A student is asked to multiply a number by `11/23` . By mistake, he divides the number by `11/23` and get result 1224 more. Find the number?

Options

A. 253
B. 506
C. 759
D. 1012

Answer

To solve the problem step by step, let's denote the unknown number as \( X \).

### Step 1: Understand the operations
The student was supposed to multiply \( X \) by \( \frac{11}{23} \), but instead, he divided \( X \) by \( \frac{11}{23} \).

### Step 2: Write the equations
1. The correct operation (multiplication) would result in:
\[ \text{Correct Result} = X \times \frac{11}{23} \]
2. The incorrect operation (division) would result in:
\[ \text{Incorrect Result} = X \div \frac{11}{23} = X \times \frac{23}{11} \]

### Step 3: Set up the equation
According to the problem, the incorrect result is 1224 more than the correct result:
\[ X \times \frac{23}{11} = X \times \frac{11}{23} + 1224 \]

### Step 4: Clear the fractions
To eliminate the fractions, multiply the entire equation by \( 253 \) (which is the least common multiple of 11 and 23):
\[ 253 \times \left( X \times \frac{23}{11} \right) = 253 \times \left( X \times \frac{11}{23} + 1224 \right) \]

### Step 5: Simplify the equation
This gives:
\[ 23 \times 23X = 11 \times 11X + 253 \times 1224 \]
\[ 529X = 121X + 309072 \]

### Step 6: Rearrange the equation
Now, rearranging the equation gives:
\[ 529X - 121X = 309072 \]
\[ 408X = 309072 \]

### Step 7: Solve for \( X \)
Now, divide both sides by 408:
\[ X = \frac{309072}{408} \]

### Step 8: Calculate the value
Calculating this gives:
\[ X = 759 \]

### Conclusion
The number is \( 759 \).