Rani had 98 buttons more than Lyvia. After Rani gave Lisa 13 buttons, Rani had three times as many buttons as Lyvia. How many buttons did Rani have at first?

Let's start by defining the number of buttons Lyvia had as \( L \). According to the problem, Rani had \( L + 98 \) buttons initially. After Rani gave Lisa 13 buttons, Rani's buttons became \( (L + 98 - 13) \) or \( L + 85 \). The problem states that now Rani has three times as many buttons as Lyvia, which gives us the equation: \[ L + 85 = 3L \] Now, let's solve for \( L \): 1. Rearranging the equation, we get: \[ L + 85 = 3L \] \[ 85 = 3L - L \] \[ 85 = 2L \] \[ L = \frac{85}{2} = 42.5 \] Wait a second! This answer doesn't make sense in the context of counting buttons since we can't have half a button. Let's look again at the steps, as there might be a calculation error, or possibly, the problem needs integers. Let’s analyze the original equations again: 1. After giving away buttons, we know: \[ Rani's\ remaining\ buttons = L + 85 \] \[ Rani's\ buttons\ should\ satisfy: L + 85 = 3L \] This rearranges correctly to: \[ 85 = 2L \] \[ 2L = 85 \] But then for counting, it implies Lyvia doesn't start with buttons consistent with integers and requires reviewing or re-checking the conditions. Assuming Lyvia should have integer buttons. This suggests isolating correctly the first condition as potentially each being zero starting from how precisely they count buttons. In conclusion, through checking equations precisely and engaging with integers for button counting, adjusting assumptions could yield improved resolution based on potential context.