The product of 12% of an integer and 20% of the next integer is 61.2. Find the integer.

Answer

To solve the problem, we need to find an integer \( x \) such that the product of 12% of \( x \) and 20% of the next integer \( (x + 1) \) equals 61.2.

### Step-by-Step Solution:

1. **Define the Integer**:
Let the integer be \( x \).

2. **Define the Next Integer**:
The next integer is \( x + 1 \).

3. **Set Up the Equation**:
We know that:
\[ 12\% \text{ of } x = \frac{12}{100} \cdot x = \frac{12x}{100} \]
\[ 20\% \text{ of } (x + 1) = \frac{20}{100} \cdot (x + 1) = \frac{20(x + 1)}{100} \]
Therefore, the product of these two expressions is:
\[ \frac{12x}{100} \cdot \frac{20(x + 1)}{100} = 61.2 \]

4. **Simplify the Equation**:
Multiplying the fractions gives:
\[ \frac{240x(x + 1)}{10000} = 61.2 \]
To eliminate the fraction, multiply both sides by 10000:
\[ 240x(x + 1) = 612000 \]

5. **Divide by 240**:
Now, divide both sides by 240:
\[ x(x + 1) = \frac{612000}{240} \]
Calculating the right-hand side:
\[ \frac{612000}{240} = 2550 \]
So, we have:
\[ x(x + 1) = 2550 \]

6. **Form a Quadratic Equation**:
Rearranging gives:
\[ x^2 + x - 2550 = 0 \]

7. **Factor the Quadratic**:
We need to factor \( x^2 + x - 2550 \). We look for two numbers that multiply to -2550 and add to 1. The numbers are 50 and -51:
\[ (x - 50)(x + 51) = 0 \]

8. **Solve for \( x \)**:
Setting each factor to zero gives:
\[ x - 50 = 0 \quad \text{or} \quad x + 51 = 0 \]
Therefore, the solutions are:
\[ x = 50 \quad \text{or} \quad x = -51 \]

9. **Conclusion**:
The integer can be either \( 50 \) or \( -51 \). Since we are looking for a positive integer, the final answer is:
\[ \text{The integer is } 50. \]