A person spend `1//3^(rd)` of his income on food, clothes and education in the ratio 3:2: 4. He spent 2/5 part of his income on transport and entertainment in the ratio 4 : 5 and he deposit Rs 2700 in the bank then find the amount which is spent on food, clothes and transport.

Options

A. 3675
B. 3600
C. 3800
D. 3725

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understanding the Income Distribution
The person spends \( \frac{1}{3} \) of his income on food, clothes, and education in the ratio \( 3:2:4 \).

Let the total income be \( I \). Then, the amount spent on food, clothes, and education is:
\[ \text{Amount spent} = \frac{1}{3} I \]

### Step 2: Finding the Total Parts of the Ratio
The total parts of the ratio \( 3:2:4 \) is:
\[ 3 + 2 + 4 = 9 \text{ parts} \]

### Step 3: Finding the Value of Each Part
If \( \frac{1}{3} I \) corresponds to 9 parts, then the value of each part is:
\[ \text{Value of each part} = \frac{\frac{1}{3} I}{9} = \frac{I}{27} \]

### Step 4: Finding the Amount Spent on Each Category
Now we can find the amount spent on food, clothes, and education:
- Amount spent on food:
\[ \text{Food} = 3 \times \frac{I}{27} = \frac{I}{9} \]
- Amount spent on clothes:
\[ \text{Clothes} = 2 \times \frac{I}{27} = \frac{2I}{27} \]
- Amount spent on education:
\[ \text{Education} = 4 \times \frac{I}{27} = \frac{4I}{27} \]

### Step 5: Total Amount Spent on Food, Clothes, and Education
The total amount spent on food, clothes, and education is:
\[ \text{Total} = \frac{I}{9} + \frac{2I}{27} + \frac{4I}{27} \]
To add these fractions, we need a common denominator, which is 27:
\[ \text{Total} = \frac{3I}{27} + \frac{2I}{27} + \frac{4I}{27} = \frac{9I}{27} = \frac{I}{3} \]

### Step 6: Amount Spent on Transport and Entertainment
The person spends \( \frac{2}{5} \) of his income on transport and entertainment in the ratio \( 4:5 \).

The amount spent on transport and entertainment is:
\[ \text{Amount spent} = \frac{2}{5} I \]

### Step 7: Finding the Total Parts of the Transport Ratio
The total parts of the ratio \( 4:5 \) is:
\[ 4 + 5 = 9 \text{ parts} \]

### Step 8: Finding the Value of Each Part for Transport
If \( \frac{2}{5} I \) corresponds to 9 parts, then the value of each part is:
\[ \text{Value of each part} = \frac{\frac{2}{5} I}{9} = \frac{2I}{45} \]

### Step 9: Finding the Amount Spent on Transport and Entertainment
Now we can find the amount spent on transport and entertainment:
- Amount spent on transport:
\[ \text{Transport} = 4 \times \frac{2I}{45} = \frac{8I}{45} \]
- Amount spent on entertainment:
\[ \text{Entertainment} = 5 \times \frac{2I}{45} = \frac{10I}{45} = \frac{2I}{9} \]

### Step 10: Total Amount Spent on Transport and Entertainment
The total amount spent on transport and entertainment is:
\[ \text{Total} = \frac{8I}{45} + \frac{10I}{45} = \frac{18I}{45} = \frac{2I}{5} \]

### Step 11: Total Income Calculation
The total income \( I \) can be calculated using the deposit in the bank:
The total amount spent on food, clothes, education, transport, and entertainment is:
\[ \frac{1}{3} I + \frac{2}{5} I \]
Finding a common denominator (15):
\[ \frac{5I}{15} + \frac{6I}{15} = \frac{11I}{15} \]
The remaining amount after spending is:
\[ I - \frac{11I}{15} = \frac{4I}{15} \]
This remaining amount is equal to Rs 2700 (the amount deposited in the bank):
\[ \frac{4I}{15} = 2700 \]
Solving for \( I \):
\[ I = 2700 \times \frac{15}{4} = 10125 \]

### Step 12: Finding the Amount Spent on Food, Clothes, and Transport
Now we can find the amounts spent:
- Amount spent on food:
\[ \text{Food} = \frac{I}{9} = \frac{10125}{9} = 1125 \]
- Amount spent on clothes:
\[ \text{Clothes} = \frac{2I}{27} = \frac{2 \times 10125}{27} = 750 \]
- Amount spent on transport:
\[ \text{Transport} = \frac{8I}{45} = \frac{8 \times 10125}{45} = 1800 \]

### Final Calculation
Now, we sum the amounts spent on food, clothes, and transport:
\[ \text{Total} = 1125 + 750 + 1800 = 3675 \]

### Final Answer
The total amount spent on food, clothes, and transport is Rs 3675.

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