10% of Vincent's savings was $32 more than 40% of Luke's savings. After Vincent spent ¹₂ of his savings and Luke spent 40% of his savings, Vincent had $258 more than Luke. How much was Vincent's saving at first?

	Luke	Vincent
Before	5 u	20 u + 320
Change	- 2 u	- 10 u - 160
After	3 u	10 u + 160
Comparing Luke and Vincent in the end		258 more

10% = ¹⁰₁₀₀ = ¹₁₀

40% = ⁴⁰₁₀₀ = ²₅

¹₁₀ Vincent = ²₅ Luke + 32 

Make the numerators the same. LCM of 1 and 2 is 2.
 
1x210x2 Vincent = 2x15x1 Luke + 32 
220 Vincent = 25 Luke + 32 

Simplify the fractions.
¹₂₀ Vincent = ¹₅ Luke + 16

Let ¹₅ Luke be 1 u.

Luke's savings at first = 5 u

¹₂₀ Vincent = ¹₅ Luke + 16
Let ¹₂₀ Vincent be 1 u + 16.

Vincent's savings at first
= 20 x (1 u + 16) 
= 20 u + 320

Savings that Luke spent
= 40% x 5 u
= ⁴⁰₁₀₀ x 5 u
= 2 u

Savings that Vincent spent 
= ¹₂ x (20 u + 320)
= 10 u + 160 

Vincent had $258 more than Luke in the end. If another $258 was given to Luke, the amount that each had in the end would be the same.

10 u + 160 = 3 u + 258 
10 u - 3 u = 258 - 160
7 u = 98

1 u = 98 ÷ 7 = 14

Savings that Vincent had at first
= 20 u + 320
= 20 x 14 + 320
= 280 + 320
= $600

Answer(s): $600