Reduce and Solve gives:
eq = Rationalize[{-101.81 a - 47.62*b - 36.42*c -
32.51*d == -61.91, -12.48*a - 7.58*b - 6.11*c - 5.63*d == -8.91,
a + b + c + d == 1}, 0]
(*{-((10181 a)/100) - (2381 b)/50 - (1821 c)/50 - (3251 d)/100 == -(
6191/100), -((312 a)/25) - (379 b)/50 - (611 c)/100 - (563 d)/
100 == -(891/100), a + b + c + d == 1}*)
Reduce[First@eq && a >= 0 && b >= 0 && c >= 0 && d >= 0, {a, b, c, d}]
(*0 <= a <= 6191/
10181 && ((0 <= b < (6191 - 10181 a)/4762 &&
0 <= c <= (6191 - 10181 a - 4762 b)/3642) || (b == (
6191 - 10181 a)/4762 && c == 0)) &&
d == (6191 - 10181 a - 4762 b - 3642 c)/3251*)
Solve[First@eq && a >= 0 && b >= 0 && c >= 0 && d >= 0, {a, b, c, d}, MaxExtraConditions -> All]
(*{{a -> ConditionalExpression[(6191 - 4762*b - 3642*c - 3251*d)/10181,
Inequality[0, LessEqual, b, LessEqual, 6191/4762] && c >= 0 && -6191 + 4762*b + 3642*c < 0 && d >= 0 &&
-6191 + 4762*b + 3642*c + 3251*d <= 0]}, {a -> ConditionalExpression[0, 0 <= b <= 6191/4762],
c -> ConditionalExpression[(6191 - 4762*b)/3642, 0 <= b <= 6191/4762],
d -> ConditionalExpression[0, 0 <= b <= 6191/4762]}}*)
Looks like solution exist.