A cross-country team had a total of 10 practices last week, with each practice being in the morning or the afternoon. During each morning practice, they ran 5 miles, and during each afternoon practice, they ran 6 miles. If they ran a total of 57 miles last week, how many morning practices did they have?

Question

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Answer 1

Let \( m \) be the number of morning practices and \( a \) be the number of afternoon practices. We have the following two equations based on the problem:

1. The total number of practices: \[ m + a = 10 \]

2. The total miles run: \[ 5m + 6a = 57 \]

We can use the first equation to express \( a \) in terms of \( m \): \[ a = 10 - m \]

Now, we can substitute this expression for \( a \) into the second equation: \[ 5m + 6(10 - m) = 57 \]

Expanding this gives: \[ 5m + 60 - 6m = 57 \]

Simplifying this, we get: \[ -1m + 60 = 57 \]

Now, solving for \( m \): \[ -1m = 57 - 60 \] \[ -1m = -3 \] \[ m = 3 \]

Thus, the number of morning practices \( m \) is \( \boxed{3} \).

To verify, we can find \( a \): \[ a = 10 - 3 = 7 \]

Now check the miles: \[ 5m + 6a = 5(3) + 6(7) = 15 + 42 = 57 \]

The calculations confirm that the solution is correct, and the number of morning practices is indeed \( \boxed{3} \).