During the race, Bernie ran 9 meters every 4 seconds, while Barnaby ran 2 meters evey second and got a 10 meter head start. If the race is 70 meters long, did Bernie ever catch with Barnaby? If so, when?

I have to use a proportion to solve this.

The proportion can be the speed: 9/4 = 2.25 m/s for Bernie and 20.00 m/s for Barnaby.

The distance from the starting line at time t after the start is
X1 (Barnaby) = 10 + 2 t
X2 (Benie) = 2.25 t
Bernie passes Barnaby when X1 = X2. Solve for the X where that happens.
10 + 2t = 2.25 t
10 = 0.25 t
t = 40 seconds

X1 = X2 = 90 m

To solve this problem, you can use a proportion to compare the distances covered by Bernie and Barnaby over time.

Let's first calculate how long it takes Barnaby to cover the total race distance of 70 meters:

Distance = Speed × Time

We know that Barnaby runs at a speed of 2 meters per second, meaning he covers 2 meters every second. Let's represent the time it takes for Barnaby to cover the full distance as t:

70 meters = 2 meters/second × t

Now, let's calculate the total time it takes for Bernie to catch up with Barnaby.

Bernie runs 9 meters every 4 seconds. So, the speed at which Bernie covers distance can be represented as:

Speed = Distance / Time

In this case, distance for Bernie is (70 - 10) meters (as Barnaby has a 10 meter head start), and the time is the same t that we found earlier:

Bernie's speed = (70 - 10) meters / t seconds
Bernie's speed = 60 meters / t seconds

Now we have two different expressions for Bernie's and Barnaby's speeds:

Bernie's speed = 60 meters / t seconds
Barnaby's speed = 2 meters / second

To determine if Bernie ever catches up with Barnaby, we can set up a proportion and solve for t:

(60 meters / t seconds) = (2 meters / second)

To solve for t, we cross-multiply:

60 meters × second = 2 meters × t seconds

Dividing both sides of the equation by 2 meters, we get:

30 seconds = t

Therefore, it takes 30 seconds for Bernie to catch up with Barnaby.

To determine when Bernie catches up with Barnaby, we need to find the distance each has covered after 30 seconds:

Bernie's distance covered in 30 seconds: Bernie's speed × time
Bernie's distance covered = 9 meters / 4 seconds × 30 seconds
Bernie's distance covered = 9 × 7.5 meters

Barnaby's distance covered in 30 seconds: Barnaby's speed × time
Barnaby's distance covered = 2 meters / second × 30 seconds
Barnaby's distance covered = 2 × 30 meters

Therefore, Bernie catches up with Barnaby after 30 seconds, when they have both covered a distance of 9 × 7.5 meters = 67.5 meters.

So, Bernie does catch up with Barnaby before reaching the finish line of 70 meters.

To determine if and when Bernie caught up with Barnaby, we can set up a proportion and use it to find the time it takes for both to cover the same distance.

Let's assume t is the time it takes for Bernie to catch up with Barnaby.

Based on the given information, we know that Bernie runs at a rate of 9 meters every 4 seconds, which can be written as:
9 meters / 4 seconds = 2.25 meters / 1 second

Barnaby, on the other hand, runs at a rate of 2 meters every second.

Now, let's calculate the distances covered by both Bernie and Barnaby after time t:
Bernie's distance = 2.25 meters/second * t seconds
Barnaby's distance = 2 meters/second * t seconds + 10 meters (due to his 10-meter head start)

Since we want to find the time Bernie catches up with Barnaby, we can set up the following equation based on the distances covered by both runners:
2.25t = 2t + 10

Let's solve for t:

2.25t - 2t = 10
0.25t = 10
t = 10 / 0.25
t = 40

Therefore, Bernie catches up with Barnaby after 40 seconds.