1. Stating the problem: Given a frequency distribution with class intervals and frequencies \(7, p, 10, 9, 13\), find \(p\) if the mean is 54. 2. Assign midpoints to each class interval: \[\text{Midpoints} = 10, 30, 50, 70, 90\] 3. Write the mean formula for grouped data: \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\] where \(f_i\) are frequencies and \(x_i\) are midpoints. 4. Plug given values and unknown \(p\): \[54 = \frac{7 \times 10 + p \times 30 + 10 \times 50 + 9 \times 70 + 13 \times 90}{7 + p + 10 + 9 + 13}\] 5. Calculate sums: Numerator: \[7 \times 10 = 70\] \[p \times 30 = 30p\] \[10 \times 50 = 500\] \[9 \times 70 = 630\] \[13 \times 90 = 1170\] Sum = \(70 + 30p + 500 + 630 + 1170 = 2370 + 30p\) Denominator: \[7 + p + 10 + 9 + 13 = 39 + p\] 6. Write the equation: \[54 = \frac{2370 + 30p}{39 + p}\] 7. Multiply both sides by \(39 + p\): \[54 (39 + p) = 2370 + 30p\] \[2106 + 54p = 2370 + 30p\] 8. Rearrange terms: \[54p - 30p = 2370 - 2106\] \[24p = 264\] 9. Solve for \(p\): \[p = \frac{264}{24} = 11\] Final answer: \(p = 11\)