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The problem appears to be a race scenario between Ivan and Jeremy, where Jeremy completes the race, and Ivan has run a portion of the distance.

### Problem Breakdown:
- **Jeremy's Time**: Jeremy has completed the race.
- **Ivan's Distance**: When Jeremy completed the race, Ivan had only run \( \frac{2}{3} \) of the distance.
- **Ivan's Speed**: Ivan's speed is \( \frac{2}{3} \) of Jeremy's speed.

The questions are:
1. **Part (a)**: Find Ivan's speed in terms of Jeremy's speed.
2. **Part (b)**: Find the total distance of the race.

### Let's solve the questions step-by-step:

#### Part (a): Ivan's Speed
From the problem, we know:
- Ivan's speed is \( \frac{2}{3} \) of Jeremy's speed.

So, if Jeremy's speed is \( S_j \), then Ivan's speed \( S_i \) is given by:
\[
S_i = \frac{2}{3} S_j
\]
Thus, Ivan's speed is \( \frac{2}{3} \) of Jeremy's speed.

#### Part (b): Distance of the Race
Let's denote:
- \( d \) as the total distance of the race.
- \( t_j \) as the time Jeremy took to complete the race.

Since Ivan ran \( \frac{2}{3} \) of the distance when Jeremy finished, we know:
\[
\text{Distance Ivan covered} = \frac{2}{3}d
\]
Now, using the formula \( \text{distance} = \text{speed} \times \text{time} \):
- Jeremy's distance equation: \( d = S_j t_j \)
- Ivan's distance equation: \( \frac{2}{3}d = S_i t_j \)

Substitute \( S_i = \frac{2}{3}S_j \) into the equation for Ivan:
\[
\frac{2}{3}d = \frac{2}{3}S_j t_j
\]
Simplifying both sides:
\[
d = S_j t_j
\]

This confirms that the total distance of the race is the same as what Jeremy ran in time \( t_j \), so the distance is fully determined by Jeremy’s speed and time.

### Final Answers:
- (a) Ivan's speed \( S_i = \frac{2}{3} S_j \).
- (b) The total distance of the race \( d = S_j t_j \).

Would you like further clarification or details?

### Related Questions:
1. If Ivan’s speed was \( \frac{3}{4} \) of Jeremy’s speed, how would that change the outcome?
2. What if Jeremy ran the race in 15 minutes—how long would it take Ivan to complete the race?
3. Can you express the time taken by Ivan to complete the race?
4. What is the ratio of their times to finish the race if both start at the same time?
5. How does the relative speed affect the total time Ivan would take to catch up?

### Tip:
When solving problems with relative speed or distances, always use a clear variable assignment and express each part (speed, time, and distance) in terms of one another to avoid confusion.

Ivan and Jeremy ran in a race. When Jeremy had completed the race in t minutes, Ivan had only run 2/3 of the distance. Ivan's speed for the race was 2/3 of Jeremy's speed. (a) Find Ivan's speed in terms of Jeremy's speed. (b) Find the distance of the race.

Learn how to solve a problem involving two runners, Ivan and Jeremy, where Ivan runs 2/3 of the distance and has a speed that is 2/3 of Jeremy's speed. This step-by-step guide covers key algebraic concepts, including ratios, proportions, and distance-speed-time relations. Suitable for students in Grades 6-8.

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