1. **Problem statement:** Find the missing frequency in the data set where the values of $x$ are 2, 6, 7, 8, 9 and their corresponding frequencies $f$ are 4, 6, 12, $x$, and 8 respectively, given that the arithmetic mean (A M) is 7.3. 2. **Formula for arithmetic mean:** $$\text{A M} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ are frequencies and $x_i$ are the values. 3. **Calculate known sums:** Sum of known frequencies: $4 + 6 + 12 + 8 = 30$ (excluding the missing frequency $x$) Sum of known frequency times value: $$4 \times 2 + 6 \times 6 + 12 \times 7 + 8 \times 9 = 8 + 36 + 84 + 72 = 200$$ 4. **Set up equation with missing frequency $x$:** Total frequency: $30 + x$ Total sum of $f_i x_i$: $200 + 8x$ Given arithmetic mean: $$7.3 = \frac{200 + 8x}{30 + x}$$ 5. **Solve for $x$:** Multiply both sides by $(30 + x)$: $$7.3(30 + x) = 200 + 8x$$ Expand: $$219 + 7.3x = 200 + 8x$$ Bring terms involving $x$ to one side: $$219 - 200 = 8x - 7.3x$$ Simplify: $$19 = 0.7x$$ Divide both sides by 0.7: $$x = \frac{19}{0.7} = 27.14$$ 6. **Interpretation:** The missing frequency is approximately $27.14$. Since frequency must be a whole number, it is likely $27$. **Final answer:** The missing frequency is $27$.