Fix to_reduced_units with dimensionless units

CHANGES

@@ -26,6 +26,7 @@ Pint Changelog
 - Add `pint.testing` with functions to compare pint objects in tests (Issue #1421).
 - Fix handling modulo & floordiv operator in pint_eval (Issue #1470)
 - Fix `to_compact` and `infer_base_unit` for non-float non_int_type.
+- Fix `to_reduced_units` to work with dimensionless units. (Issue #919)
 - Fix parsing of units string with same canonalized name (Issue #1441 & #1142)
 
 ### Breaking Changes

pint/registry.py

@@ -830,7 +830,9 @@ def _get_dimensionality_ratio(self, unit1, unit2):
             return 1
 
         dim1, dim2 = (self.get_dimensionality(unit) for unit in (unit1, unit2))
-        if not dim1 or not dim2 or dim1.keys() != dim2.keys():  # not comparable
+        if dim1 == dim2:
+            return 1
+        elif not dim1 or not dim2 or dim1.keys() != dim2.keys():  # not comparable
             return None
 
         ratios = (dim2[key] / val for key, val in dim1.items())

pint/testsuite/test_quantity.py

@@ -665,6 +665,13 @@ def test_to_reduced_units(self):
         q = self.Q_(0.5, "g*t/kg")
         helpers.assert_quantity_equal(q.to_reduced_units(), self.Q_(0.5, "kg"))
 
+    def test_to_reduced_units_dimensionless(self):
+        ureg = UnitRegistry(preprocessors=[lambda x: x.replace("%", " percent ")])
+        ureg.define("percent = 0.01 count = %")
+        Q_ = ureg.Quantity
+        reduced_quantity = (Q_("1 s") * Q_("5 %") / Q_("1 count")).to_reduced_units()
+        assert reduced_quantity == ureg.Quantity(0.05, ureg.second)
+
     @pytest.mark.parametrize(
         ("unit_str", "expected_unit"),
         [