Olivia bought some school supplies. In the first store she spent half the money she had plus $6. In the second store she spent one third of what was left plus $6.In the last store she spent ¼ of what was left and came home with $6.How much did she start out with?
$X Initially.
1. X - (x/2+6) = x/2 - 6.
2. (x/2-6)-1/3(x/2-6)-6 = 2/3(x/2-6)-6 =2x/6-4-6 = x/3-10.
3. (x/3-10) - 1/4(x/3-10) = 3/4(x/3-10)= x/4-30/4 = (x-30)/4 = Bal. = $6.
(x-30)/4 = 6.
x-30 = 24.
X = 24 + 30 = $54. = The amt. she started out with.
To solve this problem, let's break it down step by step.
Let's assume Olivia started out with X amount of money.
1. In the first store, she spent half the money she had plus $6. This means she spent (X/2) + $6.
2. After spending money in the first store, Olivia has (X - (X/2 + $6)) = (X/2 - $6) left.
3. In the second store, she spent one third of what was left plus $6. This means she spent (1/3) * (X/2 - $6) + $6.
4. After spending money in the second store, Olivia has (X/2 - $6) - [(1/3) * (X/2 - $6) + $6] = (X/6 - $2) left.
5. In the last store, she spent 1/4 of what was left and came home with $6. This means she spent (1/4) * (X/6 - $2) and came home with $6.
6. Setting up the equation, we have: (X/6 - $2) - (1/4) * (X/6 - $2) = $6.
Now, we can solve this equation to find the value of X.
7. Simplifying the equation: X/6 - $2 - (1/4) * X/6 + $1/2 = $6.
8. Combining like terms: X/6 - (1/4) * X/6 = $6 - $2 - $1/2.
9. Simplifying further: (4X - X)/24 = $6 - $2 - $1/2.
10. Combining like terms: 3X/24 = $6 - $2 - $1/2.
11. Simplifying: 3X/24 = $4 - $1/2.
12. Combining like terms: 3X/24 = $3.50.
13. Now, cross-multiplying: 3X * 1 = 24 * $3.50.
14. Simplifying: 3X = $84.
15. Dividing both sides by 3: X = $28.
Therefore, Olivia started out with $28.