Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\%$ of the arithmetic, $40\%$ of the algebra, and $60\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\%$ of the problems right. How many more problems would she have needed to answer correctly to earn a $60\%$ passing grade? 

$ \text{(A)}\ 1\qquad\text{(B)}\ 5\qquad\text{(C)}\ 7\qquad\text{(D)}\ 9\qquad\text{(E)}\ 11 $
1. First, calculate the total number of problems Tori needs to answer correctly to pass the test. Since she needs at least $60\%$ of the problems correct:
   \[
   75 \times 0.6 = 45
   \]
   Therefore, Tori needs to answer $45$ questions correctly to pass.

2. Next, calculate the number of problems Tori answered correctly in each category:
   - Arithmetic: She answered $70\%$ of the $10$ arithmetic problems correctly:
     \[
     0.7 \times 10 = 7
     \]
   - Algebra: She answered $40\%$ of the $30$ algebra problems correctly:
     \[
     0.4 \times 30 = 12
     \]
   - Geometry: She answered $60\%$ of the $35$ geometry problems correctly:
     \[
     0.6 \times 35 = 21
     \]

3. Sum the number of problems she answered correctly in each category:
   \[
   7 + 12 + 21 = 40
   \]
   Therefore, Tori answered $40$ questions correctly.

4. Determine how many more problems Tori needs to answer correctly to reach the passing grade of $45$:
   \[
   45 - 40 = 5
   \]

Conclusion:
Tori needs to answer $5$ more problems correctly to pass the test.

The final answer is $\boxed{5}$