In a library `15%` of the books are in Hindi. `25%` of the remaining are in French and `16(2)/(3)%` of the remaining are in English and rest `3434` books are in Punjabi. Find no. of books in library.

Options

A. `8484`
B. `6464`
C. `4444`
D. `7474`

Answer

To solve the problem step by step, we will follow the information provided in the question regarding the distribution of books in the library.

### Step 1: Define the total number of books
Let the total number of books in the library be \( x \).

### Step 2: Calculate the number of Hindi books
Since 15% of the books are in Hindi, we can calculate the number of Hindi books as:
\[ \text{Hindi books} = 15\% \text{ of } x = \frac{15}{100} \times x = \frac{3x}{20} \]

### Step 3: Calculate the remaining books after Hindi
The remaining books after accounting for the Hindi books are:
\[ \text{Remaining books} = x - \frac{3x}{20} = \frac{20x}{20} - \frac{3x}{20} = \frac{17x}{20} \]

### Step 4: Calculate the number of French books
25% of the remaining books are in French, so:
\[ \text{French books} = 25\% \text{ of remaining books} = \frac{25}{100} \times \frac{17x}{20} = \frac{17x}{80} \]

### Step 5: Calculate the remaining books after Hindi and French
Now we calculate the remaining books after accounting for both Hindi and French books:
\[ \text{Remaining books} = \frac{17x}{20} - \frac{17x}{80} \]
To perform this subtraction, we need a common denominator:
\[ \frac{17x}{20} = \frac{68x}{80} \quad \text{(converting to a common denominator)} \]
Thus,
\[ \text{Remaining books} = \frac{68x}{80} - \frac{17x}{80} = \frac{51x}{80} \]

### Step 6: Calculate the number of English books
Next, we need to find out how many books are in English. Given that \( 16\frac{2}{3}\% \) of the remaining books are in English, we convert this mixed number to an improper fraction:
\[ 16\frac{2}{3}\% = \frac{50}{3}\% \]
Now, we calculate the number of English books:
\[ \text{English books} = \frac{50}{3} \times \frac{1}{100} \times \frac{51x}{80} = \frac{50 \times 51x}{3 \times 100 \times 80} = \frac{2550x}{24000} = \frac{17x}{160} \]

### Step 7: Calculate the total number of books
Now we have the number of books in Hindi, French, English, and the remaining books in Punjabi (which is given as 3434). We can set up the equation:
\[ \text{Hindi books} + \text{French books} + \text{English books} + \text{Punjabi books} = x \]
Substituting the values we calculated:
\[ \frac{3x}{20} + \frac{17x}{80} + \frac{17x}{160} + 3434 = x \]

### Step 8: Solve the equation
To solve for \( x \), we first need a common denominator for the fractions. The least common multiple of 20, 80, and 160 is 160. Rewriting the equation:
\[ \frac{24x}{160} + \frac{34x}{160} + \frac{17x}{160} + 3434 = x \]
Combining the fractions:
\[ \frac{75x}{160} + 3434 = x \]
Rearranging gives:
\[ x - \frac{75x}{160} = 3434 \]
This simplifies to:
\[ \frac{85x}{160} = 3434 \]
Now, solving for \( x \):
\[ x = 3434 \times \frac{160}{85} \]
Calculating this gives:
\[ x = 6464 \]

### Final Answer
The total number of books in the library is \( 6464 \).

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