Two friends enjoy competing with each other to see who has the best time in running a mile. Initially (before they ever raced each other), the first friend runs a mile in 7 minutes, and for each race that they run, his time decreases by 11 seconds. Initially, the second friend runs a mile in 7 minutes and 20 seconds, and for each race that they run, his time decreases by 14 seconds. Which will be the first race in which the second friend beats the first?
race number 

Let's define the problem and set up the equations:

- The first friend's initial time is 7 minutes, which is 420 seconds (since \( 7 \times 60 = 420 \) seconds). His time decreases by 11 seconds for each race.
- The second friend's initial time is 7 minutes and 20 seconds, which is 440 seconds (since \( 7 \times 60 + 20 = 440 \) seconds). His time decreases by 14 seconds for each race.

### First Friend's Time:
The first friend's time after \( n \) races can be expressed as:
\[
T_1(n) = 420 - 11n
\]

### Second Friend's Time:
The second friend's time after \( n \) races can be expressed as:
\[
T_2(n) = 440 - 14n
\]

### Goal:
We want to find the first race number \( n \) such that the second friend's time is less than the first friend's time. That is:
\[
T_2(n) < T_1(n)
\]

Substitute the expressions for \( T_1(n) \) and \( T_2(n) \):
\[
440 - 14n < 420 - 11n
\]

### Solving the inequality:
1. Subtract \( 420 \) from both sides:
\[
440 - 14n - 420 < - 11n
\]
\[
20 - 14n < -11n
\]

2. Add \( 14n \) to both sides:
\[
20 < 3n
\]

3. Divide both sides by 3:
\[
n > \frac{20}{3}
\]
\[
n > 6.67
\]

Since \( n \) must be a whole number, the first race in which the second friend beats the first is **race 7**.

Would you like more details on the solution or have any questions?

Here are 5 relative questions to explore:

1. What would happen if the second friend improved at a slower rate?
2. How can we model the times of the two friends with a different starting time?
3. What if there was a third friend joining the race?
4. Can this pattern continue indefinitely, or will there be a limit to how fast they can run?
5. How would a delay in improvement affect the race outcomes?

**Tip**: When dealing with sequences, always check for the first integer solution that satisfies inequalities, as fractional results aren't practical in discrete events like races.

Two friends enjoy competing with each other to see who has the best time in running a mile. Initially (before they ever raced each other), the first friend runs a mile in 7 minutes, and for each race that they run, his time decreases by 11 seconds. Initially, the second friend runs a mile in 7 minutes and 20 seconds, and for each race that they run, his time decreases by 14 seconds. Which will be the first race in which the second friend beats the first?

Explore how to solve a real-world problem involving two friends competing in races. Learn how to use algebraic linear equations and inequalities to determine when the second friend will beat the first. Perfect for students in Grades 8-10, this problem covers sequences, linear equations, and solving inequalities.

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