Final answer:
To solve the problem, set up equations based on the given information. The smallest number can be represented as x. Write equations for the relationships between the numbers and solve them simultaneously to find the value of x.
Step-by-step explanation:
To solve the problem, we need to set up equations based on the given information. Let the smallest number be x, the middle number be y, and the largest number be z. We are told that the largest number is 12 less than 5 times the smallest number, so we can write the equation: z = 5x - 12. We are also told that one less than one-half the largest number is the middle number, so we can write the equation: y = 1/2z - 1. Finally, we are told that the sum of the three numbers is 32, so we can write the equation: x + y + z = 32.
Now, we can solve these equations simultaneously to find the value of x. Substituting the first equation into the second equation, we get: y = 1/2(5x - 12) - 1. Simplifying this equation, we get: y = 5/2x - 7. Substituting the values of x and y into the third equation, we get: x + (5/2x - 7) + (5x - 12) = 32. Simplifying this equation, we get: 11.5x - 19 = 32. Adding 19 to both sides and dividing by 11.5, we find that x = 3.
Therefore, the smallest number is 3.