Question

A family consists of the father, the mother, two sons and the youngest daughter. The father's age is four times the age of the second son. The first son's age is in the ratio \(3:1\) with that of his sister. The mother is \(3.5\) times as old as the second son. The second son's age is \(\frac{2}{3}\) of the first son's age. The youngest daughter is 5 years old. Find the sum of all their ages.

• A. 115 yrs
• B. 105 yrs
• C. 205 yrs
• D. 210 yrs

Hint:

Start from the daughter's age (5 years), use the 3:1 ratio to find the first son, then work out the second son, father and mother one by one before adding all five ages.

Answer 1

The Correct Option is B

Step 1: Name the ages and note what is given.
Let the father's age be \(F\), the mother's age be \(M\), the first son's age be \(S_1\), the second son's age be \(S_2\), and the daughter's age be \(D\).
We are told \(D = 5\) years.

Step 2: Use the ratio between the first son and his sister.
The first son's age is in the ratio 3:1 with his sister's age, so
\[ S_1 : D = 3 : 1 \implies S_1 = 3D = 3\times 5 = 15 \]

Step 3: Find the second son's age.
The second son's age is \(\frac{2}{3}\) of the first son's age:
\[ S_2 = \frac{2}{3}S_1 = \frac{2}{3}\times 15 = 10 \]

Step 4: Find the father's age.
The father's age is four times the second son's age:
\[ F = 4\times S_2 = 4\times 10 = 40 \]

Step 5: Find the mother's age.
The mother is 3.5 times as old as the second son:
\[ M = 3.5\times S_2 = 3.5\times 10 = 35 \]

Step 6: Add up all five ages.
\[ F+M+S_1+S_2+D = 40+35+15+10+5 = 105 \]

Step 7: Check the other options.
Options (1), (3) and (4), that is 115, 205 and 210, do not match when we add the five ages we derived directly from the given ratios.
Only summing the five correctly derived ages gives 105 years.

Final Answer:
The sum of the ages of all five family members is 105 years. \[ \boxed{105 \text{ years}} \]