10% of Henry's savings was $40 more than 40% of Liam's savings. After Henry spent ¹₂ of his savings and Liam spent 40% of his savings, Henry had $291 more than Liam. How much was Henry's saving at first?

	Liam	Henry
Before	5 u	20 u + 400
Change	- 2 u	- 10 u - 200
After	3 u	10 u + 200
Comparing Liam and Henry in the end		291 more

10% = ¹⁰₁₀₀ = ¹₁₀

40% = ⁴⁰₁₀₀ = ²₅

¹₁₀ Henry = ²₅ Liam + 40 

Make the numerators the same. LCM of 1 and 2 is 2.
 
1x210x2 Henry = 2x15x1 Liam + 40 
220 Henry = 25 Liam + 40 

Simplify the fractions.
¹₂₀ Henry = ¹₅ Liam + 20

Let ¹₅ Liam be 1 u.

Liam's savings at first = 5 u

¹₂₀ Henry = ¹₅ Liam + 20
Let ¹₂₀ Henry be 1 u + 20.

Henry's savings at first
= 20 x (1 u + 20) 
= 20 u + 400

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

Savings that Henry spent 
= ¹₂ x (20 u + 400)
= 10 u + 200 

Henry had $291 more than Liam in the end. If another $291 was given to Liam, the amount that each had in the end would be the same.

10 u + 200 = 3 u + 291 
10 u - 3 u = 291 - 200
7 u = 91

1 u = 91 ÷ 7 = 13

Savings that Henry had at first
= 20 u + 400
= 20 x 13 + 400
= 260 + 400
= $660

Answer(s): $660