Tenants with one spare bedroom have had a payment reduction of 14% and those deemed to have two or more spare a reduction of 25%.
Lawyers for the 10 cases say that, unless the families move from their homes into smaller properties, they face building up rent arrears and being forced out any way.
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Thousands of people downstream of a dam have been evacuated, but authorities are unable to reach many on the hurricane-hit island.
Only half of the 4.2 million of eligible couples are applying for a marriage allowance that could be saving them hundreds each year.
Analysts said the outlook for the UK's public finances had "weakened significantly" with Brexit likely to put more pressure on the economy.