Certainly! To solve for the second number given that the product of two rational numbers is [tex]\(\frac{9}{11}\)[/tex] and one of the numbers is [tex]\(-33\)[/tex], follow these steps:
1. Let's denote the first number as [tex]\( \text{number1} \)[/tex] and the second number as [tex]\( \text{number2} \)[/tex].
2. According to the problem, the product of [tex]\(\text{number1}\)[/tex] and [tex]\(\text{number2}\)[/tex] is given by:
[tex]\[
\text{number1} \times \text{number2} = \frac{9}{11}
\][/tex]
3. We know that:
[tex]\[
\text{number1} = -33
\][/tex]
4. Substitute [tex]\(\text{number1}\)[/tex] into the equation:
[tex]\[
-33 \times \text{number2} = \frac{9}{11}
\][/tex]
5. To solve for [tex]\(\text{number2}\)[/tex], divide both sides of the equation by [tex]\(-33\)[/tex]:
[tex]\[
\text{number2} = \frac{\frac{9}{11}}{-33}
\][/tex]
6. When dividing by [tex]\(-33\)[/tex], it's equivalent to multiplying by [tex]\(\frac{1}{-33}\)[/tex]:
[tex]\[
\text{number2} = \frac{9}{11} \times \frac{1}{-33}
\][/tex]
7. Simplifying the multiplication:
[tex]\[
\text{number2} = \frac{9 \times 1}{11 \times (-33)} = \frac{9}{-363}
\][/tex]
8. Further simplification gives:
[tex]\[
\text{number2} = -\frac{9}{363}
\][/tex]
9. Reducing the fraction by dividing both the numerator and the denominator by the greatest common divisor, which here is 3:
[tex]\[
\text{number2} = -\frac{9 \div 3}{363 \div 3} = -\frac{3}{121}
\][/tex]
10. Converting this fraction into its decimal form, we get:
[tex]\[
\text{number2} \approx -0.024793388429752067
\][/tex]
Therefore, the second number is approximately [tex]\(-0.024793388429752067\)[/tex].
