拉维先生花了他所有钱的 \frac{1}{4} 买了一件衬衫，并用剩余钱的 \frac{5}{6} 买了一个羽毛球拍、一条短裤和两双鞋。羽毛球拍的价格是一双鞋的 4 倍，而那条短裤的价格是一双鞋的 \frac{2}{3}。衬衫的价格比短裤贵 $84.30。

1. Find the fraction of the total money spent on the racket, shorts, and 2 pairs of shoes.

Fraction of money remaining after buying the shirt = $1 - \frac{1}{4} = \frac{3}{4}$
Fraction of total money spent on the racket, shorts, and 2 pairs of shoes = $\frac{5}{6} \times \frac{3}{4} = \frac{5}{8}$

2. Express the cost of the items in terms of a common unit.

Let the cost of a pair of shoes be $3$ units.
Cost of the badminton racket = $4 \times 3 = 12$ units.
Cost of the pair of shorts = $\frac{2}{3} \times 3 = 2$ units.
Cost of 2 pairs of shoes = $2 \times 3 = 6$ units.
Total cost of these items = $12 + 2 + 6 = 20$ units.

3. Find the fraction of the total money spent on the pair of shorts.

$20$ units represents $\frac{5}{8}$ of the total money.
$1$ unit represents $\frac{5}{8} \div 20 = \frac{1}{32}$ of the total money.
The pair of shorts costs $2$ units, which represents $2 \times \frac{1}{32} = \frac{1}{16}$ of the total money.

1. Determine the fraction of money spent on the shirt and the remaining money

Fraction spent on shirt = $\frac{1}{4}$
Remaining fraction = $1 - \frac{1}{4} = \frac{3}{4}$

2. Determine the fraction of total money spent on the group of items (racket, shorts, shoes)

Fraction spent on racket, shorts, and shoes = $\frac{5}{6}$ of remaining
$= \frac{5}{6} \times \frac{3}{4} = \frac{5}{8}$

3. Express the cost of the items in terms of units of shoes

Let the cost of a pair of shoes be $1$ unit ($1u$).
Cost of racket = $4u$
Cost of shorts = $\frac{2}{3}u$
Cost of 2 pairs of shoes = $2u$
Total cost of these items = $4u + \frac{2}{3}u + 2u = 6\frac{2}{3}u = \frac{20}{3}u$

4. Relate the units to the fraction of total money

The total cost of these items represents $\frac{5}{8}$ of the total money.
So, $\frac{20}{3}u$ corresponds to $\frac{5}{8}$ of the total money.
Therefore, $1u$ corresponds to $\frac{5}{8} \div \frac{20}{3} = \frac{5}{8} \times \frac{3}{20} = \frac{3}{32}$ of the total money.

5. Find the fraction of total money spent on the shorts

Fraction spent on shorts = $\frac{2}{3}u = \frac{2}{3} \times \frac{3}{32} = \frac{1}{16}$ of the total money.

6. Use the given price difference to find the total money

Fraction spent on shirt = $\frac{1}{4} = \frac{4}{16}$
Difference in fraction between shirt and shorts = $\frac{4}{16} - \frac{1}{16} = \frac{3}{16}$
We are given that the shirt cost $\char36{}84.30$ more than the shorts.
So, $\frac{3}{16}$ of total money = $\char36{}84.30$
Total money = $\char36{}84.30 \div 3 \times 16 = \char36{}28.10 \times 16 = \char36{}449.60$

7. Calculate the total amount spent altogether

Total fraction spent = Fraction for shirt + Fraction for (racket, shorts, shoes)
$= \frac{1}{4} + \frac{5}{8} = \frac{2}{8} + \frac{5}{8} = \frac{7}{8}$
Total amount spent = $\frac{7}{8} \times \char36{}449.60 = 7 \times \char36{}56.20 = \char36{}393.40$