ALL THE EXTRA CLOTHING WE COULD CARRY. THE ACFT COMPASS IS WHAT WE USED TO NAV TO THE ROAD AND OUR VFR MAPS. SUMMATION OF FLT: WE ENCOUNTERED HEADWINDS OF 20-30 KTS ALONG THE ENTIRE RTE; WX HAD DROPPED DOWN TO AT OR BELOW MINIMUMS; FORCING US TO GO TO A DIFFERENT ARPT WHICH WAS AT OUR MAX RANGE (WITH 30 MIN RESERVE). AT THE PREDICTED 11 GPH THAT THE HANDBOOK STATED; THE TRIP SHOULD HAVE ONLY TAKEN 4.4 HRS TO CPR. IT ACTUALLY TOOK A LITTLE OVER 5 HRS. THE HANDBOOK STATES AT 12000 FT FUEL CONSUMPTION IS 10 GPH. WE USED 11 GPH FOR CALCULATIONS. OUR FLT ORIGINALLY WAS FOR ONLY 3.3 HRS TO RAP.