Christian had 25% more buttons than Lucas. Joshua had 50% fewer buttons than Christian. Christian and Lucas gave Joshua some buttons in the ratio of 3 : 4. As a result, Joshua had 60% more buttons than before. Given that Lucas had 48 less buttons than Joshua in the end, how many buttons did Lucas give to Joshua?

Christian	Lucas	Joshua
5x2	4x2
2x5		1x5
10	8	5

100% + 25% = 125%
125% = ¹²⁵₁₀₀ = ⁵₄

Christian : Lucas = 5 : 4

100% - 50% = 50%
50% = ⁵⁰₁₀₀ = ¹₂

Christian : Joshua = 2 : 1

The number of buttons that Christian had at first is repeated. Make the number of buttons that Christian had at first the same.  LCM of 5 and 2 is 10.

	Christian	Lucas	Joshua
Before	10x7 =  70 u	8x7 = 56 u	5x7 = 35 u
Change			+ 3x7 = + 21 u
Comparing the changes in the number of buttons	- 3x3 = - 9 u	- 4x3 = - 12 u	+ 7x3 = + 21 u
After	61 u	44 u	8x7 = 56 u

60% x 5 
= ⁶⁰₁₀₀ x 5 
= 3 

The total number of buttons that Christian and Lucas gave to Joshua is repeated.  Make the total number of buttons that Christian and Lucas gave to Joshua the same.  LCM of 7 and 3 is 21.

Number of buttons that Lucas had less than Joshua 
= 56 u - 44 u
= 12 u

12 u = 48

1 u = 48 ÷ 12 = 4

Number of buttons that Lucas gave to Joshua
= 12 u
= 12 x 4
= 48

Answer(s): 48