To find the other rational number when the product of two rational numbers is [tex]\(-1\)[/tex] and one of them is [tex]\(\frac{2}{5}\)[/tex], follow these steps:
1. Let the unknown rational number be [tex]\(x\)[/tex].
2. According to the problem, the product of [tex]\(\frac{2}{5}\)[/tex] and [tex]\(x\)[/tex] is [tex]\(-1\)[/tex]. This gives us the equation:
[tex]\[
\left(\frac{2}{5}\right) \times x = -1
\][/tex]
3. To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] on one side of the equation. This can be done by dividing both sides of the equation by [tex]\(\frac{2}{5}\)[/tex]. However, dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of [tex]\(\frac{2}{5}\)[/tex] is [tex]\(\frac{5}{2}\)[/tex]. Therefore, we can rewrite the equation as:
[tex]\[
x = -1 \times \frac{5}{2}
\][/tex]
4. Multiply [tex]\(-1\)[/tex] by [tex]\(\frac{5}{2}\)[/tex]:
[tex]\[
x = -\frac{5}{2}
\][/tex]
5. To express [tex]\(-\frac{5}{2}\)[/tex] as a decimal, simply divide 5 by 2 and then negate the result:
[tex]\[
-\frac{5}{2} = -2.5
\][/tex]
So, if one of the rational numbers is [tex]\(\frac{2}{5}\)[/tex], the other rational number is [tex]\(-2.5\)[/tex].
