10% of Elijah's savings was $54 more than 30% of Sean's savings. After Elijah spent ¹₃ of his savings and Sean spent 40% of his savings, Elijah had $584 more than Sean. How much was Elijah's saving at first?

	Sean	Elijah
Before	10 u	30 u + 540
Change	- 4 u	- 10 u - 180
After	6 u	20 u + 360
Comparing Sean and Elijah in the end		584 more

10% = ¹⁰₁₀₀ = ¹₁₀

30% = ³⁰₁₀₀ = ³₁₀

¹₁₀ Elijah = ³₁₀ Sean + 54 

Make the numerators the same. LCM of 1 and 3 is 3.
 
1x310x3 Elijah = 3x110x1 Sean + 54 
330 Elijah = 310 Sean + 54 

Simplify the fractions.
¹₃₀ Elijah = ¹₁₀ Sean + 18

Let ¹₁₀ Sean be 1 u.

Sean's savings at first = 10 u

¹₃₀ Elijah = ¹₁₀ Sean + 18
Let ¹₃₀ Elijah be 1 u + 18.

Elijah's savings at first
= 30 x (1 u + 18) 
= 30 u + 540

Savings that Sean spent
= 40% x 10 u
= ⁴⁰₁₀₀ x 10 u
= 4 u

Savings that Elijah spent 
= ¹₃ x (30 u + 540)
= 10 u + 180 

Elijah had $584 more than Sean in the end. If another $584 was given to Sean, the amount that each had in the end would be the same.

20 u + 360 = 6 u + 584 
20 u - 6 u = 584 - 360
14 u = 224

1 u = 224 ÷ 14 = 16

Savings that Elijah had at first
= 30 u + 540
= 30 x 16 + 540
= 480 + 540
= $1020

Answer(s): $1020