The average age of 3 children in a family is 20% of the average age of the father and the eldest child. The total age of the mother and the youngest child is 39 years. If the father’s age is 26 years, what is the age of the second child?

### Concept & Strategy To find a specific unknown variable in a system of linear equations, the number of independent equations must be at least equal to the number of unknown variables. --- ### Step-by-Step Solution * **Given Variables:** Let the three children be $C_1$ (eldest), $C_2$ (second), and $C_3$ (youngest). Let the Father be $F$ and Mother be $M$. * **Given Constraints:** 1. Average of 3 children = $20\%$ of average of father and eldest child: $$\frac{C_1 + C_2 + C_3}{3} = 0.20 \times \left(\frac{F + C_1}{2}\right)$$ 2. Total age of mother and youngest child: $$M + C_3 = 39$$ 3. Father's age: $$F = 26$$ * **Deduction:** * We have 5 variables: $C_1, C_2, C_3, F, M$. * We are provided with only 3 independent pieces of information/equations. * Substituting $F = 26$ into the first equation yields: $$\frac{C_1 + C_2 + C_3}{3} = 0.10 \times (26 + C_1)$$ $$C_1 + C_2 + C_3 = 0.3(26 + C_1)$$ $$0.7C_1 + C_2 + C_3 = 7.8$$ * We still have one equation containing $M$ and $C_3$, and another containing $C_1, C_2, C_3$. There is insufficient data to isolate $C_2$. --- ### Exam Strategy & Shortcut Scan the variables versus the equations provided. You have 5 unknown individuals ($F, M, C_1, C_2, C_3$) and only 3 mathematical conditions. Since no other relationships link the middle child uniquely to the other known values, the system is underdetermined. Mark **Cannot be determined** immediately without doing algebraic expansions. --- ### Common Pitfall Do not waste time trying to solve the algebraic equations by guessing values. If the number of variables significantly exceeds the number of independent equations, look out for the "Cannot be determined" option. --- ### Final Answer **Therefore, the correct answer is Cannot be determined.**