Question

use polyas four-step method in problem solving to solve the following problem. five runners, andy, beth, dale, ella, and tri, are in a one-mile race. beth finished 4 second(s) after andy. andy finished 2 second(s) before dale. dale finished 7 second(s) before tri. ella finished 1 second(s) after tri. in which order did the runners finish the race? first place

Explanation:

Step1: Understand the Problem

We have five runners: Andy, Beth, Dale, Ella, Tri. We need to determine their finishing order based on the time differences given. Shorter time means finishing earlier (higher place, like first is earliest).

Step2: Devise a Plan

Let's assign a variable for Andy's time, then express others' times relative to Andy. Let Andy's time be \( A \). Then:
- Beth: \( B = A + 4 \) (finishes 4s after Andy, so slower, higher time)
- Dale: \( D = A - 2 \) (finishes 2s before Andy, so faster, lower time)
- Tri: Dale finished 7s before Tri, so \( T = D + 7 = (A - 2) + 7 = A + 5 \)
- Ella: Ella finished 1s after Tri, so \( E = T + 1 = (A + 5) + 1 = A + 6 \)

Now we have all times in terms of \( A \):
- Andy: \( A \)
- Beth: \( A + 4 \)
- Dale: \( A - 2 \)
- Tri: \( A + 5 \)
- Ella: \( A + 6 \)

Step3: Carry Out the Plan

Now we order the times from smallest (earliest finish) to largest (latest finish):
- Dale: \( A - 2 \) (smallest, first place candidate)
- Andy: \( A \) (next)
- Beth: \( A + 4 \) (next)
- Tri: \( A + 5 \) (next)
- Ella: \( A + 6 \) (largest, last place)

Let's verify the relationships:
- Beth (A+4) is 4s after Andy (A): correct.
- Andy (A) is 2s before Dale? Wait, no: Dale is \( A - 2 \), so Andy is 2s after Dale. Wait, the problem says "Andy finished 2 second(s) before Dale" – wait, that would mean Andy's time is less than Dale's? Wait, I made a mistake here. Let's re-express:

If Andy finished 2s before Dale, that means Andy's time is \( D - 2 \), so \( A = D - 2 \) → \( D = A + 2 \). Oh! I had the direction wrong. Let's correct:

Let's redefine:

Let Andy's time be \( A \).

- Beth finished 4s after Andy: \( B = A + 4 \)
- Andy finished 2s before Dale: so Dale's time is \( A + 2 \) (because Andy is faster, so Dale is 2s slower than Andy)
- Dale finished 7s before Tri: so Tri's time is \( D + 7 = (A + 2) + 7 = A + 9 \)
- Ella finished 1s after Tri: \( E = T + 1 = (A + 9) + 1 = A + 10 \)

Wait, no, "finished before" means faster (smaller time). So if Dale finished 7s before Tri, Tri's time is Dale's time + 7. Correct. But Andy finished 2s before Dale: so Andy's time is Dale's time - 2 → \( A = D - 2 \) → \( D = A + 2 \). So Dale is 2s slower than Andy.

Now let's list all times:

- Andy: \( A \)
- Dale: \( A + 2 \)
- Beth: \( A + 4 \) (4s after Andy: \( A + 4 \), which is after Dale's \( A + 2 \), so Beth is slower than Dale)
- Tri: Dale + 7 = \( (A + 2) + 7 = A + 9 \)
- Ella: Tri + 1 = \( A + 10 \)

Wait, but now the order from smallest (earliest) to largest (latest) is:

Andy (A) → Dale (A+2) → Beth (A+4) → Tri (A+9) → Ella (A+10)? But that can't be right because Beth is 4s after Andy, but Dale is 2s after Andy, so Dale is before Beth? Wait, \( A + 2 \) is less than \( A + 4 \), so Dale (A+2) finishes before Beth (A+4). Correct.

But let's check the original problem statement again: "Andy finished 2 second(s) before Dale" – so Andy's time is less than Dale's, so Andy is faster. Then "Dale finished 7 second(s) before Tri" – Dale's time less than Tri's, Dale faster. "Ella finished 1 second(s) after Tri" – Tri's time less than Ella's, Tri faster. "Beth finished 4 second(s) after Andy" – Andy's time less than Beth's, Andy faster.

