Three boys and three girls all celebrate their birthday today, but they are each different ages. The youngest is 1 years old. The sum of the ages of the three girls is the same as the sum of the ages of the three boys. What is the smallest possible total of all six ages?
To find the smallest possible total of all six ages, we need to consider the ages of the boys and girls and their respective sums.
Let's assume that the ages of the three girls are x, y, and z, with x being the youngest. Then, the sum of their ages is x + y + z.
Since the youngest child is 1 year old, the possible values for x are 1, 2, or 3.
Let's consider the ages of the boys. Since the sum of the ages of the girls is the same as the sum of the ages of the boys, the sum of their ages is also x + y + z.
Assuming that the ages of the boys are a, b, and c, with a being the youngest, we have a + b + c = x + y + z.
Since the girls' ages cannot be the same as the boys' ages, we can assume that a > x, b > y, and c > z.
Given these conditions, we can determine the smallest possible total of all six ages by finding the values that minimize the sum x + y + z.
Let's examine the possible combinations for x = 1, 2, and 3:
If x = 1, the sum of the girls' ages is x + y + z = 1 + y + z. To minimize this sum, we set y = 2 and z = 3. The sum of the girls' ages is then 1 + 2 + 3 = 6. For the boys' ages, we can set a = 4, b = 5, and c = 6. The sum of the boys' ages is then 4 + 5 + 6 = 15.
If x = 2, the sum of the girls' ages is x + y + z = 2 + y + z. Setting y = 1 and z = 3 minimizes the sum, resulting in 2 + 1 + 3 = 6 for the girls' ages. For the boys, we can set a = 4, b = 5, and c = 6 again, resulting in a sum of 4 + 5 + 6 = 15.
If x = 3, the sum of the girls' ages is x + y + z = 3 + y + z. Setting y = 1 and z = 2 minimizes the sum, resulting in 3 + 1 + 2 = 6 for the girls' ages. For the boys, we can set a = 4, b = 5, and c = 6 once more, resulting in a sum of 4 + 5 + 6 = 15.
We can see that regardless of the values of x, the sum of the girls' ages is always 6, and the sum of the boys' ages is always 15.
Therefore, the smallest possible total of all six ages is 6 + 15 = 21.