1. **State the problem:** We have a football team with ages and frequencies given. We need to find: a) The mean age of the team. b) The age of a new player who raises the mean age to 22. 2. **Calculate the mean age of the original team:** The mean is given by the formula: $$\text{Mean} = \frac{\sum (\text{age} \times \text{frequency})}{\sum \text{frequency}}$$ Calculate the total frequency: $$5 + 1 + 3 + 0 + 3 = 12$$ Calculate the sum of age times frequency: $$19 \times 5 + 20 \times 1 + 21 \times 3 + 22 \times 0 + 23 \times 3 = 95 + 20 + 63 + 0 + 69 = 247$$ So the mean age is: $$\frac{247}{12}$$ 3. **Simplify the fraction:** $$\frac{\cancel{247}}{\cancel{12}}$$ (No common factors to cancel) Calculate the decimal value: $$\frac{247}{12} \approx 20.5833$$ Rounded to 1 decimal place: $$20.6$$ 4. **Find the age of the new player:** Let the new player's age be $x$. The new mean is 22 with one more player, so total players = 13. The total sum of ages now is: $$247 + x$$ Using the mean formula: $$22 = \frac{247 + x}{13}$$ Multiply both sides by 13: $$22 \times 13 = 247 + x$$ $$286 = 247 + x$$ Subtract 247 from both sides: $$286 - 247 = x$$ $$39 = x$$ **Final answers:** - a) Mean age = $20.6$ - b) Age of new player = $39$