### Instruction: What is the reciprocal of the sum of a half, a third, and a fourth?### Response: The first thing I notice is that the problem is asking for the reciprocal of a sum, so I want to find the sum first and then flip it over to get the reciprocal.
To find the sum of a half, a third, and a fourth, I need to have a common denominator for all the fractions.
The least common multiple of 2, 3, and 4 is 12, so I will multiply each fraction by a factor that gives them a denominator of 12.
For a half, I need to multiply by 6/6, so I get 6/12.
For a third, I need to multiply by 4/4, so I get 4/12.
For a fourth, I need to multiply by 3/3, so I get 3/12.
Now I can add the fractions with the same denominator: 6/12 + 4/12 + 3/12 = 13/12.
This is the sum I was looking for, and now I can take the reciprocal by flipping it over: 12/13.
This is the final answer.