Final answer:
There are a total of 53 positive integer pairs (x, y) that satisfy the equation 4x+12y=640. The equation simplifies to x+3y=160, and by determining the bounds of y, it is found that y can range from 1 to 53, inclusive.
Step-by-step explanation:
To find how many positive integer pairs (x,y) satisfy the equation 4x+12y=640, we can start by simplifying the equation. Dividing both sides by 4 gives us x + 3y = 160. Now, we can express x in terms of y using the equation x = 160 - 3y.
As we're looking for positive integer solutions, x must be positive. This means that x > 0 and since x is an integer, the smallest value x could take is 1. Plugging x = 1 into our equation gives 1 + 3y = 160 which simplifies to 3y = 159 and hence y = 53. So the largest value for y when x is at its smallest is 53.
If we set x as the largest possible integer such that y remains a positive integer, this would happen when y = 1. Substituting y = 1 into the equation gives x + 3(1) = 160 which simplifies to x = 157. Thus the range of possible y values when x is within the bounds of being a positive integer is from 1 to 53, inclusive.
To determine the exact number of solutions, we count the integer values of y, since for each integer value of y there is a corresponding positive integer value of x. This gives us a total of 53 possible solutions.