Sure! Let's break down the problem step-by-step.
We need to evaluate the expression:
[tex]\[
\left(1 + \frac{1}{15} - \frac{9}{10}\right) \div \left(1 \frac{1}{4} \div 1 \frac{1}{2}\right)
\][/tex]
Step 1: Simplify the numerator:
We begin with the expression inside the parentheses in the numerator:
[tex]\[
1 + \frac{1}{15} - \frac{9}{10}
\][/tex]
Convert all terms to a common denominator, which in this case is 30:
[tex]\[
1 = \frac{30}{30}, \quad \frac{1}{15} = \frac{2}{30}, \quad \frac{9}{10} = \frac{27}{30}
\][/tex]
Now we can write:
[tex]\[
1 + \frac{1}{15} - \frac{9}{10} = \frac{30}{30} + \frac{2}{30} - \frac{27}{30}
\][/tex]
Combine the fractions:
[tex]\[
\frac{30}{30} + \frac{2}{30} - \frac{27}{30} = \frac{30 + 2 - 27}{30} = \frac{5}{30} = \frac{1}{6}
\][/tex]
So, the simplified numerator is:
[tex]\[
\frac{1}{6}
\][/tex]
Step 2: Simplify the denominator:
We now need to simplify the expression inside the parentheses in the denominator:
[tex]\[
1 \frac{1}{4} \div 1 \frac{1}{2}
\][/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[
1 \frac{1}{4} = \frac{5}{4}, \quad 1 \frac{1}{2} = \frac{3}{2}
\][/tex]
Now, perform the division by multiplying by the reciprocal:
[tex]\[
\frac{5}{4} \div \frac{3}{2} = \frac{5}{4} \times \frac{2}{3} = \frac{5 \cdot 2}{4 \cdot 3} = \frac{10}{12} = \frac{5}{6}
\][/tex]
So, the simplified denominator is:
[tex]\[
\frac{5}{6}
\][/tex]
Step 3: Divide the simplified numerator by the simplified denominator:
Now, we need to divide:
[tex]\[
\frac{1}{6} \div \frac{5}{6}
\][/tex]
Perform the division by multiplying by the reciprocal:
[tex]\[
\frac{1}{6} \times \frac{6}{5} = \frac{1 \cdot 6}{6 \cdot 5} = \frac{6}{30} = \frac{1}{5}
\][/tex]
Therefore, the final result of the given expression is:
[tex]\[
\frac{1}{5} = 0.2
\][/tex]
The detailed calculations show that the answer is [tex]\(0.2\)[/tex] or [tex]\(\frac{1}{5}\)[/tex].
