To solve the expression [tex]\(237 \frac{27}{16} \div\left(-\frac{3}{8}\right)\)[/tex], follow these steps:
1. Convert the mixed number to an improper fraction: The given mixed number is [tex]\(237 \frac{27}{16}\)[/tex]. First, convert this to an improper fraction. To do so, multiply the whole number part by the denominator of the fraction part and add the numerator: [tex]\[
237 \times 16 + 27 = 3792 + 27 = 3819
\][/tex] Thus, the improper fraction is: [tex]\[
\frac{3819}{16}
\][/tex]
2. Identify the divisor and convert it properly: The divisor is [tex]\(-\frac{3}{8}\)[/tex].
3. Rewrite the division as the multiplication by the reciprocal of the divisor: Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite the division: [tex]\[
\frac{3819}{16} \div \left(-\frac{3}{8}\right) = \frac{3819}{16} \times \left(-\frac{8}{3}\right)
\][/tex]
4. Multiply the fractions: When multiplying fractions, multiply the numerators together and the denominators together: [tex]\[
\frac{3819}{16} \times \left(-\frac{8}{3}\right) = \frac{3819 \times (-8)}{16 \times 3} = \frac{-30552}{48}
\][/tex]
5. Simplify the fraction if possible: To simplify [tex]\(\frac{-30552}{48}\)[/tex], divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of [tex]\(30552\)[/tex] and [tex]\(48\)[/tex] is [tex]\(48\)[/tex]: [tex]\[
\frac{-30552 \div 48}{48 \div 48} = \frac{-636.5}{1} = -636.5
\][/tex]
Hence, the final result of the expression [tex]\(237 \frac{27}{16} \div \left(-\frac{3}{8}\right)\)[/tex] is [tex]\(-636.5\)[/tex].
