To determine by what rational number we need to divide [tex]\( \frac{22}{7} \)[/tex] to obtain [tex]\( \frac{-11}{24} \)[/tex], we can follow these steps:
1. Understand the problem:
We are trying to find a number [tex]\( x \)[/tex] such that:
[tex]\[
\frac{22}{7} \div x = \frac{-11}{24}
\][/tex]
2. Convert division into multiplication:
We can rewrite the division as multiplication by the reciprocal. The equation becomes:
[tex]\[
\frac{22}{7} \times \frac{1}{x} = \frac{-11}{24}
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we express [tex]\( x \)[/tex] as:
[tex]\[
x = \frac{\frac{22}{7}}{\frac{-11}{24}}
\][/tex]
This involves multiplying by the reciprocal of [tex]\( \frac{-11}{24} \)[/tex].
4. Rewriting the fraction and simplifying:
Simplify the fraction as follows:
[tex]\[
x = \frac{22}{7} \times \frac{24}{-11}
\][/tex]
5. Perform the multiplication:
Calculate the product of the fractions:
[tex]\[
x = \frac{22 \times 24}{7 \times -11}
\][/tex]
[tex]\[
x = \frac{528}{-77}
\][/tex]
[tex]\[
x = -\frac{528}{77}
\][/tex]
6. Simplify the fraction:
Simplify [tex]\( \frac{528}{77} \)[/tex] by dividing both the numerator and the denominator by their greatest common divisor (GCD), which in this case is 1, since 528 and 77 are coprime:
[tex]\[
x = -\frac{528}{77}
\][/tex]
[tex]\[
x \approx -0.1458333 \ldots
\][/tex]
Thus, the rational number by which [tex]\(\frac{22}{7}\)[/tex] should be divided to get [tex]\(\frac{-11}{24}\)[/tex] is approximately [tex]\(-0.14583333333333331\)[/tex].
