10% of Tim's savings was $42 more than 30% of Billy's savings. After Tim spent ¹₃ of his savings and Billy spent 40% of his savings, Billy had $448 less than Tim. How much was Billy's saving at first?

	Billy	Tim
Before	10 u	30 u + 420
Change	- 4 u	- 10 u - 140
After	6 u	20 u + 280
Comparing Billy and Tim in the end		448 more

10% = ¹⁰₁₀₀ = ¹₁₀

30% = ³⁰₁₀₀ = ³₁₀

¹₁₀ Tim = ³₁₀ Billy + 42 

Make the numerators the same. LCM of 1 and 3 is 3.
 
1x310x3 Tim = 3x110x1 Billy + 42 
330 Tim = 310 Billy + 42 

Simplify the fractions.
¹₃₀ Tim = ¹₁₀ Billy + 14

Let ¹₁₀ Billy be 1 u.

Billy's savings at first = 10 u

¹₃₀ Tim = ¹₁₀ Billy + 14
Let ¹₃₀ Tim be 1 u + 14.

Tim's savings at first
= 30 x (1 u + 14) 
= 30 u + 420

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

Savings that Tim spent 
= ¹₃ x (30 u + 420)
= 10 u + 140 

Billy had $448 less than Tim in the end. If another $448 was given to Billy, the amount that each had in the end would be the same.

20 u + 280 = 6 u + 448 
20 u - 6 u = 448 - 280
14 u = 168

1 u = 168 ÷ 14 = 12

Savings that Billy had at first
= 10 u 
= 10 x 12
= $120

Answer(s): $120