Question

Malcolm and Ravi raced each other.
The average of their maximum speeds was
\[260\text{ km/h}\]. If Malcolm's speed is doubled, Malcolm's maximum speed would be
\[80\text{ km/h}\] more than Ravi's maximum speed.
What were Malcolm's and Ravi's maximum speeds?
Malcolm's maximum speed was

\[\text{km/h}\] and Ravi's maximum speed was

\[\text{km/h}\].

Answer 1

Let \( m \) represent Malcolm's maximum speed and \( r \) represent Ravi's maximum speed. We know from the problem's statement the following information:

1. The average of their maximum speeds is \( 260 \) km/h:
\[
\frac{m + r}{2} = 260
\]
This gives us:
\[
m + r = 520 \tag{1}
\]

2. If Malcolm's speed is doubled, his maximum speed would be \( 80 \) km/h more than Ravi's maximum speed:
\[
2m = r + 80 \tag{2}
\]

Now we can solve these two equations step-by-step.

Starting with equation (2):
\[
2m - r = 80
\]

We can rearrange this to express \( r \) in terms of \( m \):
\[
r = 2m - 80 \tag{3}
\]

Next, we substitute equation (3) into equation (1):
\[
m + (2m - 80) = 520
\]

This simplifies to:
\[
3m - 80 = 520
\]

Adding \( 80 \) to both sides:
\[
3m = 600
\]

Dividing both sides by \( 3 \):
\[
m = 200 \text{ km/h}
\]

Now, we can find \( r \) using equation (3):
\[
r = 2(200) - 80
\]
\[
r = 400 - 80 = 320 \text{ km/h}
\]

Finally, we can summarize the results:
- Malcolm's maximum speed is \( \boxed{200} \) km/h.
- Ravi's maximum speed is \( \boxed{320} \) km/h.