Let

b = the number of blue ribbons

w = the number of white ribbons

r = the number of red ribbons

A music competition awarded 40 ribbons:

b + w + r = 35

The number of blue ribbons awarded was 1 less than the number of white ribbons:

b = w - 1

The number of red ribbons was 4 more than the number of blue ribbons:

r = 4 + b

We have the following system of equations:

b + w + r = 35

b = w - 1

r = 4 + b
........
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b = 10 blue ribbons

w = 11 white ribbons

r = 14 red ribbons

There were 10 blue, 11 white and 14 red ribbons awarded.