Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of 2 miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, Jose started bicycling at a constant speed of 7 miles per hour faster than Tanya ran, following the same straight road from the school to the park. All three people arrived at the park at the same time. The distance from the school to the park is nm miles, where m and n are relatively prime positive integers. Find m+n.
All of them followed the same straight road from the school to the park. dP=dT=dJ
One hour after Patrick left, Tanya started, and one hour after Tanya left, Jose started. tT=tP−1,tJ=tT−1=tP−2
Tanya is 2 miles faster than Patrick and Jose is 7 miles per hour faster than Tanya. vT=vP+2,vJ=vT+7⟹vJ=vP+9
By Speed, Distance, and Time and Hints 1, 2, and 3 v=td⟹d=vtdP=dT=dJvPtP=vTtT=vJtJvPtP=(vP+2)(tP−1)=(vP+9)(tP−2) Assume vP=v, tP=t for writing easier.
Hint 5. (v+2)(t−1)=vtvt+2t−v−2=vt2t−v−2=0(v+9)(t−2)=vtvt+9t−2v−18=vt9t−2v=189t−2(2t−2)=185t+4=18t=514v=2(514)−2=518d=vt=518×514=25252=nmm+n=252+25=277
