Gary spent 5/8 of his money on 6 identical notebooks and 8 identical files. Each file cost thrice as much as each notebook. Then, he spent 4/9 of his remaining money on a magazine. The magazine cost $16.80 more than each notebook.
(a) What fraction of Gary’s money was spent on the magazine? Give your answer in the simplest form.
(b) How much money had Gary left?
To find the fraction of Gary's money spent on the magazine, we first need to determine the total amount of money he spent. We can break it down step by step.
Let's assume Gary had x amount of money initially.
Step 1: Amount spent on notebooks and files
Gary spent 5/8 of his money on 6 identical notebooks and 8 identical files.
Each file cost thrice as much as each notebook.
Let's first find the cost of each notebook. Since Gary spent 5/8 of his money on notebooks and files, the fraction spent on notebooks is 5/8.
So, Gary spent (5/8) * x on notebooks.
Since there are 6 identical notebooks, the cost of each notebook is [(5/8) * x] / 6 = (5x/48).
And since each file costs thrice as much as each notebook, the cost of each file is 3 * (5x/48) = (15x/48) = (5x/16).
The total amount spent on notebooks and files is given by:
6 * (5x/48) + 8 * (5x/16) = (30x/48) + (40x/16) = (5x/8) + (5x/2) = (5x/8) + (20x/8) = (25x/8).
Step 2: Amount spent on the magazine
Gary spent 4/9 of his remaining money on a magazine.
The magazine cost $16.80 more than each notebook.
After buying the notebooks and files, Gary had (x - 25x/8) = (8x - 25x)/8 = (-17x/8) amount remaining.
So, the amount spent on the magazine is (4/9) * (-17x/8).
Step 3: Fraction spent on the magazine
To find the fraction spent on the magazine, we divide the amount spent on the magazine by the initial amount of money Gary had.
Fraction spent on the magazine = [(4/9) * (-17x/8)] / x
= (-68x/72x)
= -17/18
(a) Therefore, the fraction of Gary's money spent on the magazine is -17/18.
To find out how much money Gary had left, we need to subtract the amount spent on the magazine from the remaining money after buying notebooks and files.
Money Gary had left = (x - (4/9) * (-17x/8))
= (x + (17x/18))
= (18x + 17x)/18
= 35x/18
(b) Gary had (35x/18) amount of money left.