Isabella had 25% more marbles than Matthew. Ryan had 50% fewer marbles than Isabella. Isabella and Matthew gave Ryan some marbles in the ratio of 4 : 5. As a result, Ryan had 60% more marbles than before. Given that Matthew had 45 less marbles than Ryan in the end, how many marbles did Matthew give to Ryan?

Isabella	Matthew	Ryan
5x2	4x2
2x5		1x5
10	8	5

100% + 25% = 125%
125% = ¹²⁵₁₀₀ = ⁵₄

Isabella : Matthew = 5 : 4

100% - 50% = 50%
50% = ⁵⁰₁₀₀ = ¹₂

Isabella : Ryan = 2 : 1

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

	Isabella	Matthew	Ryan
Before	10x3 =  30 u	8x3 = 24 u	5x3 = 15 u
Change			+ 3x3 = + 9 u
Comparing the changes in the number of marbles	- 4x1 = - 4 u	- 5x1 = - 5 u	+ 9x1 = + 9 u
After	26 u	19 u	8x3 = 24 u

60% x 5 
= ⁶⁰₁₀₀ x 5 
= 3 

The total number of marbles that Isabella and Matthew gave to Ryan is repeated.  Make the total number of marbles that Isabella and Matthew gave to Ryan the same.  LCM of 9 and 3 is 9.

Number of marbles that Matthew had less than Ryan 
= 24 u - 19 u
= 5 u

5 u = 45

1 u = 45 ÷ 5 = 9

Number of marbles that Matthew gave to Ryan
= 5 u
= 5 x 9
= 45

Answer(s): 45