Final answer:
The rational number is 13.
Step-by-step explanation:
We are given that when we multiply a rational number by 52 and add 23 to the product, we get 712.
Let's assume the rational number is x.
Therefore, we can write the equation as 52x + 23 = 712.
To solve for x, we need to isolate x on one side of the equation.
Subtracting 23 from both sides, we get 52x = 689.
Finally, dividing both sides by 52, we find that x = 13.
Therefore, the rational number is 13.