Luis Nunez begins walking at a rate of 6 mph. Fifteen minutes after he​ leaves, his​ wife, Kristin, realizes that Luis has forgotten his wallet. Kristin gets on her bicycle and begins riding at a rate of 16 mph along the same path that Luis took.
​a) How long will it take for Kristin to catch​ Luis?
​b) How far from their house will Kristin catch​ Luis?
Type an integer or a​ decimal

Let's break down this problem step by step.

### Part a) How long will it take for Kristin to catch Luis?

We know:
- Luis walks at a rate of 6 mph.
- Kristin starts 15 minutes (or 0.25 hours) after Luis and rides at 16 mph.

In that 0.25 hours, Luis has already walked a distance of:
\[
\text{Distance covered by Luis in 15 minutes} = 6 \, \text{mph} \times 0.25 \, \text{hours} = 1.5 \, \text{miles}.
\]
Now, Kristin needs to cover this 1.5 miles **plus** the distance Luis continues to walk while Kristin is riding. To determine when Kristin catches Luis, we need to set up the following equation.

Let \( t \) be the time in hours that Kristin rides her bike until she catches up to Luis.

- During this time \( t \), Kristin will cover a distance of \( 16t \) miles.
- During this same time, Luis continues to walk, covering a distance of \( 6t \) miles.

The total distance Luis has walked when Kristin catches him is the 1.5 miles he walked before Kristin started plus the distance \( 6t \) he walked after Kristin began riding. Therefore, the distance covered by Kristin and Luis must be equal when Kristin catches up:
\[
16t = 1.5 + 6t.
\]

Now, let's solve for \( t \).

\[
16t - 6t = 1.5
\]
\[
10t = 1.5
\]
\[
t = \frac{1.5}{10} = 0.15 \, \text{hours}.
\]

Thus, it will take Kristin **0.15 hours** to catch Luis.

### Part b) How far from their house will Kristin catch Luis?

Now that we know it takes Kristin 0.15 hours to catch Luis, we can calculate the distance from the house by using the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}.
\]
For Kristin:
\[
\text{Distance} = 16 \, \text{mph} \times 0.15 \, \text{hours} = 2.4 \, \text{miles}.
\]

Therefore, Kristin will catch Luis **2.4 miles** from their house.

### Summary:
- a) It will take Kristin **0.15 hours** (or 9 minutes) to catch Luis.
- b) Kristin will catch Luis **2.4 miles** from their house.

Would you like more details, or do you have any questions?

Here are 5 related questions to extend this concept:
1. How would the time change if Kristin's speed was only 12 mph?
2. What if Luis walked at 4 mph instead of 6 mph?
3. If Kristin took 20 minutes to realize Luis had forgotten his wallet, how far would she be when she caught him?
4. How would the results differ if Luis was running at 8 mph instead of walking?
5. Can you calculate how much time Luis had left to reach 3 miles before Kristin caught up?

**Tip:** When working with different units of time (minutes and hours), always convert everything to a common unit for easier calculation.

Luis Nunez begins walking at a rate of 6 mph. Fifteen minutes after he leaves, his wife, Kristin, realizes that Luis has forgotten his wallet. Kristin gets on her bicycle and begins riding at a rate of 16 mph along the same path that Luis took. a) How long will it take for Kristin to catch Luis? b) How far from their house will Kristin catch Luis?

Solve a real-life rate and distance problem where Kristin chases after Luis on her bicycle. Learn how to calculate the time it takes for Kristin to catch Luis and how far from the house she catches him. This problem involves linear equations and the formula Distance = Speed × Time, suitable for Grades 7-10.

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