To compute the given expression [tex]\(\frac{2}{5} \times \left[5 \times \frac{7}{6}\right] + \frac{1}{3} \times \left(3 \times \frac{4}{11}\right)\)[/tex], we can leverage the associative and commutative properties of multiplication and addition. Let's break down the calculation step-by-step:
### Step-by-Step Solution:
#### Step 1: Simplify [tex]\(\frac{2}{5} \times \left[5 \times \frac{7}{6}\right]\)[/tex]
1.1 Start by simplifying inside the brackets:
[tex]\[ 5 \times \frac{7}{6} = \frac{5 \times 7}{6} = \frac{35}{6} \][/tex]
1.2 Now, multiply the result by [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[ \frac{2}{5} \times \frac{35}{6} \][/tex]
1.3 Perform the multiplication of fractions:
[tex]\[ \frac{2 \times 35}{5 \times 6} = \frac{70}{30} \][/tex]
1.4 Simplify the fraction:
[tex]\[ \frac{70}{30} = \frac{7}{3} = 2.3333333333333335 \][/tex]
#### Step 2: Simplify [tex]\(\frac{1}{3} \times \left(3 \times \frac{4}{11}\right)\)[/tex]
2.1 Start by simplifying inside the parentheses:
[tex]\[ 3 \times \frac{4}{11} = \frac{3 \times 4}{11} = \frac{12}{11} \][/tex]
2.2 Now, multiply the result by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} \times \frac{12}{11} \][/tex]
2.3 Perform the multiplication of fractions:
[tex]\[ \frac{1 \times 12}{3 \times 11} = \frac{12}{33} \][/tex]
2.4 Simplify the fraction:
[tex]\[ \frac{12}{33} = 0.36363636363636365 \][/tex]
#### Step 3: Add the results from Step 1 and Step 2
3.1 Combine the simplified results:
[tex]\[ 2.3333333333333335 + 0.36363636363636365 = 2.696969696969697 \][/tex]
### Conclusion
The properties used include:
- Associative Property of Multiplication: This allowed us to group factors in different ways.
- Commutative Property of Multiplication: This allowed us to multiply numbers in any order.
- Simplification of Fractions: This involves reducing fractions to their lowest terms.
So the simplified form of the expression [tex]\(\frac{2}{5} \times \left[5 \times \frac{7}{6}\right] + \frac{1}{3} \times \left(3 \times \frac{4}{11}\right)\)[/tex] is:
[tex]\[ 2.696969696969697 \][/tex]
