Fathers Calculation: Birthday Party Chocolates

This problem requires setting up and solving a system of equations based on the given constraints about people and chocolates.

Defining Variables and Equations

Let $C$ represent the number of children, $M$ the number of mothers, and $F$ the number of fathers.

• Total Persons: The total number of people is 69.
  $C + M + F = 69$
• Total Chocolates: Each child gets 2 chocolates, each mother gets 1, and fathers get none. Total chocolates are 70.
  $2C + M = 70$
• Children-Parents Relation: The number of children equals half the sum of mothers and fathers.
  $C = \frac{1}{2}(M + F)$
  This implies $2C = M + F$.

Step-by-Step Solution

1. Find the number of children ($C$). Use the equation $2C = M + F$ and substitute it into the total persons equation $C + M + F = 69$.

   Replace $M + F$ with $2C$:
   $C + (2C) = 69$

   Combine terms and solve for $C$:
   $3C = 69$
   $C = \frac{69}{3}$
   $C = 23$

   There are 23 children.

2. Find the number of mothers ($M$). Use the total chocolates equation $2C + M = 70$ and the value $C = 23$.

   Substitute $C=23$:
   $2(23) + M = 70$
   $46 + M = 70$

   Solve for $M$:
   $M = 70 - 46$
   $M = 24$

   There are 24 mothers.

3. Find the number of fathers ($F$). Use the total persons equation $C + M + F = 69$ and the values $C = 23$ and $M = 24$.

   Substitute $C=23$ and $M=24$:
   $23 + 24 + F = 69$
   $47 + F = 69$

   Solve for $F$:
   $F = 69 - 47$
   $F = 22$

   There are 22 fathers.

Verification

Confirm the solution satisfies all conditions:

• Total Persons: $23 + 24 + 22 = 69$ (Correct)
• Total Chocolates: $2(23) + 1(24) = 46 + 24 = 70$ (Correct)
• Children relation: $23 = \frac{1}{2}(24 + 22) = \frac{1}{2}(46)$ (Correct)

The number of fathers is 22.