Non-algebra way :
$2 × 5 = $10
$5 × 2 = $10
Since they have the same amount of money at first, the ratio of notes Athena has to Frances is 5 : 2
This simplest ratio does not change no matter how many notes they actually have at first. It remains.
Next, Frances spent $60.
$60 ÷ $5 = 12
So, Frances used 12 pieces of $5 notes.
Since Athena did not spend, her number of notes remainded unchanged.
The ratio after the spending is 25 : 6. So change the ratio 5 : 2 such that Athena's units is also 25.
5 : 2 = 25 : 10 (after multiplying by 5 throughout)
Change in Frances' notes = 10 units - 6 units = 4 units
So Frances used 4 units. Which means 4 units = 12 pieces
1 unit = 12 pieces ÷ 4 = 3 pieces
Number of notes Athena has = 25 × 3 = 75
$ that she has = 75 × $2 = $150