Tom's mathematics test had $60$ questions: $10$ arithmetic questions, $30$ algebra questions and $20$ geometry questions. Although he correctly answered $70\%$ of the arithmetic questions, $50\%$ of the algebra questions and $50\%$ of the geometry questions, he did not pass the test because he got fewer than $60\%$ of the questions right. How many more questions would he have needed to answer correctly to earn a passing grade of $60\%?$
