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.

Question

There are 187 fruits in total. The ratio of apples to mangoes is \(2 : 3\), and the rest are oranges. After selling 67 apples, 26 mangoes, and half of the oranges, the ratio of the remaining apples to oranges becomes \(1 : 2\). Determine how many unsold fruits remain.

Answer 1

Let's solve this by taking the second son's age as the base variable and building every other age from it, instead of starting from the daughter as before.

We know the daughter's age directly: \(D = 5\) years.

The problem says the second son's age is two thirds of the first son's age, so if we call the second son's age \(x\), then:

\[ x = \frac{2}{3}S_1 \implies S_1 = \frac{3x}{2} \]

We are also told the first son and his sister are in the ratio 3:1, meaning \(S_1 = 3D = 15\). Since \(S_1 = \frac{3x}{2}\), we get:

\[ \frac{3x}{2} = 15 \implies x = 10 \]

So the second son's age is \(x = 10\) years, which fixes the first son at \(S_1 = 15\) years.

The father is four times the second son's age: \(F = 4x = 4 \times 10 = 40\) years.

The mother is 3.5 times the second son's age: \(M = 3.5x = 35\) years.

Adding every member's age, father (40) plus mother (35) plus first son (15) plus second son (10) plus daughter (5):

\[ 40+35+15+10+5 = 105 \]

This matches only option (2). The other totals, 115, 205 and 210, do not correspond to any consistent set of ages built from the given ratios, so they can be ruled out.

\[ \boxed{105 \text{ years}} \]

Answer 2