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?

Answer 1

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 $\boxed{(3,\ 4)}$.