Question

ned and ralph left the auditorium at the same time but headed in opposite directions. after 2 hours they were 250 miles apart. if ned drove 5 mph faster than ralph, how fast did ralph drive?

Answer 1

To solve the problem, let's define the speeds of Ned and Ralph.

Let:
- \( r \) be Ralph's speed in miles per hour.
- \( r + 5 \) be Ned's speed (since he drives 5 mph faster than Ralph).

Both of them left the auditorium at the same time and after 2 hours they were 250 miles apart. The total distance they covered is the sum of the distances each of them drove.

The distance covered by Ralph in 2 hours is:
\[
\text{Distance}_{Ralph} = r \times 2
\]

The distance covered by Ned in 2 hours is:
\[
\text{Distance}_{Ned} = (r + 5) \times 2
\]

Adding these two distances gives the total distance they are apart:
\[
2r + 2(r + 5) = 250
\]

Now, let's simplify this equation:
\[
2r + 2r + 10 = 250
\]
\[
4r + 10 = 250
\]

Next, we subtract 10 from both sides:
\[
4r = 240
\]

Now we divide both sides by 4:
\[
r = 60
\]

Thus, Ralph's speed is \( 60 \) miles per hour.

To check:
- Ralph's distance in 2 hours: \( 60 \times 2 = 120 \) miles.
- Ned's speed: \( 60 + 5 = 65 \) miles per hour.
- Ned's distance in 2 hours: \( 65 \times 2 = 130 \) miles.
- Total distance apart: \( 120 + 130 = 250 \) miles.

The calculations confirm the solution is consistent. Therefore, Ralph drove at a speed of:
\[
\boxed{60} \text{ miles per hour.}
\]