Complete the following multiplication and division calculations involving fractions (show all the steps).
[tex]$4.1 \cdot 1 \cdot \frac{4}{6} \div \frac{3}{12}$[/tex]
Answer (2)
Convert the decimal to a fraction: 4.1 = 10 41 .
Rewrite division as multiplication by the reciprocal and simplify: 6 4 ÷ 12 3 = 3 2 ⋅ 4 .
Multiply the fractions: 10 41 ⋅ 3 2 ⋅ 4 = 30 328 .
Simplify the fraction to get the final answer: 15 164 .
Explanation
Understanding the Problem We need to calculate the value of the expression 4.1 ⋅ 1 ⋅ 6 4 ÷ 12 3 . This involves multiplication and division of a decimal number and fractions. Let's convert the decimal to a fraction to make the calculations easier.
Converting to Fractions and Reciprocals First, convert the decimal number 4.1 to a fraction: 4.1 = 10 41 . Next, rewrite the division as multiplication by the reciprocal: 6 4 ÷ 12 3 = 6 4 ⋅ 3 12 .
Simplifying Fractions Simplify the fractions before performing the multiplication: 6 4 = 3 2 and 3 12 = 4 . Now, rewrite the expression as 10 41 ⋅ 1 ⋅ 3 2 ⋅ 4 .
Multiplying Fractions Multiply the fractions: 10 41 ⋅ 1 ⋅ 3 2 ⋅ 4 = 10 ⋅ 3 41 ⋅ 1 ⋅ 2 ⋅ 4 = 30 328 .
Simplifying and Converting to Decimal Simplify the resulting fraction: 30 328 = 15 164 . Now, convert the improper fraction to a decimal: 15 164 = 10.9333... Therefore, the final answer is approximately 10.93.
Examples
Understanding fraction operations is essential in many real-life scenarios. For instance, when you're cooking and need to adjust recipe quantities, you often deal with fractions. If a recipe calls for 3 2 cup of flour and you want to double the recipe, you need to multiply 3 2 by 2. Similarly, if you're splitting a pizza with friends and each person gets 8 1 of the pizza, knowing how to divide the whole (1 pizza) into equal parts is crucial. These basic fraction operations are also fundamental in more advanced fields like engineering and finance, where precise calculations are necessary.
To evaluate the expression 4.1 ⋅ 1 ⋅ 6 4 ÷ 12 3 , we convert 4.1 to a fraction, simplify the division of fractions, and then multiply the resulting fractions. The final answer simplifies to 15 164 .
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