Sure, let's evaluate the given expression step-by-step:
The original expression we need to evaluate is:
[tex]\[
\left(\frac{3}{16} - \frac{9}{16}\right) \div \frac{5}{4}
\][/tex]
1. Step 1: Subtract the fractions inside the parentheses
Since the denominators of [tex]\(\frac{3}{16}\)[/tex] and [tex]\(\frac{9}{16}\)[/tex] are the same, we can subtract the numerators directly:
[tex]\[
\frac{3}{16} - \frac{9}{16} = \frac{3 - 9}{16} = \frac{-6}{16}
\][/tex]
Next, we simplify [tex]\(\frac{-6}{16}\)[/tex] by finding the greatest common divisor (GCD) of 6 and 16, which is 2:
[tex]\[
\frac{-6}{16} = \frac{-6 \div 2}{16 \div 2} = \frac{-3}{8}
\][/tex]
So, the result of the subtraction is [tex]\(\frac{-3}{8}\)[/tex].
2. Step 2: Divide the result by [tex]\(\frac{5}{4}\)[/tex]
To divide by a fraction, we multiply by its reciprocal. Hence, dividing by [tex]\(\frac{5}{4}\)[/tex] is equivalent to multiplying by [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{-3}{8} \div \frac{5}{4} = \frac{-3}{8} \times \frac{4}{5}
\][/tex]
Multiply the numerators and denominators of the fractions:
[tex]\[
\frac{-3 \times 4}{8 \times 5} = \frac{-12}{40}
\][/tex]
Finally, we simplify [tex]\(\frac{-12}{40}\)[/tex] by finding the GCD of 12 and 40, which is 4:
[tex]\[
\frac{-12}{40} = \frac{-12 \div 4}{40 \div 4} = \frac{-3}{10}
\][/tex]
So, the result of the division is [tex]\(\frac{-3}{10}\)[/tex].
Putting it all together:
[tex]\[
\left(\frac{3}{16} - \frac{9}{16}\right) \div \frac{5}{4} = \left(\frac{-3}{8}\right) \div \frac{5}{4} = \frac{-3}{10}
\][/tex]
Therefore, the expression evaluates to [tex]\(\frac{-3}{10}\)[/tex].
