Let's denote the amount of money Hakim initially had as [tex]\( x \)[/tex].
### Step 1: Spending on the Wallet
Hakim spent [tex]\(\frac{1}{6}\)[/tex] of his money and an additional \[tex]$15 on a wallet.
- Amount spent on the wallet: \( \frac{x}{6} + 15 \)
Remaining money after buying the wallet:
\[ x - \left( \frac{x}{6} + 15 \right) \]
\[ x - \frac{x}{6} - 15 \]
\[ \frac{6x - x}{6} - 15 \]
\[ \frac{5x}{6} - 15 \]
### Step 2: Spending on the Belt
He then spent \(\frac{1}{4}\) of the remaining money and an additional \$[/tex]10 on a belt.
Let the remaining money after the wallet purchase be [tex]\( y = \frac{5x}{6} - 15 \)[/tex].
- Amount spent on the belt: [tex]\(\frac{1}{4}y + 10 \)[/tex]
Substitute [tex]\( y \)[/tex]:
[tex]\[ \frac{1}{4} \left( \frac{5x}{6} - 15 \right) + 10 \][/tex]
[tex]\[ \frac{5x}{24} - \frac{15}{4} + 10 \][/tex]
Remaining money after buying the belt:
[tex]\[ \left( \frac{5x}{6} - 15 \right) - \left( \frac{5x}{24} - \frac{15}{4} + 10 \right) \][/tex]
[tex]\[ \frac{5x}{6} - 15 - \frac{5x}{24} + \frac{15}{4} - 10 \][/tex]
Simplify the terms step by step:
[tex]\[ \frac{5x}{6} - \frac{5x}{24} - 15 + \frac{15}{4} - 10 \][/tex]
Finding a common denominator for the [tex]\( x \)[/tex]-terms, which is 24:
[tex]\[ \frac{20x}{24} - \frac{5x}{24} \][/tex]
[tex]\[ \frac{15x}{24} \][/tex]
Simplify the constant terms:
[tex]\[ \frac{15}{4} - 15 - 10 \][/tex]
[tex]\[ \frac{15}{4} - 25 \][/tex]
[tex]\[ \frac{15}{4} - \frac{100}{4} \][/tex]
[tex]\[ -\frac{85}{4} \][/tex]
So, the remaining money expression is:
[tex]\[ \frac{15x}{24} - \frac{85}{4} \][/tex]
### Step 3: Equate to the Final Amount
Hakim was left with \[tex]$65:
\[ \frac{15x}{24} - \frac{85}{4} = 65 \]
To solve for \( x \), first clear the fractions by finding a common denominator:
\[ \frac{15x}{24} = 65 + \frac{85}{4} \]
Convert \( \frac{85}{4} \):
\[ \frac{85}{4} = 21.25 \]
So, the equation becomes:
\[ \frac{15x}{24} = 65 + 21.25 \]
\[ \frac{15x}{24} = 86.25 \]
Multiply both sides by 24 to solve for \( x \):
\[ 15x = 86.25 \times 24 \]
\[ 15x = 2070 \]
Divide both sides by 15:
\[ x = \frac{2070}{15} \]
\[ x = 138 \]
### Conclusion
Hakim initially had \$[/tex]138.