So let's assign actual numbers to make it easier. Let's set Andy's time as 10 seconds (arbitrary, since we care about relative times).

- Andy: 10s
- Beth: 10 + 4 = 14s
- Dale: Andy + 2 = 12s (because Andy finished 2s before Dale: so Dale took 2s longer than Andy)
- Tri: Dale + 7 = 12 + 7 = 19s
- Ella: Tri + 1 = 20s

Now order by time (smallest first, earliest finish):

1. Andy (10s)
2. Dale (12s)
3. Beth (14s)
4. Tri (19s)
5. Ella (20s)

Wait, but that contradicts the earlier mistake. Wait, no: if Andy finished 2s before Dale, that means Andy's time is 2s less than Dale's. So if Andy is 10s, Dale is 12s (Andy is 2s before Dale: 10 is 2 less than 12). Correct. Then Dale finished 7s before Tri: Dale is 12s, Tri is 19s (12 +7). Ella is 1s after Tri: 20s. Beth is 4s after Andy: 14s. So the order is Andy (10), Dale (12), Beth (14), Tri (19), Ella (20).

But wait, let's check the problem statement again: "Five runners, Andy, Beth, Dale, Ella, and Tri". Wait, did I miss a runner? No, five: Andy, Beth, Dale, Ella, Tri. Wait, but in my first calculation, I had a mistake in Dale's time. Let's re-express with correct relative times:

Let’s define the finish times as follows (smaller time = earlier finish):

- Let Andy's time be \( t_A \)
- Beth: \( t_B = t_A + 4 \) (since she finished 4s after Andy, so her time is larger)
- Andy finished 2s before Dale: so \( t_A = t_D - 2 \) → \( t_D = t_A + 2 \) (Dale's time is 2s larger than Andy's, so Andy is faster)
- Dale finished 7s before Tri: \( t_D = t_T - 7 \) → \( t_T = t_D + 7 = (t_A + 2) + 7 = t_A + 9 \) (Tri's time is 7s larger than Dale's, so Dale is faster than Tri)
- Ella finished 1s after Tri: \( t_E = t_T + 1 = (t_A + 9) + 1 = t_A + 10 \) (Ella's time is 1s larger than Tri's, so Tri is faster than Ella)

Now, let's list all times in terms of \( t_A \):

- \( t_A \): Andy
- \( t_D = t_A + 2 \): Dale
- \( t_B = t_A + 4 \): Beth
- \( t_T = t_A + 9 \): Tri
- \( t_E = t_A + 10 \): Ella

Now, order from smallest to largest (earliest to latest):

1. Andy (\( t_A \))
2. Dale (\( t_A + 2 \))
3. Beth (\( t_A + 4 \))
4. Tri (\( t_A + 9 \))
5. Ella (\( t_A + 10 \))

Wait, but that seems odd because Dale is between Andy and Beth. Let's verify the relationships:

- Beth finished 4s after Andy: \( t_B - t_A = 4 \) → correct (14 - 10 = 4 if \( t_A = 10 \))
- Andy finished 2s before Dale: \( t_D - t_A = 2 \) → correct (12 - 10 = 2)
- Dale finished 7s before Tri: \( t_T - t_D = 7 \) → correct (19 - 12 = 7)
- Ella finished 1s after Tri: \( t_E - t_T = 1 \) → correct (20 - 19 = 1)

So the order is Andy (first), Dale (second), Beth (third), Tri (fourth), Ella (fifth).

But wait, let's check again. Maybe I made a mistake in the direction of "before". If "Andy finished 2s before Dale", that means Andy's time is less than Dale's, so Andy is faster. So Andy comes before Dale. Then Dale is next. Then Beth is 4s after Andy, so Beth is after Dale (since Dale is 2s after Andy, Beth is 4s after Andy: 2s after Andy is Dale, 4s after is Beth, so Dale is before Beth). Then Tri is 7s after Dale, so after Beth. Then Ella is 1s after Tri. So the order is Andy, Dale, Beth, Tri, Ella.

Answer:

First Place: Andy
Second Place: Dale
Third Place: Beth
Fourth Place: Tri
Fifth Place: Ella

(So the order from first to last is Andy, Dale, Beth, Tri, Ella)