When 616 is divided by a certain positive number which is `66(2)/(23)%` of the quotient it leaven 16 as the remainder .Find the divisor

Options

A. 20
B. 30
C. 24
D. 15

Answer

To solve the problem step by step, we need to find the divisor when 616 is divided by a certain positive number, which is \(66\frac{2}{23}\%\) of the quotient, leaving a remainder of 16.

### Step 1: Define the Variables
Let the divisor be \(x\) and the quotient be \(q\).

### Step 2: Set Up the Equation
According to the problem, when 616 is divided by \(x\), it leaves a remainder of 16. This can be expressed as:
\[ 616 = q \cdot x + 16 \]
Rearranging this gives:
\[ q \cdot x = 616 - 16 \]
\[ q \cdot x = 600 \]

### Step 3: Convert Percentage to Fraction
The problem states that the divisor \(x\) is \(66\frac{2}{23}\%\) of the quotient \(q\). First, we convert \(66\frac{2}{23}\%\) to a fraction:
\[ 66\frac{2}{23} = \frac{66 \times 23 + 2}{23} = \frac{1518 + 2}{23} = \frac{1520}{23} \]
Thus,
\[ x = \frac{1520}{23} \cdot \frac{q}{100} \]
This simplifies to:
\[ x = \frac{1520q}{2300} = \frac{76q}{115} \]

### Step 4: Substitute \(x\) in the Equation
Now, substitute \(x\) back into the equation \(q \cdot x = 600\):
\[ q \cdot \left(\frac{76q}{115}\right) = 600 \]
This simplifies to:
\[ \frac{76q^2}{115} = 600 \]

### Step 5: Solve for \(q^2\)
Multiply both sides by 115 to eliminate the fraction:
\[ 76q^2 = 600 \cdot 115 \]
Calculating the right side:
\[ 600 \cdot 115 = 69000 \]
So we have:
\[ 76q^2 = 69000 \]
Now, divide both sides by 76:
\[ q^2 = \frac{69000}{76} \]
Calculating this gives:
\[ q^2 = 907.8947368 \quad (\text{approximately}) \]
Taking the square root:
\[ q \approx 30.1 \]

### Step 6: Find the Divisor \(x\)
Now substitute \(q\) back into the equation for \(x\):
\[ x = \frac{76 \cdot 30.1}{115} \]
Calculating this gives:
\[ x \approx 20.0 \]

### Conclusion
Thus, the divisor \(x\) is approximately \(20\).