10% of Ian's savings was $51 more than 30% of Riordan's savings. After Ian spent ¹₁₀ of his savings and Riordan spent 40% of his savings, Ian had $522 more than Riordan. How much was Ian's saving at first?

	Riordan	Ian
Before	10 u	30 u + 510
Change	- 4 u	- 3 u - 51
After	6 u	27 u + 459
Comparing Riordan and Ian in the end		522 more

10% = ¹⁰₁₀₀ = ¹₁₀

30% = ³⁰₁₀₀ = ³₁₀

¹₁₀ Ian = ³₁₀ Riordan + 51 

Make the numerators the same. LCM of 1 and 3 is 3.
 
1x310x3 Ian = 3x110x1 Riordan + 51 
330 Ian = 310 Riordan + 51 

Simplify the fractions.
¹₃₀ Ian = ¹₁₀ Riordan + 17

Let ¹₁₀ Riordan be 1 u.

Riordan's savings at first = 10 u

¹₃₀ Ian = ¹₁₀ Riordan + 17
Let ¹₃₀ Ian be 1 u + 17.

Ian's savings at first
= 30 x (1 u + 17) 
= 30 u + 510

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

Savings that Ian spent 
= ¹₁₀ x (30 u + 510)
= 3 u + 51 

Ian had $522 more than Riordan in the end. If another $522 was given to Riordan, the amount that each had in the end would be the same.

27 u + 459 = 6 u + 522 
27 u - 6 u = 522 - 459
21 u = 63

1 u = 63 ÷ 21 = 3

Savings that Ian had at first
= 30 u + 510
= 30 x 3 + 510
= 90 + 510
= $600

Answer(s): $600