Sharing a cake unevenly: In a family, the father took 1/4 of a cake and this amount was exactly three times as much as each of the other family members received (all others got equal shares). How many family members are there in total?

Introduction / Context:
This is a classic partition problem. One person takes a fixed fraction, and that share is a simple multiple of everyone else’s equal share. We build an equation for the total and solve for the number of people.

Given Data / Assumptions:

• Father’s share = 1/4 of the cake.
• Each other member’s share = x (equal for all others).
• Father’s share is 3 times any other member’s share: 1/4 = 3x.
• Let total members be n (1 father + n − 1 others).

Concept / Approach:
From 1/4 = 3x, find x. Then write the total cake as father’s share plus the sum of the others’ shares. Solve for n using a simple linear equation.

Step-by-Step Solution:
From 1/4 = 3x ⇒ x = 1/12.Total cake = father + others = 1/4 + (n − 1)*(1/12).Set sum to 1: 1/4 + (n − 1)/12 = 1.Multiply by 12: 3 + (n − 1) = 12 ⇒ n − 1 = 9 ⇒ n = 10.

Verification / Alternative check:
With 10 members: father has 1/4; each of the 9 others has 1/12; total = 3/12 + 9/12 = 12/12 = 1, consistent.

Why Other Options Are Wrong:
3, 7, 9, and 12 do not satisfy both the equality of the others’ shares and the 3-times relationship with a 1/4 father’s share.

Common Pitfalls:
Assuming the father’s share is three times the sum of the others (it is three times each other’s share); miscounting the number of “other” members.

Final Answer:
10