10% of Reggie's savings was $34 more than 40% of Gabriel's savings. After Reggie spent ⁷₁₀ of his savings and Gabriel spent 40% of his savings, Gabriel had $111 less than Reggie. How much was Gabriel's saving at first?

	Gabriel	Reggie
Before	5 u	20 u + 340
Change	- 2 u	- 14 u - 238
After	3 u	6 u + 102
Comparing Gabriel and Reggie in the end		111 more

10% = ¹⁰₁₀₀ = ¹₁₀

40% = ⁴⁰₁₀₀ = ²₅

¹₁₀ Reggie = ²₅ Gabriel + 34 

Make the numerators the same. LCM of 1 and 2 is 2.
 
1x210x2 Reggie = 2x15x1 Gabriel + 34 
220 Reggie = 25 Gabriel + 34 

Simplify the fractions.
¹₂₀ Reggie = ¹₅ Gabriel + 17

Let ¹₅ Gabriel be 1 u.

Gabriel's savings at first = 5 u

¹₂₀ Reggie = ¹₅ Gabriel + 17
Let ¹₂₀ Reggie be 1 u + 17.

Reggie's savings at first
= 20 x (1 u + 17) 
= 20 u + 340

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

Savings that Reggie spent 
= ⁷₁₀ x (20 u + 340)
= 14 u + 238 

Gabriel had $111 less than Reggie in the end. If another $111 was given to Gabriel, the amount that each had in the end would be the same.

6 u + 102 = 3 u + 111 
6 u - 3 u = 111 - 102
3 u = 9

1 u = 9 ÷ 3 = 3

Savings that Gabriel had at first
= 5 u 
= 5 x 3
= $15

Answer(s): $15