To find the other rational number given that the product of two rational numbers is [tex]\(\frac{-9}{16}\)[/tex] and one of the numbers is [tex]\(\frac{3}{14}\)[/tex], we follow these steps:
1. Identify the given values:
- The product of the two rational numbers is [tex]\(\frac{-9}{16}\)[/tex].
- One of the rational numbers is [tex]\(\frac{3}{14}\)[/tex].
2. Set up the equation:
Let's denote the other rational number as [tex]\(x\)[/tex]. According to the problem, we have:
[tex]\[
\frac{3}{14} \cdot x = \frac{-9}{16}
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by [tex]\(\frac{3}{14}\)[/tex]. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we have:
[tex]\[
x = \frac{\frac{-9}{16}}{\frac{3}{14}}
\][/tex]
Simplify the division of two fractions:
[tex]\[
x = \frac{-9}{16} \cdot \frac{14}{3}
\][/tex]
4. Multiply the fractions:
To multiply two fractions, we multiply their numerators and their denominators:
[tex]\[
x = \frac{-9 \cdot 14}{16 \cdot 3} = \frac{-126}{48}
\][/tex]
5. Simplify the fraction:
To simplify [tex]\(\frac{-126}{48}\)[/tex], we look for the greatest common divisor (GCD) of 126 and 48. The GCD of 126 and 48 is 6. Therefore, we can simplify the fraction by dividing both the numerator and the denominator by their GCD:
[tex]\[
x = \frac{-126 \div 6}{48 \div 6} = \frac{-21}{8}
\][/tex]
Thus, the other rational number is:
[tex]\[
x = -\frac{21}{8}
\][/tex]
In decimal form, this is approximately:
[tex]\[
x \approx -2.625
\][/tex]
Therefore, the other rational number is [tex]\(-2.625\)[/tex].
