In a family, each daughter has the same number of brothers as she has sisters, and each son has twice as many sisters as he has brothers. Based on this information, how many sons are there in the family?

Introduction / Context:
This classic family puzzle is an arithmetic reasoning and basic algebra question. It describes relationships between numbers of sons and daughters using counts of brothers and sisters from each child’s perspective. By carefully translating the statements into equations and solving them, we can determine the number of sons in the family.

Given Data / Assumptions:

• Let the family have s sons and d daughters.
• Each daughter has the same number of brothers as sisters.
• Each son has twice as many sisters as he has brothers.
• We assume that all children are either sons or daughters and that there is at least one of each.

Concept / Approach:
From the point of view of any daughter, the number of brothers is s and the number of sisters is d - 1, because she does not count herself. The statement that these two numbers are equal leads to the equation s = d - 1. From the point of view of any son, the number of brothers is s - 1 and the number of sisters is d. The statement that each son has twice as many sisters as brothers gives d = 2(s - 1). Solving this small system of two equations in two unknowns yields the number of sons and daughters.

Step-by-Step Solution:
Step 1: From each daughter: number of brothers = s, number of sisters = d - 1, and these are equal. So s = d - 1. Step 2: Rearrange s = d - 1 to get d = s + 1. Step 3: From each son: number of brothers = s - 1, number of sisters = d, and each son has twice as many sisters as brothers. So d = 2(s - 1). Step 4: Substitute d = s + 1 into d = 2(s - 1): s + 1 = 2(s - 1). Step 5: Expand the right side: s + 1 = 2s - 2. Step 6: Rearrange: 1 + 2 = 2s - s, so 3 = s. Step 7: Therefore, the number of sons s is 3. The number of daughters d = s + 1 = 4, although the question asks only for the number of sons.

Verification / Alternative check:
Check from a daughter’s perspective: with s = 3 and d = 4, each daughter has 3 brothers and 4 - 1 = 3 sisters, so that condition holds. From a son’s perspective, each son has 3 - 1 = 2 brothers and 4 sisters, and 4 is twice 2, so that condition also holds. Both statements are satisfied, confirming that 3 sons is correct.

Why Other Options Are Wrong:
If s = 1 or s = 2, the equations s = d - 1 and d = 2(s - 1) do not yield consistent integer values for d that satisfy both conditions simultaneously. For s = 4, you get d = 5 from the first equation but d = 6 from the second, which is contradictory. Thus options A, B, and D lead to inconsistent family structures.

Common Pitfalls:
A frequent error is to count the child himself or herself when listing brothers or sisters, which changes the counts by one and leads to incorrect equations. Another mistake is to assume that the number of sons and daughters must be the same. Writing the relationships algebraically avoids both issues and makes the solution straightforward.

Final Answer:
The family has 3 sons.