30% of Gabriel's savings was $54 more than 40% of Luis's savings. After Gabriel spent ¹₅ of his savings and Luis spent 40% of his savings, Gabriel had $214 more than Luis. How much was Gabriel's saving at first?

	Luis	Gabriel
Before	15 u	20 u + 180
Change	- 6 u	- 4 u - 36
After	9 u	16 u + 144
Comparing Luis and Gabriel in the end		214 more

30% = ³⁰₁₀₀ = ³₁₀

40% = ⁴⁰₁₀₀ = ²₅

³₁₀ Gabriel = ²₅ Luis + 54 

Make the numerators the same. LCM of 3 and 2 is 6.
 
3x210x2 Gabriel = 2x35x3 Luis + 54 
620 Gabriel = 615 Luis + 54 

Simplify the fractions.
¹₂₀ Gabriel = ¹₁₅ Luis + 9

Let ¹₁₅ Luis be 1 u.

Luis's savings at first = 15 u

¹₂₀ Gabriel = ¹₁₅ Luis + 9
Let ¹₂₀ Gabriel be 1 u + 9.

Gabriel's savings at first
= 20 x (1 u + 9) 
= 20 u + 180

Savings that Luis spent
= 40% x 15 u
= ⁴⁰₁₀₀ x 15 u
= 6 u

Savings that Gabriel spent 
= ¹₅ x (20 u + 180)
= 4 u + 36 

Gabriel had $214 more than Luis in the end. If another $214 was given to Luis, the amount that each had in the end would be the same.

16 u + 144 = 9 u + 214 
16 u - 9 u = 214 - 144
7 u = 70

1 u = 70 ÷ 7 = 10

Savings that Gabriel had at first
= 20 u + 180
= 20 x 10 + 180
= 200 + 180
= $380

Answer(s): $380