Robin has a table full of supplies that she will use to decorate picture frames. She decorated one frame with 3 buttons and 12 bows. On her other frames, she also wants to use 2 types of items on each picture frame, but only if the 2 items have the same unit rate as the items on the first frame. Which of the pairs of items could Robin use for the other frames?
To find the unit rate for the buttons and bows, we need to divide the number of each item by the number of frames. If we let $b$ represent the number of buttons used per frame and $w$ represent the number of bows used per frame, then the unit rates for buttons and bows are $\frac{3}{1} = 3$ buttons per frame and $\frac{12}{1}= 12$ bows per frame. So the unit rates for buttons and bows are 3 and 12, respectively. We need to find two other numbers $x$ and $y$ such that the unit rates 3 and 12 are preserved. In other words, $b=3x$ and $w=12x$ for some value of $x$. Since $b$ and $w$ must be positive integers, we can see that x must be a common divisor of 3 and 12. The common divisors of 3 and 12 are 1, 3, 12, so the possible pairs $x$ and $y$ are $(x,y)=(1,12)$ or $(x,y)=(3,4)$. This means that Robin could use either 1 button and 12 bows or 3 buttons and 4 bows to decorate the other frames. The other pairs listed would not have the same unit rate as the first frame. So, the answer is $\boxed{(1,\ 12)}$ or $\boxed{(3,\ 4)}$.
