Question

naomi and shelley both hiked the discovery point trail. when shelley started hiking, naomi was already 1.3 miles from the trailhead, walking at a rate of 2.4 miles per hour. shelley started at the trailhead and jogged at a rate of 5 miles per hour. how long did it take shelley to catch up to naomi? simplify any fractions. hours

Explanation:

Step1: Set up distance - rate - time equations

Let $t$ be the time in hours it takes Shelley to catch up to Naomi. The distance formula is $d = rt$ (distance = rate×time). Naomi's distance $d_N$ from the starting - point is $d_N=1.3 + 2.4t$ (she had a 1.3 - mile head - start and walks at 2.4 miles per hour). Shelley's distance $d_S$ from the starting - point is $d_S = 5t$ (she starts at the trailhead and jogs at 5 miles per hour).

Step2: Set the two distances equal

When Shelley catches up to Naomi, $d_N=d_S$. So we set up the equation $1.3 + 2.4t=5t$.

Step3: Solve the equation for $t$

Subtract $2.4t$ from both sides of the equation:
$1.3=5t - 2.4t$.
Combine like terms: $1.3 = 2.6t$.
Then divide both sides by 2.6 to solve for $t$: $t=\frac{1.3}{2.6}$.
Simplify the fraction: $t = 0.5$ hours.

Answer:

$0.5$