Question

last saturday, erin and krysta each biked the prairie point trail, a 20 - mile path leading to the peak of prairie point mountain. both girls started at 7:00 a.m., heading toward the peak. erin started at the trailhead and biked at a constant speed of 10 miles per hour. krysta started at an entry point 3 miles ahead of the trailhead and rode at a steady pace of 8 miles per hour. if each girl kept a constant speed, which equation can you use to find h, the number of hours it took for erin to catch up to krysta?
10h = 8h + 3
10 = 3h + 8
how long did it take for erin to catch up to krysta?
simplify any fractions.
hours

Explanation:

Step1: Start with the correct equation

We know the correct equation from the problem context (the first equation given: \(10h = 8h + 3\) is incorrect, wait, no—wait, let's re - evaluate. Wait, Erin's distance: she starts 3 miles ahead and goes at 8 mph? No, wait, no—wait, Krysta starts at trailhead (0 miles) and goes at 10 mph. Erin starts 3 miles ahead and goes at 8 mph? Wait, no, the problem says: "Erin started at the trailhead and biked at a constant speed of 10 miles per hour. Krysta started at an entry point 3 miles ahead of the trailhead and rode at a steady pace of 8 miles per hour." Wait, I misread earlier. So Erin: speed \(v_E=10\) mph, starts at 0 miles. Krysta: speed \(v_K = 8\) mph, starts at 3 miles. We want to find \(h\) when their distances are equal. Distance for Erin: \(d_E=10h\). Distance for Krysta: \(d_K = 8h+3\). So the correct equation is \(10h=8h + 3\). Now solve for \(h\).

Step2: Subtract \(8h\) from both sides

Subtract \(8h\) from both sides of the equation \(10h=8h + 3\).
\(10h-8h=8h + 3-8h\)
Simplify both sides: \(2h=3\)

Step3: Divide both sides by 2

Divide both sides of the equation \(2h = 3\) by 2.
\(h=\frac{3}{2}=1.5\)

Answer:

\(1.5\) (or \(\frac{3}{2}\))