Sandra spent one-fifth pf her money and loaned one-third of the remainder to her brother Sam. She then had $24. How much money did she have at first?

m - m/5 - (4m/5)/3 = 24

m - m/5 - 4m/15 = 24
8m/15 = 24
m = 24(15/8) = 45

check:
started with 45
spent 9, leaving 36
lent 12, leaving 24

To solve this problem, we can work backward step by step.

Let's assume Sandra's initial amount of money is represented by 'x'.

According to the information given, Sandra spent one-fifth of her money, which is equivalent to (1/5)x. Now, she has (4/5)x remaining.

Next, she loaned one-third of the remainder to her brother Sam. This means she loaned (1/3)*(4/5)x, leaving her with (2/3)*(4/5)x.

Finally, we are told that she has $24 left. So, we have the equation:

(2/3)*(4/5)x = $24

To find the value of 'x', multiply the numerators and denominators:

(2/3)*(4/5) = (8/15)

Now, we can solve for 'x':

(8/15)x = $24

Multiply both sides of the equation by 15/8:

x = ($24)*(15/8)
x = $45

Therefore, Sandra had $45 initially.