20% of Ian's savings was $15 more than 30% of Rael's savings. After Ian spent ¹₃ of his savings and Rael spent 40% of his savings, Ian had $102 more than Rael. How much was Ian's saving at first?

	Rael	Ian
Before	10 u	15 u + 75
Change	- 4 u	- 5 u - 25
After	6 u	10 u + 50
Comparing Rael and Ian in the end		102 more

20% = ²⁰₁₀₀ = ¹₅

30% = ³⁰₁₀₀ = ³₁₀

¹₅ Ian = ³₁₀ Rael + 15 

Make the numerators the same. LCM of 1 and 3 is 3.
 
1x35x3 Ian = 3x110x1 Rael + 15 
315 Ian = 310 Rael + 15 

Simplify the fractions.
¹₁₅ Ian = ¹₁₀ Rael + 5

Let ¹₁₀ Rael be 1 u.

Rael's savings at first = 10 u

¹₁₅ Ian = ¹₁₀ Rael + 5
Let ¹₁₅ Ian be 1 u + 5.

Ian's savings at first
= 15 x (1 u + 5) 
= 15 u + 75

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

Savings that Ian spent 
= ¹₃ x (15 u + 75)
= 5 u + 25 

Ian had $102 more than Rael in the end. If another $102 was given to Rael, the amount that each had in the end would be the same.

10 u + 50 = 6 u + 102 
10 u - 6 u = 102 - 50
4 u = 52

1 u = 52 ÷ 4 = 13

Savings that Ian had at first
= 15 u + 75
= 15 x 13 + 75
= 195 + 75
= $270

Answer(s): $270