Sure! Let's break down and solve the expressions step by step.
### Expression 1: [tex]\(\frac{8}{13} \times \left( \frac{5}{12} + \frac{2}{24} \right)\)[/tex]
1. Inner Expression [tex]\(\frac{5}{12} + \frac{2}{24}\)[/tex]:
- To add the fractions, first find a common denominator. The least common multiple (LCM) of 12 and 24 is 24.
- Convert [tex]\(\frac{5}{12}\)[/tex] to have 24 as the denominator:
[tex]\[
\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}
\][/tex]
- Now, add the fractions [tex]\(\frac{10}{24} + \frac{2}{24}\)[/tex]:
[tex]\[
\frac{10}{24} + \frac{2}{24} = \frac{10 + 2}{24} = \frac{12}{24} = \frac{1}{2}
\][/tex]
2. Now compute the multiplication [tex]\(\frac{8}{13} \times \frac{1}{2}\)[/tex]:
- Multiply the numerators: [tex]\(8 \times 1 = 8\)[/tex]
- Multiply the denominators: [tex]\(13 \times 2 = 26\)[/tex]
- Simplify the fraction [tex]\(\frac{8}{26}\)[/tex]:
[tex]\[
\frac{8}{26} = \frac{8 \div 2}{26 \div 2} = \frac{4}{13}
\][/tex]
Thus, [tex]\(\frac{8}{13} \times \left( \frac{5}{12} + \frac{2}{24} \right) = \frac{4}{13}\)[/tex].
### Expression 2: [tex]\(\frac{0.15}{0.5} \times \left( \frac{-0.16}{1.2} \right)\)[/tex]
1. Evaluate [tex]\(\frac{0.15}{0.5}\)[/tex]:
- Convert the division into multiplication by the reciprocal: [tex]\(\frac{0.15}{0.5} = 0.15 \times \frac{1}{0.5} = 0.15 \times 2 = 0.3\)[/tex]
2. Evaluate [tex]\(\frac{-0.16}{1.2}\)[/tex]:
- Similarly, convert into multiplication by the reciprocal: [tex]\(\frac{-0.16}{1.2} = -0.16 \times \frac{1}{1.2} = -0.16 \times \frac{1}{1.2} = -0.16 \times \frac{10}{12} = -0.16 \times \frac{5}{6} = -\frac{0.8}{6} = -0.1333\)[/tex]
3. Multiply the results [tex]\(0.3 \times -0.1333\)[/tex]:
[tex]\[
0.3 \times -0.1333 = -0.04
\][/tex]
Thus, [tex]\(\frac{0.15}{0.5} \times \left( \frac{-0.16}{1.2} \right) = -0.04\)[/tex].
### Summary of Results:
- [tex]\(\frac{8}{13} \times \left( \frac{5}{12} + \frac{2}{24} \right) = \frac{4}{13}\)[/tex]
- [tex]\(\frac{0.15}{0.5} \times \left( \frac{-0.16}{1.2} \right) = -0.04\)[/tex]
Therefore:
- The first expression evaluates to approximately [tex]\(0.3076923076923077\)[/tex] when interpreted as a decimal.
- The second expression evaluates to [tex]\(-0.04\)[/tex].
