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Carina is driving from her home in Anaheim to Berkeley on the same day her brother is driving from Berkeley to Anaheim, so they decide to meet for lunch along the way in Buttonwillow. The distance from Anaheim to Berkeley is 410 miles. It takes Carina 3 hours to get to Buttonwillow, while her brother drives 4 hours to get there. The average speed Carina’s brother drove was 15 miles per hour faster than Carina’s average speed. Find Carina’s and her brother’s average speeds.

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Answer:

Let's denote Carina's average speed as \(s\) (in miles per hour). Carina's brother drove 15 miles per hour faster, so his speed is \(s + 15\).

Carina's distance to Buttonwillow is \(3s\) (since she drove for 3 hours), and her brother's distance is \(4(s + 15)\ (since he drove for 4 hours at a faster speed).

The sum of their distances should be the distance between Anaheim and Berkeley, which is 410 miles:

\[3s + 4(s + 15) = 410\]

Now, solve for \(s\) to find Carina's average speed. Once you have \(s\), you can find Carina's brother's speed by adding 15.

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