Mr. Shameem invested `33.(1)/(3)%` of his savings in `20%` `50rs` shares quoted at `60rs` and the remainder of the savings in `10%` `100rs` shares quoted at `110rs`. If his total income from these investments is `9200rs` find
(i) his total savings
(ii) the number of `50rs ` shares
the number of `100rs` shares

Answer

To solve the problem, we will break it down step by step.

### Step 1: Define the total savings
Let Mr. Shameem's total savings be \( x \).

### Step 2: Calculate the investment in 50 Rs shares
Mr. Shameem invested \( 33.\overline{3}\% \) of his savings in 20% shares of 50 Rs.
This can be expressed as:
\[ \text{Investment in 50 Rs shares} = \frac{33.\overline{3}}{100} \times x = \frac{100}{3} \times \frac{x}{100} = \frac{x}{3} \]

### Step 3: Calculate the investment in 100 Rs shares
The remainder of his savings is invested in 10% shares of 100 Rs:
\[ \text{Investment in 100 Rs shares} = x - \frac{x}{3} = \frac{2x}{3} \]

### Step 4: Calculate the number of shares purchased
1. **For 50 Rs shares:**
The price of each share is 60 Rs. The number of shares purchased is:
\[ \text{Number of 50 Rs shares} = \frac{\text{Investment in 50 Rs shares}}{\text{Price per share}} = \frac{\frac{x}{3}}{60} = \frac{x}{180} \]

2. **For 100 Rs shares:**
The price of each share is 110 Rs. The number of shares purchased is:
\[ \text{Number of 100 Rs shares} = \frac{\text{Investment in 100 Rs shares}}{\text{Price per share}} = \frac{\frac{2x}{3}}{110} = \frac{2x}{330} = \frac{x}{165} \]

### Step 5: Calculate the total income from investments
The total income from both investments is given as 9200 Rs. We can express this as:
\[ \text{Income from 50 Rs shares} + \text{Income from 100 Rs shares} = 9200 \]
Calculating the income from each investment:
1. **Income from 50 Rs shares:**
\[ \text{Income} = \text{Number of shares} \times \text{Dividend} = \frac{x}{180} \times 20\% \times 50 = \frac{x}{180} \times \frac{20}{100} \times 50 = \frac{x}{180} \times 10 = \frac{10x}{180} = \frac{x}{18} \]

2. **Income from 100 Rs shares:**
\[ \text{Income} = \text{Number of shares} \times \text{Dividend} = \frac{x}{165} \times 10\% \times 100 = \frac{x}{165} \times 10 = \frac{10x}{165} = \frac{2x}{33} \]

### Step 6: Set up the equation
Combining both incomes:
\[ \frac{x}{18} + \frac{2x}{33} = 9200 \]

### Step 7: Solve for \( x \)
To solve the equation, we need a common denominator. The least common multiple of 18 and 33 is 198. Rewriting the equation:
\[ \frac{11x}{198} + \frac{12x}{198} = 9200 \]
Combining the fractions:
\[ \frac{23x}{198} = 9200 \]
Cross-multiplying gives:
\[ 23x = 9200 \times 198 \]
Calculating \( 9200 \times 198 \):
\[ 9200 \times 198 = 1821600 \]
Thus,
\[ x = \frac{1821600}{23} = 79200 \]

### Step 8: Calculate the number of shares
1. **Number of 50 Rs shares:**
\[ \text{Number of 50 Rs shares} = \frac{x}{180} = \frac{79200}{180} = 440 \]

2. **Number of 100 Rs shares:**
\[ \text{Number of 100 Rs shares} = \frac{x}{165} = \frac{79200}{165} = 480 \]

### Final Answers
(i) Total savings = 79200 Rs
(ii) Number of 50 Rs shares = 440
(iii) Number of 100 Rs shares = 480