The average age of husband, wife and their child $3$ years ago was $27$ years and that of wife and the child $5$ years ago was $20$ years. The present age of the husband is

### Concept & Strategy Break the problem down by bringing different historical data points to the present timeline separately. Once all sums are in the present tense, finding the missing individual is simple subtraction. ### Step-by-Step Solution * **Given:** * Average age of Husband ($H$), Wife ($W$), and Child ($C$) $3$ years ago = $27$ years * Average age of Wife ($W$) and Child ($C$) $5$ years ago = $20$ years * **Calculation:** **Step 1: Process the 3-member group** Sum of ages of $H, W,$ and $C$ ($3$ members) $3$ years ago: $$ 27 \times 3 = 81 \text{ years} $$ Present sum of $H, W, C$ (add $3$ years for each of the $3$ members): $$ \text{Present Sum } (H+W+C) = 81 + (3 \times 3) = 81 + 9 = 90 \text{ years} $$ **Step 2: Process the 2-member group** Sum of ages of $W$ and $C$ ($2$ members) $5$ years ago: $$ 20 \times 2 = 40 \text{ years} $$ Present sum of $W$ and $C$ (add $5$ years for each of the $2$ members): $$ \text{Present Sum } (W+C) = 40 + (2 \times 5) = 40 + 10 = 50 \text{ years} $$ **Step 3: Isolate the Husband's age** $$ H = (H + W + C) - (W + C) $$ $$ H = 90 - 50 = 40 \text{ years} $$ ### Exam Strategy & Shortcut Operate entirely on the present averages. Present average of $(H+W+C) = 27 + 3 = 30$. Total = $30 \times 3 = 90$. Present average of $(W+C) = 20 + 5 = 25$. Total = $25 \times 2 = 50$. Husband's age = $90 - 50 = 40$. This mental math method takes less than 15 seconds. ### Common Pitfall Mixing up the timeframes when adding the elapsed years. Students sometimes add $3 \times 2 = 6$ years to the second group instead of $5 \times 2 = 10$, misreading the conditions. Always pair the correct historical timeframe with the correct group. ### Final Answer **Therefore, the correct answer is 40 years.**