To solve for the other number when the product of two rational numbers is [tex]\(-\frac{2}{3}\)[/tex] and one of the numbers is [tex]\(\frac{3}{7}\)[/tex], follow these steps:
1. Understand the Problem:
- We have two rational numbers, let's call them [tex]\( \text{num1} \)[/tex] and [tex]\( \text{num2} \)[/tex].
- Their product is given: [tex]\(\text{num1} \times \text{num2} = -\frac{2}{3}\)[/tex].
- One of these numbers, [tex]\(\text{num1}\)[/tex], is known: [tex]\(\text{num1} = \frac{3}{7}\)[/tex].
2. Set up the Equation:
- Substitute the known value of [tex]\(\text{num1}\)[/tex] into the product equation:
[tex]\[
\frac{3}{7} \times \text{num2} = -\frac{2}{3}
\][/tex]
3. Solve for the Unknown Number ([tex]\(\text{num2}\)[/tex]):
- To isolate [tex]\(\text{num2}\)[/tex], you need to divide both sides of the equation by [tex]\(\frac{3}{7}\)[/tex]. Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\text{num2} = \left(-\frac{2}{3}\right) \div \left(\frac{3}{7}\right) = \left(-\frac{2}{3}\right) \times \left(\frac{7}{3}\right)
\][/tex]
4. Perform the Multiplication:
- Multiply the numerators together and the denominators together:
[tex]\[
\text{num2} = \frac{-2 \times 7}{3 \times 3} = \frac{-14}{9}
\][/tex]
5. Simplify the Fraction:
- The fraction [tex]\(\frac{-14}{9}\)[/tex] is already in its simplest form.
Therefore, the other number is:
[tex]\[
\text{num2} = -\frac{14}{9}
\][/tex]
Let's also summarize the values given:
- The product, [tex]\(\text{product} = -\frac{2}{3}\)[/tex].
- The known number, [tex]\(\text{num1} = \frac{3}{7}\)[/tex].
- The other number, [tex]\(\text{num2} = -\frac{14}{9}\)[/tex].
In decimal form, these values are approximately:
- [tex]\(\text{product} \approx -0.6666666666666666\)[/tex].
- [tex]\(\text{num1} \approx 0.42857142857142855\)[/tex].
- [tex]\(\text{num2} \approx -1.5555555555555556\)[/tex].
