Two motorists set out at the same time to go from Toledo to Ft. Wayne, a distance of 100 miles. They both followed the same route and traveled at different, though uniform, speeds of an integral number of miles per hour. The difference in their speeds was a prime number of miles per hour. After they had been driving for two hours, the distance of the slower car from Toledo was five times the distance of the faster car from Ft. Wayne. How fast did the two motorists drive?
M1 is the slower motorist, traveling at rate r1.
M2 is the faster motorist, traveling at r1 + p, where p is prime.
In 2 hours, M1 travels 2r1 miles and M2 travels 2(r1 + p) miles.
M2 is the faster motorist, traveling at r1 + p, where p is prime.
In 2 hours, M1 travels 2r1 miles and M2 travels 2(r1 + p) miles.
Now,
2r1 = 5(100 – 2(r1 + p))
2r1 = 500 – 10r1 – 10p
10p = 500 – 12r1
p = 50 – 6r1/5
2r1 = 500 – 10r1 – 10p
10p = 500 – 12r1
p = 50 – 6r1/5
So 50 – 6r1/5 is prime.
We note that r1 is a multiple of 5, but more importantly, we note that 6r1/5 is even. Thus 50 – 6r1/5 is even, so p is an even prime.
Ha: It’s the only even prime, namely 2.
So 50 – 6r1/5 = 2 → 6r1/5 = 48 → 6r1 = 240 → r1 = 40 and r1 + 2 = 42.
Ha: It’s the only even prime, namely 2.
So 50 – 6r1/5 = 2 → 6r1/5 = 48 → 6r1 = 240 → r1 = 40 and r1 + 2 = 42.
Check: 2 · 40 = 80 = 5 · 16 → 16 ≟ 100 – 2(42) = 100 – 84 = 16, yup!