32) Park, Jack, and Galvin distributed prize money of $$ x $$ dollars among themselves. Park received $$ 3 / 10 $$ of what Jack and Galvin together received. Jack received $$ 3 / 11 $$ of what Park and Galvin together received. What is the ratio of the amount received by Park to the amount received by Jack? (A) $$ 7: 8 $$ (B) $$ 8: 7 $$ (C) $$ 10: 11 $$ (D) $$ 14: 13 $$ 33) Of Ibrahim's salary, $$ \frac{1}{3} $$ is spent for clothing, $$ \frac{2}{5} $$ for food, and $$ \frac{1}{10} $$ for transportation. What part of Ibrahim's salary is left for other expenditures and savings? a) $$ \frac{1}{6} $$ b) $$ \frac{5}{6} $$ c) $$ \frac{2}{9} $$ d) $$ \frac{2}{7} $$

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Let's solve both problems step by step.

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### **Problem 32**

**Given:**
- Park (P), Jack (J), and Galvin (G) share a prize of $$ x $$ dollars.
- **Park's share:** $$ P = \frac{3}{10}(J + G) $$
- **Jack's share:** $$ J = \frac{3}{11}(P + G) $$

**Objective:** Find the ratio $$ P : J $$.

**Solution:**

1. **Express Equations:**
   $$
   \begin{cases}
   P = \frac{3}{10}(J + G) \quad \text{(1)} \
   J = \frac{3}{11}(P + G) \quad \text{(2)}
   \end{cases}
   $$

2. **Solve Equation (1) for $$ G $$:**
   $$
   10P = 3(J + G) \
   10P = 3J + 3G \
   3G = 10P - 3J \
   G = \frac{10P - 3J}{3} \quad \text{(3)}
   $$

3. **Substitute $$ G $$ from Equation (3) into Equation (2):**
   $$
   J = \frac{3}{11}\left(P + \frac{10P - 3J}{3}\right) \
   11J = 3P + 10P - 3J \
   11J + 3J = 13P \
   14J = 13P \
   \frac{P}{J} = \frac{14}{13}
   $$

**Answer:** The ratio of Park's amount to Jack's amount is **$$ \boxed{14 : 13} $$** (Option D).

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### **Problem 33**

**Given:**
- Ibrahim spends:
  - $$ \frac{1}{3} $$ on clothing,
  - $$ \frac{2}{5} $$ on food,
  - $$ \frac{1}{10} $$ on transportation.

**Objective:** Determine the remaining part of the salary for other expenditures and savings.

**Solution:**

1. **Calculate Total Spent:**
   $$
   \text{Total} = \frac{1}{3} + \frac{2}{5} + \frac{1}{10}
   $$

2. **Find a Common Denominator (30):**
   $$
   \frac{1}{3} = \frac{10}{30} \
   \frac{2}{5} = \frac{12}{30} \
   \frac{1}{10} = \frac{3}{30}
   $$
   $$
   \text{Total} = \frac{10}{30} + \frac{12}{30} + \frac{3}{30} = \frac{25}{30} = \frac{5}{6}
   $$

3. **Calculate Remaining Part:**
   $$
   1 - \frac{5}{6} = \frac{1}{6}
   $$

**Answer:** The remaining part of Ibrahim's salary is **$$ \boxed{\frac{1}{6}} $$** (Option a).

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**Final Answers:**
- **Problem 32:** Option **(D) $$ 14 : 13 $$**
- **Problem 33:** Option **(a) $$ \frac{1}{6} $$**
**Problem 32:** The ratio of Park's amount to Jack's amount is **14:13** (Option D). **Problem 33:** The remaining part of Ibrahim's salary is **1/6** (Option a).

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