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Mitchell earned a total score of 55 points. Twenty percent of the baskets he made were worth three-points each. If four-fifths of the two-point baskets Mitchell attempted were successful and he made one-third of the three-point baskets he attempted, what is the total number of baskets that Mitchell attempted?

You don't mention his free throws made. If that number is zero, solve these equations:
x = number of 3-point shots made
y = number of 2-point shots made

3x + 2y = 55
x = 0.2(x+y)

5x = x+y
y = 4x
3x + 8x = 55
x = 5
y = 20

Total Shots attempted = 3x + (5/4) y
= 15 + 25 = 40

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