In a 200-m race, Andrew beats Benzie by 8 seconds. On the contrary, if Andrew would give Benzie a head start of 32 meters, then they would finish the race at the same time. How long does Andrew take to run the 200-meter race?

Let the time Andrew takes to run the 200-meter race be \( t_A \) seconds. 

We can break this problem into two parts based on the information provided:

### 1. **Andrew beats Benzie by 8 seconds**
Let \( v_A \) be Andrew's speed and \( v_B \) be Benzie's speed. We know that:

\[
v_A = \frac{200}{t_A}
\]

Since Andrew beats Benzie by 8 seconds, Benzie's time to finish the 200-meter race is:

\[
t_B = t_A + 8
\]

Thus, Benzie's speed can be expressed as:

\[
v_B = \frac{200}{t_A + 8}
\]

### 2. **Head start of 32 meters**
In the second scenario, Andrew gives Benzie a head start of 32 meters, meaning Benzie only needs to run 168 meters. Both finish the race at the same time, so their times are equal in this case:

\[
\frac{168}{v_B} = \frac{200}{v_A}
\]

Now, substitute \( v_A \) and \( v_B \) in terms of \( t_A \) from the earlier expressions:

\[
\frac{168}{\frac{200}{t_A + 8}} = \frac{200}{\frac{200}{t_A}}
\]

Simplifying this equation:

\[
168 \cdot \frac{t_A + 8}{200} = t_A
\]

\[
168(t_A + 8) = 200 t_A
\]

\[
168 t_A + 1344 = 200 t_A
\]

\[
200 t_A - 168 t_A = 1344
\]

\[
32 t_A = 1344
\]

\[
t_A = \frac{1344}{32} = 42
\]

Thus, Andrew takes **42 seconds** to run the 200-meter race.

Would you like more details or have any questions? Here are five related questions to explore:

1. What is Benzie's time to finish the 200-meter race?
2. What are Andrew's and Benzie's speeds during the race?
3. How much of a head start would Benzie need to beat Andrew by 2 seconds?
4. If Andrew's speed increased by 10%, how long would he take to finish the race?
5. What if the race were 400 meters instead—how long would each take then?

**Tip**: Always ensure units are consistent (time in seconds, distance in meters) when solving rate problems.

Discover how to solve a 200-meter race problem where Andrew beats Benzie by 8 seconds, but with a 32-meter head start, they finish at the same time. This detailed solution covers algebraic concepts such as rates, proportions, and time-speed relationships. Suitable for students in Grades 9-11.

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