Dan spent 1/5 of his money and an additional $4 on a CD. He then spent 1/3 of his
remaining money and an additional $8 on some comics. Given that he was left with $24
how much money did he have at first?
Had d
after Cd he had d - d/5 - 4 = 4d/5-4
after comics he had (4d/5-4) - (4d/5-4)/3 - 8
(4d/5-4) - (4d/5-4)/3 - 8 =24
(4d/5-4) - (4d/5-4)/3 =32
3(4d/5-4) - (4d/5-4) = 96
2 d/5 - 8 = 96
2 d/5 = 104
d/5 = 52
d = 260
start with x
spends x/5 + 4
amount left = x - (x/5 + 4) = 4x/5 - 4
spends 1/3 of that +8
= (1/3)(4x/5 - 4) + 8
= 4x/15 - 4/3 + 8
= 4x/15 + 20/3
4x/5 - 4 - (4x/15 + 20/3) = 24
times 15
12x - 60 - 4x - 100 = 360
8x = 520
x = 65
check:
amount spent at first = (1/5)(65) + 4 = 17
amount left = 48
spends 1/3 of that +8 = 24
amount left = 48-24 = 24
To solve this problem, we can break it down into steps and use equations for each step.
Let's suppose Dan's initial amount of money is represented by x.
Step 1: Dan spent 1/5 of his money and an additional $4 on a CD.
After this step, the amount of money Dan had remaining is (1 - 1/5) * x - 4.
Step 2: Dan spent 1/3 of his remaining money and an additional $8 on comics.
After this step, the amount of money Dan had remaining is (1 - 1/3) * ((1 - 1/5) * x - 4) - 8.
Given that Dan was left with $24, we can set up the equation:
(1 - 1/3) * ((1 - 1/5) * x - 4) - 8 = 24.
Now, let's solve this equation step by step:
Step 1: Simplify the equation within the innermost brackets.
(1 - 1/5) * x - 4 = (4/5) * x - 4.
Step 2: Multiply the simplified equation by (1 - 1/3).
(1 - 1/3) * ((4/5) * x - 4) = (2/3) * ((4/5) * x - 4).
Step 3: Simplify the equation within the outermost brackets.
(2/3) * ((4/5) * x - 4) = (8/15) * x - (8/3).
Step 4: Subtract 8 from both sides of the equation.
(8/15) * x - (8/3) - 8 = 24 - 8.
Step 5: Simplify both sides of the equation.
(8/15) * x - 8/3 - 8 = 16.
Step 6: Combine like terms.
(8/15) * x - 32/3 = 16.
Step 7: Add (32/3) to both sides of the equation.
(8/15) * x - 32/3 + 32/3 = 16 + 32/3.
Step 8: Simplify both sides of the equation.
(8/15) * x = 16 + (32/3).
Step 9: Combine like terms.
(8/15) * x = (48/3) + (32/3).
Step 10: Simplify the right side of the equation.
(8/15) * x = 80/3.
Step 11: Multiply both sides of the equation by (15/8) to solve for x.
x = (80/3) * (15/8).
Step 12: Simplify the right side of the equation.
x = 200/1.
Finally, we find that x = $200.
Therefore, Dan had $200 initially.