Bobby had ³₄ as many buttons as Luis. Luis had ⁴₇ as many buttons as Mary. After Bobby and Mary gave 16 buttons to Luis, Luis had thrice as many buttons as Bobby. Mary then had the same number of buttons as Luis.

1. What was the ratio of the number of buttons Mary had to the number of buttons Bobby had in the beginning? Answer in simplest form.
2. How many buttons did Luis have in the end?

Bobby	Luis	Mary
3	4
	4	7
3	4	7

(a)
The number of buttons that Luis is repeated.  

Ratio of the number of buttons that Mary had to the number of buttons that Bobby had in the beginning = 7 : 3

	Bobby	Luis	Mary	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 16	+ 32	- 16
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

The total number of buttons at first and in the end remains unchanged. Make the total number of buttons the same. LCM of 14 and 7 is 14.

Number of buttons that Luis received from Bobby and Mary
= 2 x 16
= 32

Number of buttons that Luis received from Bobby and Mary
= 2 x 1 u
= 2 u

2 u = 16

1 u = 16 ÷ 2 = 8

Number of buttons that Ashley had in the end
= 6 u
= 6 x 8
= 48